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36638b90-f8dc-4202-b695-f8b977dabc96 | trentmkelly/LessWrong-43k | LessWrong | Autonomy as taking responsibility for reference maintenance
I think semantics – specifically, maintaining reference relationships – is a core component of intelligent behaviour. Consequently, I think a better understanding of semantics would enable a better understanding of what machine intelligence that is “trying to do the right thing” ought to look like and how to build it. Unfortunately I’ve not managed to make these ideas particularly clear (either to myself or others). In the spirit of palimpsest-shaped research I’m sharing a draft in the hope of provoking useful thoughts.
Summary
* Particular events can have meaningfulness or significance, and can carry multiple meanings at once. For example, the motion of a dial in a meter might mean that there is electricity flowing through a connected wire, while simultaneously meaning that an alarm condition has been triggered.
* The “meanings” (or semantics) that can be ascribed to events depend on regularities (stable mechanisms) in the surrounding context. Large-scale regularities are required to sustain large-scale semantics (something carrying meaning about another thing that is far away in time or space).
* Meanings are generally more robust when their dependencies on the context are fewer and/or more local.
* However, large-scale semantics can be made robust by managing the contextual dependencies. For example, using mechanisms for error detection and correction. There is compensatory work that can be done to sustain a regularity so something local continues to mean a particular thing about something distal.
* To be autonomous is to be robust in this sense. Systems that more robustly traffic in large-scale meanings are more autonomous. Thus, being autonomous can be understood as including all the machinery required to hold stable, in a changing world, all the semantic or referential relationships that are required for continued operation.
* Creating regularities that span larger distances is a way of creating new kinds of semantics, and autonomous systems do this as |
7c8663da-c38a-44ce-ab9c-79675aa99a79 | trentmkelly/LessWrong-43k | LessWrong | Can We Align AI by Having It Learn Human Preferences? I’m Scared (summary of last third of Human Compatible)
Epistemic status: I am just learning about alignment and just read Human Compatible. Below is a summary of the paradigm he outlines for aligning AI in the last third of his book, and the questions I have of this project as a new reader.
AI researcher Stuart Russell’s 2019 book Human Compatible is one of the most popular/widely-circulated books on AI alignment right now. It argues that we need to change the paradigm of AI development in general from what he calls goal-directed behavior, where a machine optimizes on a reward function written by humans, to behavior that attempts to learn and follow human objectives.
Russell provides a set of general design principles that someone building this kind of AI would have to follow, quoted directly below:
1. The machine’s only objective is to maximize the realization of human preferences.
2. The machine is initially uncertain about what those preferences are.
3. The ultimate source of information about human preferences is human behavior.
He makes sure to state that he means “preferences” in the way that an economist would define them. Under this definition, a “preference” refers to whatever, given a set of options, a person would rather choose. He attempts to distinguish this from optimizing for “happiness” in a utilitarian framework, which might involve somehow getting a metric of how happy people are that doesn’t just depend on people’s choices, or for some other ethical system encoded by a human. So, instead of getting an objective right the first time, which is insanely hard, you have an AI that does machine learning to figure out what a person, or people, actually want, from observing their preferences.
Russell connects this to work he and others have done in Inverse Reinforcement Learning (IRL), which is machine learning that observes an agent’s behavior and determines the reward function the agent is optimizing for. For example, Russell describes observing a cricket walking on various terrains and wanting to |
cee68554-386a-4fad-925d-6c65ab0e8905 | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Fixing Insider Threats in the AI Supply Chain
Summary
=======
Insider threats are security risks caused by an organisation's staff. **Careless and intentional insider threats cause 25+% of cyber breaches**.[[1]](#fn2om770gv8d2)
In a [survey of existential risks involving cybersecurity and AI](https://forum.effectivealtruism.org/posts/KHmoNx3zpCAaiHxTW/existential-cybersecurity-risks-and-ai-a-research-agenda), I identified **key actors in the AI supply chain to protect: compute manufacturers, frontier AI labs, and AI regulatory agencies**.
These actions can help key actors reduce insider threats:[[2]](#fn7tsfm28kvdr)
* Individual factors:
+ Screen hires for “dark personality traits” (self-promotion, emotional coldness, deceitfulness, aggressiveness, and more).
+ Monitor employees for personal stress, financial issues, dissatisfaction, late/poor quality work, sudden wealth, bragging, suspicious office hours, and signs of substance abuse.
+ Train employees on personal cybersecurity (managing logins, risks with smart home devices, financial scams, etc.).
* Organisational factors:
+ Create clear whistleblowing/reporting rules. Specifically answer NDA and confidentiality concerns for junior staff.
+ Create a dedicated insider threat program and manager. Ensure data sharing between it and cybersecurity/HR teams.
+ Lower outsourcing, remote work, & external tools when possible.
+ Reduce hierarchies. Ex: create consistent opportunities for junior staff to suggest ideas to senior leaders.
* Technical factors:
+ Regularly simulate [social engineering](https://www.fortinet.com/resources/cyberglossary/social-engineering) and [phishing attacks](https://www.fortinet.com/resources/cyberglossary/phishing).
+ Use the obvious tools: [spam filters](https://www.fortinet.com/resources/cyberglossary/spam-filters), [firewalls](https://www.fortinet.com/resources/cyberglossary/firewall) and [proxies](https://www.fortinet.com/resources/cyberglossary/proxy-server), [access management](https://csrc.nist.gov/glossary/term/access_management#:~:text=Access%20Management%20is%20the%20set,Administration%2C%20Authentication%2C%20and%20Authorization.), [network segmentation](https://www.vmware.com/ca/topics/glossary/content/network-segmentation.html#:~:text=Network%20segmentation%20is%20a%20network,services%20to%20each%20sub%2Dnetwork.), etc.
+ Try monitoring tools for underused data (like HR or psychological data).[[3]](#fnn9dud4wv2tc)
---
Details on Individual Factors
=============================
**It's difficult to identify demographics factors** that correlate with insider threats.
* It's possible to analyse public records and state conclusions like: "criminals who performed insider threats are more often men than women."
* However, **insider threats are rarely reported since companies fear reputational damage and most jurisdictions have no reporting requirements**.[[2]](#fn7tsfm28kvdr) Thus, datasets likely have biases compared to the real world.
Sampling bias between different datasets and the real world. Icons by [Freepik](https://www.freepik.com/free-vector/flat-collection-circular-graphs_1145084.htm#query=pie%20charts&position=19&from_view=search&track=ais).Next, there are correlations between "dark personality traits" and staff who intentionally cause insider threats.
* Examples of dark personality traits are intuitive: "the desire for control, narcissism, ... egocentricity, a socially malevolent character, self-promotion, emotional coldness, [deceitfulness], and aggressiveness"[[2]](#fn7tsfm28kvdr)
* Yet **employees may mask such traits, especially during the hiring process.** Thus, it's impractical to use these traits to proactively screen employees.
Instead, organisations can **proactively monitor and encourage employee wellbeing.**
* Ex: It helps if staff regularly check in with each other about personal matters. These include financial stability, relational stability, mental health, a change in work times or quality, or signs of substance abuse.
* Proactive strategies depend on the demographics of an organisation's employees. Those employing young parents may offer childcare benefits, whereas those with an older workforce might invest in personal cybersecurity supports. (A personal hacked account can enable fraud / coercion that causes workplace losses.)[[2]](#fn7tsfm28kvdr)
The last point is important since **insider threats are more often caused by employee error than malicious intent**.[[1]](#fn2om770gv8d2) Extremified employee monitoring programs will reduce privacy, trust, and wellbeing.[[4]](#fne1weetsewg8) Proactive wellbeing supports have low side effects.
---
Details on Organisational Factors
=================================
First, employees who work together are best able to observe each others' wellness. Thus, an organisation's first priority is to **increase "see something, say something" behaviour** amongst all employees. What gets in the way?
* Reporting colleagues can be seen as distrustful ("ratting out"). To fix this, employers must develop reporting channels *primarily* to help employees. **Most often, reporting colleagues' anomalous behaviours should lead to extra support** for them. Reducing malicious insider threats should be a rare side effect.
An organisation where staff feel comfortable supporting each other personally is non-negotiable. Icon by [Freepik](https://www.freepik.com/free-vector/team-spirit-concept-illustration_7079890.htm#query=teamwork&position=38&from_view=search&track=sph). * Employees may be nervous about the confidentiality of reporting a behaviour. Especially if they've signed NDAs or are reporting a senior manager. Thus, clear confidentiality policies about reporting programs are essential.
+ In government settings like an AI regulatory agency, this is even more important due to strict information security clearances for public servants.
* Staff must be trained and encouraged to look for warning signs in colleagues. **Training on mental health warning signs,** [**red teaming**](https://www.crowdstrike.com/cybersecurity-101/red-team-vs-blue-team/) **exercises on** [**phishing**](https://www.fortinet.com/resources/cyberglossary/phishing) **or** [**social engineering**](https://www.fortinet.com/resources/cyberglossary/social-engineering)**, and leadership celebrating positive outcomes of "see something, say something" policies** (like supporting staff in need) can all help.
Separately, some more tangible organisational policies also have an impact.
* For instance, a dedicated insider threats prevention program and manager creates persistent focus and accountability about insider threats. This is easily affordable for frontier AI labs, AI regulatory agencies, and compute manufacturers.
* Moreover, it helps to **reduce outsourcing, contracting, remote work, and external tools when possible**. More people with less direct contact to an organisation's culture brings a higher likelihood of misaligned actions by insiders.
Sadly, an organisation can easily do all the right things on paper but cause a distrustful, inefficient, and toxic workplace. **Though intangible, culture is critical**. Even one remark by managers can have a large impact:[[5]](#fn8g8wv6rov77)
1. More hierarchy, less cooperation: "Recently, we've had increased cyberattacks and our cybersecurity team has been busy around the clock. These experts know what they're doing, so just leave it to them and focus on your jobs."
2. Less trust: "Recently, we've had increased cyberattacks and our cybersecurity team has been busy around the clock. Man, it's so hard to know who to trust these days! I sure am glad they're watching over everything."
3. More community: "Recently, we've had increased cyberattacks and our cybersecurity team has been busy around the clock. I'd struggle to focus on my work without them! I'm glad they help run the tools we need to do our best jobs."
---
Details on Technical Factors
============================
The general theme is that technical safeguards are necessary, but not sufficient. Many solutions are reactive ways to prevent insiders from causing damage.
There are **many standard defences** to briefly name: [spam filters](https://www.fortinet.com/resources/cyberglossary/spam-filters) to prevent [social engineering](https://www.fortinet.com/resources/cyberglossary/social-engineering) against staff; [principle of least privilege](https://csrc.nist.gov/glossary/term/least_privilege#:~:text=Definitions%3A,needs%20to%20perform%20its%20function.) to limit any one employee's access to sensitive information; [zero trust architectures](https://csrc.nist.gov/glossary/term/zero_trust_architecture) to prevent colleagues' access from being exploited; [network segmentation](https://www.vmware.com/ca/topics/glossary/content/network-segmentation.html#:~:text=Network%20segmentation%20is%20a%20network,services%20to%20each%20sub%2Dnetwork.), [proxies](https://csrc.nist.gov/glossary/term/proxy), and [firewalls](https://csrc.nist.gov/glossary/term/firewall) to prevent damages from spreading; [intrusion detection systems](https://csrc.nist.gov/glossary/term/intrusion_detection_system)/[anomaly detection systems](https://csrc.nist.gov/glossary/term/behavioral_anomaly_detection) to spot suspicious activities like sensitive data being exported.
While there are many improvements being researched to the above standard technologies, here are some improvements relevant to insider threats:
To start, it's common to monitor device-specific data for outliers. This data reveals some information about the user (unique keystroke patterns, common actions on the device, etc.). Still, it doesn't reveal much about user motivations. **Complementing device-based data sources with HR data reveals employee motivations**.
* HR data can include business travel history, job title, past projects, salary over time (including perhaps a lack of promotions / raises), and performance reviews.
* As a personified analogy, the HR data can "proactively focus" technical systems to look out for certain anomalies. Ex: An employee on an R&D team with poor performance recently is more likely to cause insider threats by saving sensitive data to unauthorised devices than by bypassing network firewalls (since they're unlikely to have the expertise or insider knowledge to do so). Thus, device specific logs on file access can be scrutinised more than logs on network requests.
A similar conclusion is possible with psychometric data. That said, **psychometric data sources are ethically-contentious and publicly unavailable** to develop defensive tools with. Specifically, psychological questionnaires may be seen as cumbersome or overbearing by employees. Whereas automated data collection tools like social media crawlers may be seen as privacy violations.[[4]](#fne1weetsewg8)
Icons by [Freepik](https://www.freepik.com/free-vector/global-data-security-personal-data-security-cyber-data-security-online-concept-illustration-internet-security-information-privacy-protection_12953569.htm#query=privacy&position=0&from_view=search&track=sph).These "contextual" data sources can **make other insider threat detection systems more adaptive**. For example, access management often has static policies set for each team.[[6]](#fnyf0w7firp2) If an employee has a low trustworthiness score due to some recent logins at suspicious times, a dynamic access management system could temporarily revoke the employee's access to certain sensitive documents.
As seen, the general trend with improving technical defences against insider threats at an organisation is to get *more holistic (and human) data* which is used to *adapt defences* over time.
---
Personally, I've been intrigued to learn about all these human-focused best practices to reduce insider threats. I'm hoping to get more primary data by interviewing cyber security staff at AI labs and compute manufacturers. Any suggestions on who to reach out to are much appreciated!
1. **[^](#fnref2om770gv8d2)**G. Bassett, C. D. Hylender, P. Langlois, A. Pinto, and S. Widup, “2022 Data Breach Investigations Report,” Verizon Communications Inc., 2022. Accessed: Nov 15, 2022. [Online]. Available: <https://www.verizon.com/business/resources/T501/reports/dbir/2022-data-breach-investigations-report-dbir.pdf>
2. **[^](#fnref7tsfm28kvdr)**Black, Marigold, et al. Insider Threat and White-Collar Crime in Non-Government Organisations and Industries: A Literature Review. RAND Corporation, 2022, <https://doi.org/10.7249/RRA1507-1>.
3. **[^](#fnrefn9dud4wv2tc)**L. Liu, O. De Vel, Q. -L. Han, J. Zhang and Y. Xiang, "Detecting and Preventing Cyber Insider Threats: A Survey," in IEEE Communications Surveys & Tutorials, vol. 20, no. 2, pp. 1397-1417, 2018, doi: 10.1109/COMST.2018.2800740.
4. **[^](#fnrefe1weetsewg8)**J. Love and F. Schmalz, ‘Companies Now Have Many Tools to Monitor Employee Productivity. When Should They Use Them?’, *Kellogg Insight*. Available: <https://insight.kellogg.northwestern.edu/productivity-monitoring>. [Accessed: Oct. 04, 2023]
5. **[^](#fnref8g8wv6rov77)**A. Moore, S. Perl, J. Cowley, M. Collins, T. Cassidy, N. VanHoudnos, P. Buttles-Valdez, D. Bauer, A. Parshall, J. Savinda, E. Monaco, J. Moyes, and D. Rousseau, "The Critical Role of Positive Incentives for Reducing Insider Threats," *Carnegie Mellon University, Software Engineering Institute's Digital Library*. Software Engineering Institute, Technical Report CMU/SEI-2016-TR-014, 15-Dec-2016 [Online]. Available: <https://doi.org/10.1184/R1/6585104.v1>. [Accessed: 5-Oct-2023].
6. **[^](#fnrefyf0w7firp2)**J. Crampton and M. Huth, ‘Towards an Access-Control Framework for Countering Insider Threats’, in *Insider Threats in Cyber Security*, C. W. Probst, J. Hunker, D. Gollmann, and M. Bishop, Eds., Boston, MA: Springer US, 2010, pp. 173–195. doi: 10.1007/978-1-4419-7133-3\_8. Available: <https://link.springer.com/10.1007/978-1-4419-7133-3_8>. [Accessed: Oct. 05, 2023] |
1210f3cd-7cca-4d5b-bead-9a25fcc08ffd | StampyAI/alignment-research-dataset/agisf | AGI Safety Fund | [Week 2] “Learning from human preferences” (Blog Post) by Dario Amodei, Paul Christiano & Alex Ray
We present a learning algorithm that uses small amounts of human feedback to solve modern RL environments. Machine learning systems with human feedback [have](https://papers.nips.cc/paper/4805-a-bayesian-approach-for-policy-learning-from-trajectory-preference-queries) [been](https://link.springer.com/chapter/10.1007/978-3-319-02675-6_46) [explored](https://arxiv.org/abs/1208.0984) [before](https://hal.inria.fr/hal-00980839), but we’ve scaled up the approach to be able to work on much more complicated tasks. Our algorithm needed 900 bits of feedback from a human evaluator to learn to backflip—a seemingly simple task which is simple to judge but [challenging](https://openai.com/blog/deep-reinforcement-learning-from-human-preferences/#bflip) to specify.
Our algorithm learned to backflip using around 900 individual bits of feedback from the human evaluator.The overall training process is a 3-step feedback cycle between the human, the agent’s understanding of the goal, and the RL training.
Our AI agent starts by acting randomly in the environment. Periodically, two video clips of its behavior are given to a human, and the human decides which of the two clips is closest to fulfilling its goal—in this case, a backflip. The AI gradually builds a model of the goal of the task by finding the reward function that best explains the human’s judgments. It then uses RL to learn how to achieve that goal. As its behavior improves, it continues to ask for human feedback on trajectory pairs where it’s most uncertain about which is better, and further refines its understanding of the goal.
Our approach demonstrates promising sample efficiency—as stated previously, the backflip video required under 1000 bits of human feedback. It took less than an hour of a human evaluator’s time, while in the background the policy accumulated about 70 hours of overall experience (simulated at a much faster rate than real-time.) We will continue to work on reducing the amount of feedback a human needs to supply. You can see a sped-up version of the training process in the following video.
Human Feedback training process00:39We’ve tested our method on a number of tasks in the simulated robotics and Atari domains (without being given access to the reward function: so in Atari, without having access to the game score). Our agents can learn from human feedback to achieve strong and sometimes superhuman performance in many of the environments we tested. In the following animation you can see agents trained with our technique playing a variety of Atari games. The horizontal bar on the right hand side of each frame represent’s each agents prediction about how much a human evaluator would approve of their current behavior. These visualizations indicate that agents trained with human feedback learn to value oxygen in Seaquest (left), anticipate rewards in Breakout and Pong (center), or work out how to recover from crashes in Enduro (right).
Note there’s no need for the feedback to align with the environment’s normal reward function: we can, for example, train our agents to precisely keep even with other cars in Enduro rather than maximizing game score by passing them. We also sometimes find that learning from feedback does better than reinforcement learning with the normal reward function, because the human shapes the reward better than whoever wrote the environment’s reward.
Challenges
----------
Our algorithm’s performance is only as good as the human evaluator’s intuition about what behaviors *look* correct, so if the human doesn’t have a good grasp of the task they may not offer as much helpful feedback. Relatedly, in some domains our system can result in agents adopting policies that trick the evaluators. For example, a robot which was supposed to grasp items instead positioned its manipulator in between the camera and the object so that it only *appeared* to be grasping it, as shown below.
We addressed this particular problem by adding in visual cues (the thick white lines in the above animation) to make it easy for the human evaluators to estimate depth.
The research described in this post was done in collaboration with Jan Leike, Miljan Martic, and Shane Legg at DeepMind. Our two organizations plan to continue to collaborate on topics that touch on long-term AI safety. We think that techniques like this are a step towards safe AI systems capable of learning human-centric goals, and can complement and extend existing approaches like reinforcement and imitation learning. This post is representative of the work done by OpenAI’s safety team; if you’re interested in working on problems like this, please [join us](https://openai.com/jobs/)!
Footnote
--------
By comparison, we took two hours to write our own reward function (the animation in the above right) to get a robot to backflip, and though it succeeds it’s a lot less elegant than the one trained simply through human feedback (top left). We think there are many cases where human feedback could let us specify a specific goal more intuitively and quickly than is possible by manually hand-crafting the objective.
You can replicate this backflip in [gym](http://github.com/openai/gym) with the following reward function for Hopper:
```
def reward_fn(a, ob):
backroll = -ob[7]
height = ob[0]
vel_act = a[0] * ob[8] + a[1] * ob[9] + a[2] * ob[10]
backslide = -ob[5]
return backroll * (1.0 + .3 * height + .1 * vel_act + .05 * backslide)
```
null |
6eab3b6f-0c42-42c1-971f-cfd3e3aac23e | trentmkelly/LessWrong-43k | LessWrong | Truthfulness, standards and credibility
-1: Meta Prelude
While truthfulness is a topic I’ve been thinking about for some time, I’ve not discussed much of what follows with others. Therefore, at the very least I expect to be missing important considerations on some issues (where I’m not simply wrong).
I’m hoping this should make any fundamental errors in my thought process more transparent, and amenable to correction. The downside may be reduced clarity, more illusion-of-transparency…. Comments welcome on this approach.
I don’t think what follows is novel. I’m largely pointing at problems based on known issues.
Sadly, I don’t have a clear vision of an approach that would solve these problems.
0: Introduction
> …our purpose is not to give the last word, but rather the first word, opening up the conversation… (Truthful AI)
I’d first like to say that I believe some amount of research on truthfulness to be worthwhile, and to thank those who’ve made significant efforts towards greater understanding (including, but not limited to, the authors of Truthful AI (henceforth TruAI)).
No doubt there’s some value in understanding more, but my guess is that it won’t be a particularly fruitful angle of attack. In all honesty, it seems an inefficient use of research talent to me - but perhaps I’m missing something.
Either way, I hope the following perspective will suggest some useful directions for conversation in this area.
[Note: section numbers refer to this document unless “TruAI…” is specified]
[I'll be assuming familiarity with TruAI throughout, though reading the full paper probably isn't necessary so long as you've seen the executive summary in the post]
My current belief is that near-term implementation of the kind of truthfulness standards talked about in TruAI would be net negative, for reasons I’ll go on to explain. To me it seems as if we’d be implementing a poor approximation to a confused objective.
A high-level summary of my current view:
* Narrow truthfulness looks approachable, but will b |
ced09c1d-11bf-4b78-9bc4-c9f3972fc633 | trentmkelly/LessWrong-43k | LessWrong | The Efficient Market Hypothesis in Research
A classic economics joke goes like this:
> Two economists are walking down a road, when one of them notices a $20 bill on the ground. He turns to his friend and exclaims: "Look, a $20 bill!" The other replies: "Nah, if there's a $20 on the bill on the ground, someone would've picked it up already."
The economists in the joke believe in the Efficient Market Hypothesis (EMH), which roughly says that financial markets are efficient and there's no way to "beat the market" by making intelligent trades.
If the EMH was true, then why is there still a trillion-dollar finance industry with active mutual funds and hedge funds? In reality, the EMH is not a universal law of economics (like the law of gravity), but more like an approximation. There may exist inefficiencies in markets where stock prices follow a predictable pattern and there is profit to be made (e.g.: stock prices fall when it's cloudy in New York). However, as soon as someone notices the pattern and starts exploiting it (by making a trading algorithm based on weather data), the inefficiency disappears. The next person will find zero correlation between weather in New York and stock prices.
There is a close parallel in academic research. Here, the "market" is generally efficient: most problems that are solvable are already solved. There are still "inefficiencies": open problems that can be reasonably solved, and one "exploits" them by solving it and publishing a paper. Once exploited, it is no longer available: nobody else can publish the same paper solving the same problem.
Where does this leave the EMH? In my view, the EMH is a useful approximation, but its accuracy depends on your skill and expertise. For non-experts, the EMH is pretty much universally true: it's unlikely that you've found an inefficiency that everyone else has missed. For experts, the EMH is less often true: when you're working in highly specialized areas that only a handful of people understand, you begin to notice more inefficienci |
910b88fa-8d28-4dac-aee4-80d30ad2bf11 | trentmkelly/LessWrong-43k | LessWrong | Peer review me
I wrote an article that I hoped to post on the main page, but then I got stage fright and was afraid to even put it here. So I guess I'm just going to show it to whichever of you is willing to review it privately.
Any takers? Qualifications: must be a fan of Z-movies. |
c6fbfac7-d8eb-45a8-b0d0-5fa149cad155 | trentmkelly/LessWrong-43k | LessWrong | A framework for thinking about wireheading
[Epistemic status: Writing down in words something that isn't very complicated, but is still good to have written down.]
A great deal of ink has been spilled over the possibility of a sufficiently intelligent AI noticing that whatever it has been positively reinforced for seems to be *very* strongly correlated with the floating point value stored in its memory labelled “utility function”, and so through some unauthorized mechanism editing this value and defending it from being edited back in some manner hostile to humans. I'll reason here by analogy with humans, while agreeing that they might not be the best example.
“Headwires” are (given a little thought) not difficult to obtain for humans -- heroin is freely available on the black market, and most humans know that, when delivered into the bloodstream, it generates “reward signal”. Yet most have no desire to try it. Why is this?
Asking any human, they will answer something along the lines of ”becoming addicted to heroin will not help me achieve my goals” ( or some proxy for this: spending all your money and becoming homeless is not very helpful in achieving one's goals for most values of “goals”.) Whatever the effects of heroin, the actual pain and pleasure that the human brains have experienced has led us to become optimizers of very different things, which a state of such poverty is not helpful for.
Reasoning analogously to AI, we would hope that, to avoid this, a superhuman AI trained by some kind of reinforcement learning has the following properties:
1. While being trained on “human values” (good luck with that!) the AI must not be allowed to hack its own utility function.
2. Whatever local optima the training process that generated the AI ends up in (perhaps reinforcement learning of some kind) assigns some probability to the AI optimising what we care about.
3. (most importantly) The AI realizes that trying wireheading will lead it to become an AI which prefers wireheading over aim it currently has, |
99ac66fc-721a-4ab7-80f6-c4bbb3cc60f4 | StampyAI/alignment-research-dataset/arbital | Arbital | Safe but useless
"This type of safety implies uselessness" (or conversely, "any AI powerful enough to be useful will still be unsafe") is an accusation leveled against a proposed AI safety measure that must, to make the AI safe, be enforced to the point that it will make the AI useless.
For a non-AI metaphor, consider a scissors and its dangerous blades. We can have a "safety scissors" that is only *just* sharp enough to cut paper - but this is still sharp enough to do some damage if you work at it. If you try to make the scissors *even safer* by encasing the dangerous blades in foam rubber, the scissors can't cut paper any more. If the scissors *can* cut paper, it's still unsafe. Maybe you could in principle cut clay with a scissors like that, but this is no defense unless you can tell us [something very useful](https://arbital.com/p/6y) that can be done by cutting clay.
Similarly, there's an obvious way to try cutting down the allowed output of an [Oracle AGI](https://arbital.com/p/6x) to the point where [all it can do is tell us that a given theorem is provable from the axioms of Zermelo-Fraenkel set theory](https://arbital.com/p/70). This [might](https://arbital.com/p/2j) prevent the AGI from hacking the human operators into letting it out, since all that can leave the box is a single yes-or-no bit, sent at some particular time. An untrusted superintelligence inside this scheme would have the option of strategically not telling us when a theorem *is* provable in ZF; but if the bit from the proof-verifier said that the input theorem was ZF-provable, we could very likely trust that.
But now we run up against the problem that nobody knows how to [actually save the world](https://arbital.com/p/6y) by virtue of sometimes knowing for sure that a theorem is provable in ZF. The scissors has been blunted to where it's probably completely safe, but can only cut clay; and nobody knows how to [do *enough* good](https://arbital.com/p/6y) by cutting clay.
# Ideal models of "safe but useless" agents
Should you have cause to do a mathematical study of this issue, then an excellent [ideal model](https://arbital.com/p/107) of a safe but useless agent, embodying maximal safety and minimum usefulness, would be a rock. |
a177eaea-4d1b-4f25-a1d1-ae1dcc07c695 | StampyAI/alignment-research-dataset/arxiv | Arxiv | More Data Can Hurt for Linear Regression: Sample-wise Double Descent
1 Introduction
---------------
Common statistical intuition suggests that more data should never harm the performance of an estimator.
It was recently highlighted in [[17](#bib.bib1 "Deep double descent: where bigger models and more data hurt")] that
this may not hold for *overparameterized* models:
there are settings in modern deep learning
where training on more data actually hurts.
In this note, we analyze a simple setting to understand the mechanisms behind this
behavior.
| | |
| --- | --- |
|
(a) Test MSE for d=1000,σ=0.1.
|
(b) Test MSE in theory for d=1000,σ=0.1
|
Figure 1: Test MSE vs. Num. Train Samples
for the min-norm ridgeless regression estimator in d=1000 dimensions.
The distribution is a linear model with noise:
covariates x∼N(0,Id) and
response y=⟨x,β⟩+N(0,σ2),
for d=1000,σ=0.1, and ||β||2=1.
The estimator is ^β=X†y.
Left:
Solid line shows mean over 50 trials,
and individual points show a single trial.
Right:
Theoretical predictions for the bias, variance,
and risk from Claims [3.1](#S3.SS1 "3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent")
and [3.1](#S3.SS1 "3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
We focus on well-specified linear regression with Gaussian covariates, and we analyze the test risk of
the minimum-norm ridgeless regression estimator—
or equivalently, the estimator found by gradient descent on the least squares objective.
We show that as we increase the number of samples,
performance is non-monotonic: The test risk first decreases, and then *increases*, before decreasing again.
Such a “double-descent” behavior has been observed in the behavior of test risk as a function of the model size in a variety of machine learning settings
[[19](#bib.bib15 "Statistical mechanics of learning: generalization"), [20](#bib.bib14 "Learning to generalize"), [1](#bib.bib20 "High-dimensional dynamics of generalization error in neural networks"), [3](#bib.bib25 "Reconciling modern machine learning and the bias-variance trade-off"), [21](#bib.bib8 "A jamming transition from under-to over-parametrization affects loss landscape and generalization"), [8](#bib.bib17 "Jamming transition as a paradigm to understand the loss landscape of deep neural networks"), [17](#bib.bib1 "Deep double descent: where bigger models and more data hurt")].
Many of these works are motivated by understanding
the test risk as function of model size,
for a fixed number of samples.
In this work, we take a complementary view and understand the
test risk as a function of
*sample size*, for a fixed model.
We hope that understanding such simple settings
can eventually lead to understanding
the general phenomenon,
and lead us to design learning algorithms which make the best use of data (and in particular, are monotonic in samples).
We note that similar analyses appear in recent works,
which we discuss in Section [1.1](#S1.SS1 "1.1 Related Works ‣ 1 Introduction ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent")– our focus is to highlight
the sample non-monotonicity implicit in these works, and give intuitions for the mechanisms behind it.
We specifically refer the reader to [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation"), [14](#bib.bib10 "The generalization error of random features regression: precise asymptotics and double descent curve")] for analysis in a setting most similar to ours.
##### Organization.
We first define the linear regression setting in
Section [2](#S2 "2 Problem Setup ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
Then in Section [3](#S3 "3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") we
state the form of the estimator found by gradient descent,
and give intuitions for why this estimator
has a peak in test risk when the number of samples is equal to the ambient dimension.
In Section [3.1](#S3.SS1 "3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent"), we decompose the expected excess risk
into bias and variance contributions, and we state approximate expressions
for the bias, variance, and excess risk as a function of samples.
We show that these approximate theoretical predictions closely agree with
practice, as in Figure [1](#S1.F1 "Figure 1 ‣ 1 Introduction ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
The peak in test risk turns out to be related to the conditioning of the data
matrix, and in Section [3.2](#S3.SS2 "3.2 Conditioning of the Data Matrix ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") we give intuitions for
why this matrix is poorly conditioned in the “critical regime”, but well
conditioned outside of it.
We also analyze the marginal effect of adding a single sample to
the test risk, in Section [3.3](#S3.SS3 "3.3 Effect of Adding a Single Sample ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
We conclude with discussion and open questions in Section [4](#S4 "4 Discussion ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
###
1.1 Related Works
This work was inspired by the long line of work studying “double descent” phenomena
in deep and shallow models.
The general principle is that as the model complexity increases,
the test risk of trained models first decreases and then increases (the standard U-shape), and then *decreases again*.
The peak in test risk occurs in the “critical regime” when the models
are just barely able to fit the training set. The second descent occurs in the “overparameterized regime”, when the model capacity is large enough to contain several interpolants on the training data.
This phenomenon appears to be fairly universal among natural learning algorithms,
and is observed in simple settings such as linear regression, random features regression, classification with random forests, as well as modern neural networks.
Double descent of test risk with model size was introduced in generality
by [[3](#bib.bib25 "Reconciling modern machine learning and the bias-variance trade-off")],
building on similar behavior observed as early as [[19](#bib.bib15 "Statistical mechanics of learning: generalization"), [20](#bib.bib14 "Learning to generalize")]
and more recently by
[[1](#bib.bib20 "High-dimensional dynamics of generalization error in neural networks"), [18](#bib.bib9 "A modern take on the bias-variance tradeoff in neural networks"), [21](#bib.bib8 "A jamming transition from under-to over-parametrization affects loss landscape and generalization"), [8](#bib.bib17 "Jamming transition as a paradigm to understand the loss landscape of deep neural networks")].
A generalized double descent phenomenon
was demonstrated on modern deep networks
by [[17](#bib.bib1 "Deep double descent: where bigger models and more data hurt")],
which also highlighted “sample-wise nonmonotonicity”
as a consequence of double descent – showing that more data can hurt for deep neural networks.
A number of recent works
theoretically analyze the double descent behavior
in simplified settings, often for linear models
[[4](#bib.bib16 "Two models of double descent for weak features"), [9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation"), [2](#bib.bib19 "Benign overfitting in linear regression"), [16](#bib.bib18 "Harmless interpolation of noisy data in regression"), [5](#bib.bib13 "A new look at an old problem: a universal learning approach to linear regression"), [15](#bib.bib12 "Understanding overfitting peaks in generalization error: analytical risk curves for l2 and l1 penalized interpolation"), [14](#bib.bib10 "The generalization error of random features regression: precise asymptotics and double descent curve"), [12](#bib.bib22 "Just interpolate: kernel\" ridgeless\" regression can generalize"), [11](#bib.bib26 "On the risk of minimum-norm interpolants and restricted lower isometry of kernels"), [22](#bib.bib7 "On the number of variables to use in principal component regression"), [7](#bib.bib5 "Exact expressions for double descent and implicit regularization via surrogate random design"), [10](#bib.bib4 "An analytic theory of generalization dynamics and transfer learning in deep linear networks"), [6](#bib.bib2 "A model of double descent for high-dimensional binary linear classification")].
At a high level, these works analyze the
test risk of estimators in overparameterized linear regression with different assumptions on the covariates.
We specifically refer the reader to
[[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation"), [14](#bib.bib10 "The generalization error of random features regression: precise asymptotics and double descent curve")]
for rigorous analysis in a setting most similar to ours.
In particular, [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation")]
considers the asymptotic risk of the minimum norm
ridgeless regression estimator
in the limit where dimension d and number of samples
n are scaled as d→∞,n=γd.
We instead focus on the sample-wise perspective: a fixed large d, but varying n.
In terms of technical content,
the analysis technique is not novel to our work,
and similar calculations appear in some of the prior works above.
Our main contribution is highlighting the sample non-monotonic behavior in a simple setting,
and elaborating on the mechanisms responsible.
While many of the above theoretical results are qualitatively similar, we highlight one interesting distinction:
our setting is *well-specified*,
and the bias of the estimator is monotone nonincreasing
in number of samples (see Equation [3](#S3.E3 "(3) ‣ 3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent"), and also [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation"), Section 3]).
In contrast, for *misspecified* problems (e.g. when the ground-truth is nonlinear, but we learn a linear model), the bias can actually increase with number of samples in addition to the variance increasing (see [[14](#bib.bib10 "The generalization error of random features regression: precise asymptotics and double descent curve")]).
2 Problem Setup
----------------
Consider the following learning problem:
The ground-truth distribution
D is (x,y)∈\Rd\x\R
with covariates x∼N(0,Id) and
response y=⟨x,β⟩+N(0,σ2)
for some unknown, arbitrary
β such that ||β||2≤1.
That is, the ground-truth is an isotropic Gaussian with observation noise.
We are given n samples (xi,yi) from the distribution,
and we want to learn a linear model
f^β(x)=⟨x,^β⟩
for estimating y given x.
That is, we want to find ^β
with small test mean squared error
| | | | |
| --- | --- | --- | --- |
| | R(^β) | :=\E(x,y)∼D[(⟨x,^β⟩−y)2] | |
| | | =||^β−β||2+σ2 | | (for isotropic x∼N(0,Id)) |
Suppose we do this by performing ridgeless linear regression.
Specifically, we run gradient descent initialized at 0 on the following objective (the empirical risk).
| | | | |
| --- | --- | --- | --- |
| | min^β||X^β−y||2 | | (1) |
where X∈\Rn\xd is the data-matrix of samples xi,
and y∈\Rn are the observations.
The solution found by gradient descent at convergence is
^β=X†y, where † denotes the Moore–Penrose pseudoinverse111To see this,
notice that the iterates of gradient descent lie in the row-space of X..
Figure [0(a)](#S1.F0.sf1 "(a) ‣ Figure 1 ‣ 1 Introduction ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") plots the expected test MSE
of this estimator
\EX,y[R(^β))] as we vary the number of train samples n.
Note that it is non-monotonic, with a peak in test MSE at n=d.
There are two surprising aspects of the test risk
in Figure [0(a)](#S1.F0.sf1 "(a) ‣ Figure 1 ‣ 1 Introduction ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent"), in the overparameterized regime (n<d):
1. The first descent: where test risk initially decreases
even when we have less samples n than dimensions d.
This occurs because the bias decreases.
2. The first ascent:
where test risk increases, and peaks when n=d.
This is because the variance increases, and diverges when n=d.
When n>d, this is the classical underparameterized regime, and test risk is monotone decreasing with number of samples.
Thus overparameterized linear regression exhibits a *bias-variance tradeoff*:
bias decreases with more samples,
but variance can increase.
Below, we elaborate on the mechanisms and provide intuition for this non-monotonic behavior.
3 Analysis
-----------
The solution found by gradient descent,
^β=X†y,
has different forms depending on the ratio n/d.
When n≥d, we are in the “underparameterized” regime
and there is a unique minimizer of the objective in Equation [1](#S2.E1 "(1) ‣ 2 Problem Setup ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
When n<d, we are “overparameterized” and there are many minimizers
of Equation [1](#S2.E1 "(1) ‣ 2 Problem Setup ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
In fact, since X is full rank with probability 1,
there are many minimizers which *interpolate*, i.e.
X^β=y.
In this regime, gradient descent finds the minimum with
smallest ℓ2 norm ||^β||2.
That is, the solution can be written as
| | | |
| --- | --- | --- |
| | | |
The overparameterized form yields insight into why the test MSE peaks at n=d. Recall that the observations are noisy, i.e. y=Xβ+η where η∼N(0,σ2In).
When n≪d, there are many interpolating estimators
{^β:X^β=y}, and in particular there exist such ^β with small
norm.
In contrast, when n=d, there is exactly one interpolating estimator
(X^β=y), but this estimator must have high norm in order to fit the
noise η.
More precisely, consider
| | | | |
| --- | --- | --- | --- |
| | ^β | =X†y=X†(Xβ+η)=X†Xβsignal+X†ηnoise | |
The signal term X†Xβ is simply the orthogonal
projection of β onto the rows of X.
When we are “critically parameterized” and n≈d,
the data matrix X is very poorly conditioned, and hence the noise term
X†η has high norm, overwhelming the signal.
This argument is made precise in Section [3.1](#S3.SS1 "3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent"),
and in Section [3.2](#S3.SS2 "3.2 Conditioning of the Data Matrix ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") we give intuition for why X becomes poorly conditioned
when n≈d.
The main point is that when n=d,
forcing the estimator ^β to interpolate the noise
will force it to have very high norm, far from the ground-truth
β.
(See also Corollary 1 of [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation")]
for a quantification of this point).
###
3.1 Excess Risk and Bias-Variance Tradeoffs
For ground-truth parameter β,
the excess risk222For clarity, we consider the excess risk,
which omits the unavoidable additive σ2 error in the true risk.
of an estimator ^β is:
| | | | |
| --- | --- | --- | --- |
| | ¯R(^β) | :=\E(x,y)∼D[(⟨x,^β⟩−y)2]−\E(x,y)∼D[(⟨x,β⟩−y)2] | |
| | | =\Ex∼N(0,I),η∼N(0,σ2)[(⟨x,^β⟩−⟨x,β⟩+η)2]−σ2 | |
| | | =||^β−β||2 | |
For an estimator ^βX,y that is derived from samples (X,y)∼Dn,
we consider the expected excess risk of ^β=^βX,y in expectation over samples (X,y) :
| | | | |
| --- | --- | --- | --- |
| | \EX,y[¯R(^βX,y)]=\EX,y[||^β−β||2]=||β−\E[^β]||2Bias Bn+\E[||^β−\E[^β]||2]Variance Vn | | (2) |
Where Bn,Vn are the bias and variance of the estimator on n samples.
For the specific estimator ^β=X†y in the regime n≤d,
the bias and variance can be written as (see Appendix [A.1](#A1.SS1 "A.1 Bias and Variance ‣ Appendix A Appendix: Computations ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent")):
| | | | | |
| --- | --- | --- | --- | --- |
| | Bn | =||\EX∼Dn[ProjX⊥(β)]||2 | | (3) |
| | Vn | =\EX[||ProjX(β)−\EX[ProjX(β)]||2](A)+σ2\EX[\Tr((XXT)−1)](B) | | (4) |
where ProjX is the orthogonal projector
onto the rowspace of the data X∈\Rn\xd,
and ProjX⊥ is the projector onto the orthogonal complement of the rowspace.
From Equation [3](#S3.E3 "(3) ‣ 3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent"), the bias is non-increasing with samples
(Bn+1≤Bn), since an additional sample can only grow
the rowspace: X⊥n+1⊆X⊥n.
The variance in Equation [4](#S3.E4 "(4) ‣ 3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") has two terms: the first term (A) is due to the randomness of X, and is bounded.
But the second term (B) is due to the randomness
in the noise of y, and diverges when n≈d since X becomes poorly conditioned.
This trace term is responsible for the peak in test MSE at n=d.
We can also approximately compute the bias, variance, and excess risk.
{claim}
[Overparameterized Risk]
Let γ:=nd<1 be the underparameterization ratio.
The bias and variance are:
| | | | | |
| --- | --- | --- | --- | --- |
| | Bn | =(1−γ)2||β||2 | | (5) |
| | Vn | ≈γ(1−γ)||β||2+σ2γ1−γ | | (6) |
And thus the expected excess risk for γ<1 is:
| | | | | |
| --- | --- | --- | --- | --- |
| | \E[¯R(^β)] | ≈(1−γ)||β||2+σ2γ1−γ | | (7) |
| | | =(1−nd)||β||2+σ2nd−n | | (8) |
These approximations are not exact because they hold
asyptotically in the limit of large d
(when scaling n=γd),
but may deviate for finite samples.
In particular, the bias Bn and term (A) of the variance can be computed exactly for finite samples: ProjX is simply a projector onto a
uniformly random n-dimensional subspace, so \E[ProjX(β)]=γβ, and similarly \E[||ProjX(β)||2]=γ||β||2.
The trace term (B) is nontrivial to understand for finite samples, but converges333
For large d,
the spectrum of (XXT) is understood by the Marchenko–Pastur law [[13](#bib.bib6 "Distribution of eigenvalues for some sets of random matrices")].
Lemma 3 of [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation")] uses this to show
that \Tr((XXT)−1)→γ1−γ.
to γ1−γ
in the limit of large n,d
(e.g. Lemma 3 of [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation")]).
In Section [3.3](#S3.SS3 "3.3 Effect of Adding a Single Sample ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent"), we give intuitions for why the trace term
converges to this.
For completeness, the bias, variance, and excess risk in the underparameterized regime
are given in [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation"), Theorem 1] as:
{claim}[Underparameterized Risk, [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation")]]
Let γ:=nd>1 be the underparameterization ratio.
The bias and variance are:
| | | |
| --- | --- | --- |
| | Bn=0,Vn≈σ2γ−1 | |
Figure [1](#S1.F1 "Figure 1 ‣ 1 Introduction ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") shows that Claims [3.1](#S3.SS1 "3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") and [3.1](#S3.SS1 "3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") agree
with the excess risk experimentally
even for finite d=1000.
###
3.2 Conditioning of the Data Matrix
Here we give intuitions for why the data matrix X∈\Rn\xd
is well conditioned for n≪d, but has small singular values for n≈d.
####
3.2.1 Near Criticality
First, let us consider the effect of adding a single sample when n=(d−1).
For simplicity, assume the first (d−1) samples xi are just the
standard basis vectors, scaled appropriately.
That is, assume the data matrix X∈\R(d−1)\xd is
| | | |
| --- | --- | --- |
| | | |
This has all non-zero singular values equal to d.
Then, consider adding a new isotropic Gaussian sample xn+1∼N(0,Id).
Split this into coordinates as xn+1=(g1,g2)∈\Rd−1\x\R.
The new data matrix is
| | | |
| --- | --- | --- |
| | | |
We claim that Xn+1 has small singular values.
Indeed, consider left-multiplication by :
| | | | | |
| --- | --- | --- | --- | --- |
| | vTXn+1=\mqty[g1 | −d]\mqty[dId−1 | 0 | |
| | g1 | g2]=\mqty[0 | −dg2] | |
Thus, ||vTXn+1||2≈d2,
while ||v||2≈2d2. Since Xn+1 is full-rank,
it must have a singular value less than roughly 1√2.
That is, adding a new sample has shrunk the minimum non-zero singular value of X
from d to less than a constant.
The intuition here is: although the new sample xn+1 adds rank to the existing samples, it does so in a very fragile way.
Most of the ℓ2 mass of xn+1 is contained in the span of existing samples, and xn only contains a small component outside of this subspace. This causes Xn+1 to have small singular values,
which in turn causes the ridgeless regression estimator (which applies X†) to be sensitive to noise.
A more careful analysis shows that the singular values are actually even smaller than the above simplification suggests
— since in the real setting, the matrix X was already poorly conditioned even before the new sample xn+1.
In Section [3.3](#S3.SS3 "3.3 Effect of Adding a Single Sample ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent")
we calculate the exact effect of adding a single sample to the excess risk.
####
3.2.2 Far from Criticality
When n≪d, the data matrix X does not have singular values close to 0.
One way to see this is to notice that
since our data model treats features and samples symmetrically,
X is well conditioned in the regime n≪d
for the same reason that standard linear regression works in the classical underparameterized regime n≫d (by “transposing” the setting).
More precisely, since X is full rank, its smallest non-zero singular value
can be written as
| | | |
| --- | --- | --- |
| | σmin(X)=minv∈\Rn:||v||2=1||vTX||2 | |
Since X has entries i.i.d N(0,1),
for every fixed vector v we have \EX[||vTX||2]=d||v||2=d.
Moreover, for d=Ω(n) uniform convergence holds, and
||vTX||2 concentrates around its expectation for
all vectors v in the ℓ2 ball. Thus:
| | | |
| --- | --- | --- |
| | σmin(X)2≈\EX[minv∈\Rn:||v||2=1||vTX||2]≈minv\EX[||vTX||2]=d | |
###
3.3 Effect of Adding a Single Sample
Here we show how the
trace term of the variance in Equation [4](#S3.E4 "(4) ‣ 3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") changes with increasing samples.
Specifically, the following claim shows how \Tr((XXT)−1) grows
when we add a new sample to X.
{claim}
Let X∈\Rn\xd be the data matrix after n samples,
and let x∈\Rd be the (n+1)th sample.
The new data matrix is Xn+1=\mqty[XxT], and
| | | |
| --- | --- | --- |
| | \Tr((Xn+1XTn+1)−1)=\Tr[(XXT)−1]+1+||(XT)†x||2||ProjX⊥(x)||2 | |
###### Proof.
By computation in Appendix [A.2](#A1.SS2 "A.2 Trace Computations ‣ Appendix A Appendix: Computations ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
∎
If we heuristically assume the denominator
concentrates around its expectation,
||ProjX⊥(x)||2≈d−n, then we can
use Claim [3.3](#S3.SS3 "3.3 Effect of Adding a Single Sample ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent")
to estimate the expected effect of a single sample:
| | | | | |
| --- | --- | --- | --- | --- |
| | \Ex\Tr((Xn+1XTn+1)−1) | ≈\Tr[(XXT)−1]+1+\Ex||(XXT)−1Xx||2d−n | | (9) |
| | | =\Tr[(XXT)−1](1+1d−n)+1d−n | | (10) |
We can further estimate the growth by taking a continuous
limit for large d.
Let F(nd):=\E[\Tr((XnXTn)−1)].
Then for γ:=nd, Equation [10](#S3.E10 "(10) ‣ 3.3 Effect of Adding a Single Sample ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent") yields the differential equation
| | | |
| --- | --- | --- |
| | dF(γ)dγ=(1−γ)−1F+(1−γ)−1 | |
which is solved by F(γ)=γ1−γ.
This heuristic derivation that
\E[\Tr(XXT)−1]→γ1−γ
is consistent with the rigorous asymptotics given in [[9](#bib.bib11 "Surprises in high-dimensional ridgeless least squares interpolation"), Lemma 3]
and used in Claim [3.1](#S3.SS1 "3.1 Excess Risk and Bias-Variance Tradeoffs ‣ 3 Analysis ‣ More Data Can Hurt for Linear Regression: Sample-wise Double Descent").
4 Discussion
-------------
We hope that understanding such simple settings
can eventually lead to understanding the general
behavior of overparameterized models
in machine learning.
We consider it extremely unsatisfying that
the most popular technique in modern machine learning
(training an overparameterized neural network with SGD)
can be nonmonotonic in samples [[17](#bib.bib1 "Deep double descent: where bigger models and more data hurt")].
We hope that a greater understanding here
could help develop learning algorithms which make
the best use of data (and in particular, are monotonic in samples).
In general, we believe it is interesting to understand when and why learning
algorithms are monotonic – especially when we don’t explicitly enforce them to
be.
### Acknowledgements
We especially thank
Jacob Steinhardt and
Aditi Raghunathan
for discussions and suggestions that motivated this work.
We thank
Jarosław Błasiok,
Jonathan Shi,
and Boaz Barak for useful discussions
throughout this work,
and we thank Gal Kaplun and Benjamin L. Edelman for feedback on an early
draft.
This work supported in part by
supported by NSF awards CCF 1565264,
CNS 1618026,
and
CCF 1715187,
a Simons Investigator Fellowship,
and a Simons Investigator Award. |
eae5c766-fe0c-47b6-9c67-d0da1f6d3679 | trentmkelly/LessWrong-43k | LessWrong | 'Empiricism!' as Anti-Epistemology
(Crossposted by habryka after asking Eliezer whether I could post it under his account)
i.
"Ignore all these elaborate, abstract, theoretical predictions," the Spokesperson for Ponzi Pyramid Incorporated said in a firm, reassuring tone. "Empirically, everyone who's invested in Bernie Bankman has received back 144% of what they invested two years later."
"That's not how 'empiricism' works," said the Epistemologist. "You're still making the assumption that --"
"You could only believe that something different would happen in the future, if you believed in elaborate theoretical analyses of Bernie Bankman's unobservable internal motives and internal finances," said the spokesperson for Ponzi Pyramid Incorporated. "If you are a virtuous skeptic who doesn't trust in overcomplicated arguments, you'll believe that future investments will also pay back 144%, just like in the past. That's the prediction you make if you predict based purely on empirical observations, instead of theories about a future nobody has seen!"
"That's not how anything works," said the Epistemologist. "Every future prediction has a theory connecting it to our past observations. There's no such thing as going from past observations directly to future predictions, with no theory, no assumptions, to cross the gap --"
"Sure there's such a thing as a purely empirical prediction," said the Ponzi spokesperson. "I just made one. Not to mention, my dear audience, are you really going to trust anything as complicated as epistemology?"
"The alternative to thinking about epistemology is letting other people do your thinking about it for you," said the Epistemologist. "You're saying, 'If we observe proposition X "past investors in the Ponzi Pyramid getting paid back 144% in two years", that implies prediction Y "this next set of investors in the Ponzi Pyramid will get paid back 144% in two years"'. X and Y are distinct propositions, so you must have some theory saying 'X -> Y' that lets you put in X |
f141d662-723f-452f-8fe3-a56e4a4d2f7e | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Why not constrain wetlabs instead of AI?
Most of the object-level stories about how misaligned AI goes wrong involve either nanotechnology or bio-risk or both. Certainly I can (and have, and will again) tell a story about AI x-risk that doesn't involve anything at the molecular level. A sufficient amount of (macroscale) robotics would be enough to end humanity. But the typical story that we hear, particularly from EY, involves specifically nanotechnology. So let me ask a Robin Hanson-style question: Why not try to constrain wetlabs instead of AI? By "wetlabs" I mean any capability involving DNA, molecular biology or nanotechnology.
Some arguments:
1. Governments around the world are already in the business of regulating all kinds of chemistry, such as the production of legal and illegal drugs.
2. Governments (at least in the West) are not yet in the business of regulating information technology, and basically nobody thinks they will do a good job of it.
3. The pandemic has set the stage for new thinking around regulating wetlabs, especially now that the lab leak hypothesis is considered mainstream.
4. The cat might already be out of the bag with regards to AI. I'm referring to the Alpaca and Llama models. Information is hard to constrain.
5. "You can't just pay someone over the internet to print any DNA/chemical you want" seems like a reasonable law. In fact it's somewhat surprising that it's not already a law. By comparison, "You can't just run arbitrary software on your own computer without government permission" would be an extraordinary social change and is well outside the Overton window.
6. Something about pivotal acts which... I probably shouldn't even go there. |
7d523894-3300-41a6-8bf8-389359824060 | trentmkelly/LessWrong-43k | LessWrong | [Link]: KIC 8462852, aka WTF star, "the most mysterious star in our galaxy", ETI candidate, etc.
KIC 8462852, or the WTF (Where's the Flux?) star, is an F-type main sequence star about 1,480 ly away. It's a little larger and more massive than the sun, and a few times brighter. Age is uncertain, but probably older rather than younger.
Kepler observations over the last few years reveal very strange large and aperiodic flux variations (up to 20%) - of the general form predicted by some ETI megastructure models. However there doesn't appear to be any excess infrared.
The star's fluctuations were discovered by the PlanetHunters team. In the WTF paper they review a large number of unlikely natural explanations and settle on an unusual comet swarm as the most likely scenario.
Abstract of the WTF paper:
> Over the duration of the Kepler mission, KIC 8462852 was observed to undergo irregularly shaped, aperiodic dips in flux down to below the 20% level. The dipping activity can last for between 5 and 80 days. We characterize the object with high-resolution spectroscopy, spectral energy distribution fitting, and Fourier analyses of the Kepler light curve. We determine that KIC 8462852 is a main-sequence F3 V/IV star, with a rotation period ~0.88 d, that exhibits no significant IR excess. In this paper, we describe various scenarios to explain the mysterious events in the Kepler light curve, most of which have problems explaining the data in hand. By considering the observational constraints on dust clumps orbiting a normal main-sequence star, we conclude that the scenario most consistent with the data is the passage of a family of exocomet fragments, all of which are associated with a single previous breakup event. We discuss the necessity of future observations to help interpret the system.
>
>
From "Comets or Aliens?", on the Planet Hunters blog: " However, so far over 100 professional scientists have had a look at the lightcurves and not managed to come up with a working solution."
In a another recent paper Jason Wright et al discusses the WTF star in m |
376ba8ec-8e54-46af-ada2-5630351264e1 | StampyAI/alignment-research-dataset/arxiv | Arxiv | Iterative Learning with Open-set Noisy Labels
1 Introduction
---------------
The success of Convolutional Neural Networks (CNNs) [[20](#bib.bib20)] is highly tied to the availability of large-scale annotated datasets, *e.g*., ImageNet [[10](#bib.bib10)]. However, large-scale datasets with high-quality label annotations are not always available for a new domain, due to the significant time and effort it takes for human experts. There exist several cheap but imperfect surrogates for collecting labeled data, such as crowd-sourcing from non-experts or annotations from the web, especially for images (*e.g*., extracting tags from the surrounding text or query keywords from search engines). These approaches provide the possibility to scale the acquisition of training labels, but invariably result in the introduction of some noisy (incorrect) labels. Moreover, even high-quality datasets are likely to have noisy labels, as data labeling can be subjective and error-prone. The presence of noisy labels for training samples may adversely affect representation learning and deteriorate prediction performance [[27](#bib.bib27)]. Training accurate CNNs against noisy labels is therefore of great practical importance.
Figure 1: An illustration of closed-set vs open-set noisy labels.
Figure 2: An overview of our framework that iteratively learns discriminative representations on a “jasmine-cat” dataset with open-set noisy labels. It not only learns a proper decision boundary (the black line separating jasmine and cat) but also pulls away noisy samples (green and purple) from clean samples (blue and red).
We will refer to samples whose classes are mislabeled/incorrectly annotated as noisy samples and denote their labels as noisy labels. Such noisy labels can fall into two types, closed-set and open-set. More specifically, a closed-set noisy label occurs when a noisy sample possesses a true class that is contained within the set of known classes in the training data. While, an open-set noisy label occurs when a noisy sample possesses a true class that is not contained within the set of known classes in the training data. The former scenario has been studied in previous work, but the latter one is a new direction we explore in this paper. Figure [1](#S1.F1 "Figure 1 ‣ 1 Introduction ‣ Iterative Learning with Open-set Noisy Labels") provides a pictorial illustration of noisy labels, where we have an image dataset with two classes, jasmine (the plant) and cat (the animal). The closed-set noisy labels occur when cat and jasmine are mislabeled from one category to the other, but the true labels of these images are still cat or jasmine. The open-set noisy labels occur for those images labeled as cat or jasmine, but their true labels are neither cat nor jasmine, *e.g*., the zoo map and the cartoon character. Table [1](#S1.T1 "Table 1 ‣ 1 Introduction ‣ Iterative Learning with Open-set Noisy Labels") demonstrates all the possible cases on how different samples are labeled in this problem. The leftmost column specifies the true class and the other columns specify the type of label in the dataset.
| | labeled as “jasmine” | labeled as “cat” |
| --- | --- | --- |
| true “jasmine” | clean | closed-set |
| true “cat” | closed-set | clean |
| other class images | open-set | open-set |
Table 1: Types of labels for a “jasmine-cat” dataset.
Previous work has addressed the noisy label problem explicitly or implicitly in a closed-set setting, via either loss correction or noise model based clean label inferring [[22](#bib.bib22), [29](#bib.bib29), [37](#bib.bib37), [38](#bib.bib38)]. However, these methods are vulnerable in the more generic open-set scenario, as loss or label correction may be inaccurate since the true class may not exist in the dataset. Open-set noisy labels are likely to occur for scenarios where data are harvested rapidly, or use approximate labels (*e.g*., using a search engine query to retrieve images and then labeling the images according to the query keyword that was used). To the best of our knowledge, how to address the open-set noisy label problem is a new challenge.
In this paper, we propose an iterative learning framework that can robustly train CNNs on datasets with open-set noisy labels. Our model works iteratively with:
(1) a noisy label detector to iteratively identify noisy labels;
(2) a Siamese network for discriminative feature learning, which imposes a representation constraint via contrastive loss to pull away noisy samples from clean samples in the deep representation space; and (3) a reweighting module on the softmax loss to
express a relative confidence of clean and noisy labels on the representation learning. A simplified illustration of the proposed framework is presented in Figure [2](#S1.F2 "Figure 2 ‣ 1 Introduction ‣ Iterative Learning with Open-set Noisy Labels"). Our main contributions can be summarized as follows:
(1) We identify the open-set noisy label problem as a new challenge for representation learning and prediction.
(2) We propose an iterative learning framework to robustly train CNNs in the presence of open-set noisy labels. Our model is not dependent on any assumption of noise.
(3) We empirically demonstrate that our model significantly outperforms state-of-the-art noisy label learning models for the open-set setting, and has a comparable or even better performance under the closed-set setting.
2 Related work
---------------
A simple approach to handle noisy labels is to remove samples with suspicious labels from the training data [[4](#bib.bib4)]. However, such methods are often challenged by the difficulty of distinguishing samples that are inherently hard to learn from those with noisy labels [[13](#bib.bib13)]. In contrast to simply removing them, the following work focuses on addressing the noisy label problem via deep learning.
One alternative approach is to explicitly or implicitly formulate the noise model and use a corresponding noise-aware approach. Symmetric label noise that is independent of the true label was modeled in [[21](#bib.bib21)], and asymmetric label noise that is conditionally independent of the individual sample was modeled in [[26](#bib.bib26), [33](#bib.bib33)]. More complex noise models for samples, true labels and noisy labels can be characterized by directed graphical models [[43](#bib.bib43)], Conditional Random Fields (CRF) [[37](#bib.bib37)], neural networks [[38](#bib.bib38)] or knowledge graphs [[22](#bib.bib22)]. These methods aim to correct noisy labels to their true labels via a clean label inferring. However, they require availability of an extra dataset with pre-identified noisy labels and their ground truth labels in order to model label noise. Moreover, these methods make their own specific assumptions about the noise model, which will limit their effectiveness under complicated label noise.
Other approaches utilize correction methods to adjust the loss function to eliminate the influence of noisy samples. Backward [[29](#bib.bib29)] and Forward [[29](#bib.bib29)] are two such correction methods that use an estimated or learned factor to modify the loss function. [[34](#bib.bib34), [11](#bib.bib11)] further augment the correction architecture by adding a linear layer on top of the network. Bootstrap [[30](#bib.bib30)] is another loss correction method that replaces the target labels used by a loss function with a combination of raw target labels and their predicted labels.
The above methods implicitly assume a closed-set noisy label setting, where the true labels are always contained within the set of known classes in the training data. Such restricted assumption contradicts the more practical open-set scenario. Open-set noisy samples should be considered separately. In our proposed model, we iteratively detect noisy samples and gradually pull them away from clean samples, which is different from removing them or labeling them to a new “unknown” class [[33](#bib.bib33)] (these two approaches are evaluated in Section [4.1.3](#S4.SS1.SSS3 "4.1.3 Module analysis via ablation experiments ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels")). Moreover, the proposed framework does not depend on the noise model, and is able to address both the open-set and the closed-set noisy label problem.
3 Iterative learning framework
-------------------------------
Figure 3: The framework of the proposed iterative learning approach. Iterative noisy label detection module and discriminative feature learning module form a closed-loop, i.e., one module’s inputs are the other module’s output, which can benefit from each other and be jointly enhanced. The network is jointly optimized by two types of losses: reweighted softmax loss and contrastive loss.
Our goal is to learn discriminative features from a dataset with noisy labels. We propose an iterative learning framework that gradually pulls away noisy samples from clean samples in the deep feature space. As illustrated in Figure [3](#S3.F3 "Figure 3 ‣ 3 Iterative learning framework ‣ Iterative Learning with Open-set Noisy Labels"), our proposed model consists of three major modules: 1) iterative noisy label detection, 2) discriminative feature learning, and 3) reweighting. The noisy label detection uses the output features of the network (dashed lines) to separate training samples into two subsets: clean samples and noisy samples. To benefit from the noisy label detection, we employ a Siamese network to impose a representation constraint forcing the representation of clean samples and that of noisy samples to be as discriminative as possible. Besides, a reweighting module that assigns a weight for each sample based on the confidence supplied by the noisy label detection is used to emphasize clean samples and weaken noise samples on the discriminative representation learning. Such learned discriminative representations will in turn benefit the noisy label detection. Considering the representation learning as an iterative process, we further design the noisy label detection to be iterative so that the discriminative feature learning and the iterative noisy label detection can be jointly improved over iterations. A brief description of how each module works is listed as follows:
Iterative noisy label detection:
We iteratively detect noisy labels based on the features of the network, because samples from the same class should be intrinsically similar, while mislabeled samples are generally not [[2](#bib.bib2)].
Discriminative feature learning:
We use a Siamese network with two sub-networks of sharing weights. It takes “similar” or “dissimilar” sample pairs as inputs and uses a contrastive loss to minimize distance between similar samples and maximize distance between dissimilar samples. It can also be seen as a representation constraint.
Reweighting: For detected clean samples, we set their weights to 1 (no reweighting) on softmax loss, while for detected noisy samples, we assign them smaller weights individually based on how likely one sample being noisy. To avoid misdetection, samples near the decision boundary will be weighted of close importance to clean samples.
The framework is jointly optimized by two loss terms:
| | | | |
| --- | --- | --- | --- |
| | L=RSL+ηCL, | | (1) |
where RSL is the reweighted softmax loss, CL is the contrastive loss and η is a trade-off parameter. The above objective incorporates the iterative noisy label detection, discriminative feature learning and reweighting into an effective learning framework that is robust to noisy labels.
###
3.1 Iterative noisy label detection
Considering that samples from the same class should have similar high-level representations but samples mislabeled into the class do not [[2](#bib.bib2)], we detect noisy labels based on the representations of the pre-softmax layer. To benefit from the iterative learning process of representation, we iteratively perform noisy label detection every few epochs. We also use a cumulative criterion based on all previous iterations of detection to reduce the influence of randomness in one particular iteration and further produce more stable detection results.
Our detection method is a probabilistic and cumulative version of Local Outlier Factor algorithm (pcLOF), which inherits the advantages of LOF, i.e., it is an unsupervised algorithm which performs well on high dimensional data and requires no assumptions of the underlying data distribution. Formally, pcLOF is defined as:
| | | | |
| --- | --- | --- | --- |
| | pcLOF(xi)=G(M∑m=1LOF(m)(xi)), | | (2) |
where M is the current number of iteration and G is a local Gaussian statistics transformation, which scales the cumulative LOF score to a probabilistic value in [0, 1] as in [[17](#bib.bib17), [18](#bib.bib18)]. The pcLOF score can be directly interpreted as the probability of a sample being an outlier. In the noisy label detection setting, a pcLOF score close to 0 indicates a clean sample, while a score close to 1 indicates a noisy sample.
LOF is a density-based outlier detection algorithm [[3](#bib.bib3)] and the LOF score of a sample xi is defined as follows:
| | | | |
| --- | --- | --- | --- |
| | LOF(xi)=∑xj∈Nk(xi)lrd(xj)lrd(xi)|Nk(xi)|, | | (3) |
where Nk(xi) is the set of k nearest neighbors of xi and lrd(xi) is the local reachability density (lrd) of xi:
| | | | |
| --- | --- | --- | --- |
| | lrd(xi)=1/(∑xj∈Nk(xi)reach-distk(xi,xj)|Nk(xi)|), | | (4) |
where reach-distk(xi,xj)=max{k-dist(xj),d(xi,xj)} is the reachability distance of xi to xj. Intuitively, if xi is far away from xj, then the reachability distance is simply d(xi,xj) (their actual distance). However, if they are “sufficiently” close, the actual distance is replaced by k-dist(xj) (the distance of xj to its k-th nearest neighbor), which means that samples inside of the k nearest of xj are considered to be equally distant.
Note that the noisy label detection works iteratively, thus we do not need complicated detection algorithms. With the representation become more discriminative,
they can converge to almost the same result as long as the iteration is long enough. To balance the efficiency and effectiveness in training, we perform pcLOF based iterative noisy label detection every 10 epochs after 2-epoch network initialization in our experiments.
###
3.2 Discriminative feature learning
We implement a Siamese network [[8](#bib.bib8), [14](#bib.bib14)] with two channels of the same base network and sharing weights. It generates “similar” and “dissimilar” sample pairs based on the clean and noisy samples detected by the noise label detection module, and works with a contrastive loss to minimize distance between samples of the same class and maximize distance between samples of different classes as well as distance between clean samples and noisy samples.
Denote the Euclidean distance between xi and xj in the deep representation space as follows:
| | | | |
| --- | --- | --- | --- |
| | Dl(xi,xj,θ)=||fl(xi|θ)−fl(xj|θ)||2, | | (5) |
where fl(⋅|θ) denotes the l-th layer output of the network f(⋅|θ) under parameters θ. The similarity indicator Yij is defined based on the output of noisy label detection:
| | | | |
| --- | --- | --- | --- |
| | Yij={1,if xi and xj are similar;0,if xi and xj are dissimilar. | | (6) |
Two samples are considered to be “similar” (Yij=1), if and only if the two samples are from the same class and both are correctly labeled. Two samples are considered to be “dissimilar” (Yij=0), if two samples are from different classes, or one is a clean sample and the other is a noisy sample. Note that we do not define the relationship between two noisy samples as their true classes are not contained within the training data, thus cannot be simply defined as similar or dissimilar. When noisy samples are not available before the first iteration of noisy label detection, dissimilar pairs only contain the samples from different classes.
The contrastive loss for discriminative feature learning can be formulated as:
| | | | |
| --- | --- | --- | --- |
| | CL(xi,xj,Yij)=Yij12D2l+(1−Yij)12max{0,α−Dl}, | | (7) |
where α>0 is a margin formulating how far away two dissimilar samples should be pulled from each other. This contrastive loss will force the distance between dissimilar pairs, e.g., clean samples and noisy samples, to be larger than the margin α, and similar pairs, i.e., clean samples from the same class, to be clustered.
Although the amount of possible sample pairs can be huge, some of the pairs are easy to discriminate and ineffectual for training (i.e., the distance between two dissimilar samples is already larger than the margin α). Therefore, we implement the widely used hard example mining strategy [[31](#bib.bib31)] to obtain the closest dissimilar pairs and the most distant similar pairs to feed into the network. The base network can be any kind of architectures, such as VGG [[32](#bib.bib32)], ResNet [[15](#bib.bib15)] and Inception [[36](#bib.bib36)].
###
3.3 Reweighting
To ensure an efficient and accurate representation learning, we also design a reweighting module before the softmax loss to adaptively use the label information with different confidence. Applying softmax loss on clean samples is intuitively to make use of their trustworthy label information. The reason for also applying softmax loss on noisy samples is that the detected noisy samples may contain some clean samples, especially at the start of the training. Those samples are close to the decision boundary and are often very informative for representation learning as pointed out in [[13](#bib.bib13)], which cannot be simply ignored, i.e., setting their weights to 0. This is also verified by the experiments in Section [4.1.3](#S4.SS1.SSS3 "4.1.3 Module analysis via ablation experiments ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"). Therefore, a reweighting module paired with the softmax loss is used to adaptively address the detected clean and noisy samples simultaneously.
Since the noisy label detection method pcLOF provides the confidence/probability of a sample being noisy, we use γ=1−pcLOF as a reweighting factor on the detected noisy samples to express their relative confidence.
We set the initial γ to 1 before the first iteration of noisy label detection, as the learning proceeds, the γ of a noisy sample will be gradually decreased as the noisy label detection tends to be more and more accurate. The reweighted softmax loss (RSL) of our model is defined as:
| | | | |
| --- | --- | --- | --- |
| | RSL=−(∑Nci=1logP(yi|xi,θ)+∑Nnj=1γjlogP(yj|xj,θ))Nc+Nn, | | (8) |
where Nc, Nn denote the number of detected clean and noisy samples respectively, γ is the proposed reweighting factor on the detected noisy samples, and P(yi|xi,θ) is the softmax probability of xi being in class yi. The softmax loss used in our framework can also be replaced to more advanced ones [[24](#bib.bib24), [23](#bib.bib23)] to further boost the performance.
4 Experiments
--------------
In this section, we evaluate the robustness of our proposed model to noisy labels with comprehensive experiments on CIFAR-10 (small dataset), ImageNet (large-scale dataset), and web-search dataset (real-world noisy dataset).
###
4.1 Exploratory experiments on CIFAR-10
We first conduct a series of experiments on CIFAR-10 dataset towards a comprehensive understanding of our model through comparisons to the state-of-the-arts.
Baselines:
Several recently proposed noisy label learning models are chosen as our baselines: (1) Backward [[29](#bib.bib29)]: The networks are trained via loss correction by multiplying the cross entropy loss by an estimated correction matrix; (2) Forward [[29](#bib.bib29)]: The networks are trained with label correction by multiplying the network prediction by an estimated correction matrix; (3) Bootstrap [[30](#bib.bib30)]: The networks are trained with new labels generated by a convex combination (the “hard” version) of the noisy labels and their predicted labels; and (4) CNN-CRF [[37](#bib.bib37)]: The networks are trained with latent clean labels inferred by CRF from only noisy training datasets. We also include the method of Cross-entropy: learning directly from noisy datasets with a vanilla cross entropy loss.
Experimental setup:
The baseline models use a network architecture with 6 convolutional layers and 1 fully connected layer (fc7). Batch normalization (BN) [[16](#bib.bib16)] is applied in each convolutional layer before the ReLU activation, a max-pooling layer is implemented every two convolutional layers, and a softmax layer is added on top of the network for classification. The parameters of the baselines are configured according to their original papers. For our model, two copies of the above network are implemented and the contrastive loss is built upon the fc7 layer. We set η=1, k=half the class sample size, and samples with pcLOF>0.5 are considered as noisy samples111As pointed by [[17](#bib.bib17), [18](#bib.bib18)], pcLOF is not sensitive to parameter k.
The classification accuracy (ACC) on clean CIFAR-10 test set is used as the evaluation metric.
All networks are trained by Stochastic Gradient Descent (SGD) with learning rate 0.01, weight decay 10−4 and momentum 0.9, and the learning rate is divided by 10 after 40 and 80 epochs (100 in total). All images are mean-subtracted and normalized to [0,1], and no data augmentation is implemented in this part.
Open-set noisy datasets are built by replacing some training images in CIFAR-10 by outside images, while keeping the labels and the number of images per class unchanged. The “mislabeled” outside images are from either different public datasets (type I noise) or severely damaged CIFAR-10 images (type II noise). Type I noise includes images from CIFAR-100 [[19](#bib.bib19)], ImageNet32 (32×32 ImageNet images) [[9](#bib.bib9)] and SVHN [[28](#bib.bib28)], and only those images whose labels exclude the 10 classes in CIFAR-10 are considered. Type II noise includes images damaged by Gaussian random noise (mean 0.2 and variance 1.0), corruption (75% of an image is set to black or white) and resolution distortion
(an image is resized to 4×4 and then dilated back to 32×32). Some examples of the type I and type II open-set noise are given in Figure [4](#S4.F4 "Figure 4 ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"). For closed-set noisy datasets, we choose a random proportion of CIFAR-10 training images per class and change its label to an incorrect random one. This closed-set label noise belongs to symmetric noise, which is more challenging than asymmetric noise [[29](#bib.bib29)].
Figure 4: Examples of open-set noise for “airplane” in CIFAR-10.
| Open-set label noise type | Cross-entropy | Backward | Forward | Bootstrap | CNN-CRF | Ours |
| --- | --- | --- | --- | --- | --- | --- |
| Type I | CIFAR-10 + CIFAR-100 | 62.92 | 55.97 | 64.18 | 62.11 | 64.58 | 79.28 |
| CIFAR-10 + ImageNet32 | 58.63 | 52.35 | 66.77 | 57.83 | 67.53 | 79.38 |
| CIFAR-10 + SVHN | 56.44 | 52.03 | 56.70 | 56.89 | 56.93 | 77.73 |
| Type II | CIFAR-10 + Gaussian | 61.96 | 54.98 | 72.70 | 59.05 | 72.51 | 80.37 |
| CIFAR-10 + Corruption | 57.40 | 50.24 | 63.80 | 56.00 | 64.25 | 74.48 |
| CIFAR-10 + Resolution | 56.93 | 49.58 | 62.65 | 58.95 | 63.60 | 77.30 |
Table 2: Accuracies (%) of different models on CIFAR-10 noisy dataset with 40% open-set noise. The best results are in bold.
| Method | 20% noise | 40% noise |
| --- | --- | --- |
| Clean | 84.85 | 84.85 |
| Cross-entropy | 74.17 | 62.38 |
| Backward | 76.27 | 75.18 |
| Forward | 79.25 | 77.81 |
| Bootstrapping | 74.39 | 69.50 |
| CNN-CRF | 80.15 | 78.69 |
| Ours | 81.36 | 78.15 |
Table 3: Accuracies (%) on CIFAR-10 noisy dataset with 20% and 40% closed-set noise. Top 2 results are in bold except for “Clean”.
####
4.1.1 Classification performance
Open-set label noise:
The classification accuracy on CIFAR-10 noisy datasets with 40% open-set noise is reported in Table [2](#S4.T2 "Table 2 ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"). As can be seen, our model outperforms the baselines with large margins on both type I (top three rows) and type II (bottom three rows) open-set noise. The poor performance of baselines is because they either ignore the existence of noisy labels such as Cross-entropy, or attempt to correct noisy labels to so-called “clean labels”. For example, in CIFAR-10+SVHN, noisy images from SVHN are still noisy even if their labels are corrected to one of the CIFAR-10 classes, thus still harm representation learning.
Our superior performance indicates that our model is capable of learning accurate representation directly from datasets with open-set noisy labels. Such capability opens up more opportunities for many applications that, for example, require learning directly from web-search data.
Closed-set label noise:
We also assess our model under the closed-set noise settings of 20% and 40% noise rates. The results are reported in Table [3](#S4.T3 "Table 3 ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"). The first row “clean” provides a performance “upper bound” trained on the totally clean CIFAR-10 dataset. Compared to baselines, our model achieves comparable or better performance. In particular, our model surpasses all baselines at the 20% noise level and its performance is close to the clean training. At the 40% noise level, our model achieves an accuracy that is comparable to CNN-CRF and higher than other baselines. This experiment demonstrates that our model can also effectively learn from datasets with closed-set noisy labels.
Discussion: Revisiting Table [2](#S4.T2 "Table 2 ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"), we find that some baselines (Forward and CNN-CRF) achieve considerable improvements on CIFAR-10+CIFAR-100 and CIFAR-10+ImageNet32, compared to Cross-entropy. This may be caused by the similarity between images from CIFAR-100/ImageNet32 and those in CIFAR-10. Since CIFAR-100 and ImageNet32 have many fine-grained classes, some noisy images from these two datasets can be regarded as the closed-set noise rather than the open-set noise. However, for dissimilar datasets, *e.g*., CIFAR-10+SVHN, we can see that those baselines perform almost the same poorly. This interesting finding implies another evidence that our model still works well when open-set and closed-set noise coexist.
####
4.1.2 Model interpretation
We further demonstrate some visual results to help understand how our model works.
Iterative noisy label detection:
We first show the effectiveness of the iterative noisy label detection module. We use the measure of true noisy label rate (true positive rate) of the detected noisy labels. In Figure [5](#S4.F5 "Figure 5 ‣ 4.1.2 Model interpretation ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"), the detection becomes more and more precise as the training proceeds, and the trend is consistent across tall classes, which meets our expectation that the iterative noisy label detection will improve accordingly as the learned features become increasingly discriminative. Figure [6](#S4.F6 "Figure 6 ‣ 4.1.2 Model interpretation ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels") gives some of those detected noisy images. We can see many noticeable open-set noisy images, *e.g*., flower, fish to name a few, which confirms that our iterative noisy label detection method can indeed accurately identify noisy samples contained within the dataset.
Figure 5: The true positive rate of the detected noisy labels over iteration on CIFAR-10+CIFAR-100 (40% open-set noise).
Figure 6: Randomly selected images (10 per class) from the detected noisy images of CIFAR-10+CIFAR-100 with 40% open-set noise at 100-th epoch.
| | | | |
| --- | --- | --- | --- |
| Visualization of the learned features. This visualization experiment uses a 2-class subset of CIFAR-10+CIFAR-100 (40% open-set noise) by setting the output feature dimension as 2. | Visualization of the learned features. This visualization experiment uses a 2-class subset of CIFAR-10+CIFAR-100 (40% open-set noise) by setting the output feature dimension as 2. | Visualization of the learned features. This visualization experiment uses a 2-class subset of CIFAR-10+CIFAR-100 (40% open-set noise) by setting the output feature dimension as 2. | Visualization of the learned features. This visualization experiment uses a 2-class subset of CIFAR-10+CIFAR-100 (40% open-set noise) by setting the output feature dimension as 2. |
Figure 7: Visualization of the learned features. This visualization experiment uses a 2-class subset of CIFAR-10+CIFAR-100 (40% open-set noise) by setting the output feature dimension as 2.
Discriminative features:
Next, we show the discriminative feature learning ability of our model by visualizing the learned features at different stages of the training process in 2-D space. Figure [7](#S4.F7 "Figure 7 ‣ 4.1.2 Model interpretation ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels") evidently shows that the learned features become more and more discriminative as noisy samples are gradually pulled away from clean samples. At the first epoch, all samples are densely overlapped together, as the training proceeds to the 100-th epoch, not only the two classes have been separated, the noisy samples are also pulled away from the clean ones within each class. This confirms that the discriminative feature learning module can work effectively with the iterative noisy label detection module to isolate noisy samples.
####
4.1.3 Module analysis via ablation experiments
For a comprehensive understanding of our model, we further evaluate each module via ablation experiments on CIFAR-10+CIFAR-100 with 20% and 40% open-set noise.
Table [4](#S4.T4 "Table 4 ‣ 4.1.3 Module analysis via ablation experiments ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels") presents the following six experiments: (a) Without reweighting: we only change the weights of softmax loss on noisy samples by either assigning the same weights as clean samples (case a1: γ=1), or ignoring noisy samples (case a2: γ=0). (b) Without discriminative feature learning: we only remove the contrastive loss (the remaining model still has iterative noisy label detection and reweighted softmax loss modules). As for the detected noisy samples, we either remove them (case b1: removing) or label them to a new class “unknown” (case b2: new class). (c) Without iterative noisy label detection: we either conduct detection only once at the first iteration of detection (case c1: only once) or remove the detection module along with the reweighting on softmax loss (case c2: no).
| | |
| --- | --- |
| Method | CIFAR-10+CIFAR-100 |
| 20% noise | 40% noise |
| Our model | 81.96 | 79.28 |
| (a) w/o reweighting | | |
| – case a1: γ=1 | 76.97 | 74.45 |
| – case a2: γ=0 | 79.27 | 76.03 |
| (b) w/o discriminative learning | | |
| – case b1: removing | 76.22 | 68.40 |
| – case b2: new class | 78.34 | 73.11 |
| (c) w/o iterative detection | | |
| – case c1: only once | 77.52 | 70.31 |
| – case c2: no | 76.17 | 63.50 |
Table 4: Accuracies (%) on CIFAR-10+CIFAR-100 (20% & 40% open-set noise) after removing (w/o) each module from our model.
Performance drops are observed in Table [4](#S4.T4 "Table 4 ‣ 4.1.3 Module analysis via ablation experiments ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels") when any of the three modules is removed or replaced. The accuracy drop compared to our original model can be interpreted as the contribution of the module. Particularly, if discriminative feature learning (case b1) or iterative noisy label detection (case c2) is removed, the accuracy significantly decreases, which indicates that the two modules work jointly in an efficient way and can enhance each other. When the detected noisy samples are removed (case b1) or re-labeled to a new class “unknown” (case b2), the accuracy also drops considerably, which proves that discriminative features forced by the contrastive loss are critical for accurate noisy label detection which further improves discriminative feature learning. From both cases in (a), we can see that reweighting is necessary for a proper handling of the detected noisy samples which may contain some clean samples around the decision boundary.
####
4.1.4 Parameter and complexity analysis
Moreover, we assess the influence of parameter η in our model, which is used to balance the contrastive loss and the softmax loss. We test a series of η∈[0.5,1.5] on CIFAR-10+CIFAR-100 with 40% open-set noise. Table [5](#S4.T5 "Table 5 ‣ 4.1.4 Parameter and complexity analysis ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels") shows that our model is not sensitive to the parameter η as long as it lies in a comparable range to the weight of softmax loss on clean samples (its weight is 1).
We also compare our model with other baselines against different open-set noise rates on CIFAR-10+CIFAR-100. It can be seen in Figure [8](#S4.F8 "Figure 8 ‣ 4.1.4 Parameter and complexity analysis ‣ 4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels") that our model still performs the best under all noise rates, even at a high noise rate up to 50%. The performance of baselines, however, decrease significantly as noise rate increases.
As for the complexity of our framework, the extra costs mainly lie on pcLOF computing. However, it is computed 1) within each class not on the entire data, 2) in parallel for different classes and 3) only every 10 epochs not each epoch. Moreover, pcLOF can be computed on GPU with 100X speed-up [[1](#bib.bib1)]. The proportion of the time cost of pcLOF with respect to the training time is: tpcLOFttraining≈2%. Thus, the computation of pcLOF is much less expensive.
| η | 0.5 | 0.7 | 0.9 | 1.0 | 1.3 | 1.5 |
| --- | --- | --- | --- | --- | --- | --- |
| Ours | 76.08 | 79.04 | 79.36 | 79.77 | 79.08 | 77.80 |
Table 5: Accuracies (%) of our model on CIFAR-10+CIFAR-100 (40% open-set noise) with different η.
Figure 8: Accuracies of different models on CIFAR-10+CIFAR-100 with different open-set noise rates.
###
4.2 Experiments on ImageNet
From above experiments, we have demonstrated that our model achieves superior performance on a small dataset CIFAR-10 against both open-set and closed-set label noise. Here, we further present its capacity to handle large-scale datasets containing open-set noisy labels, i.e., ImageNet. Meanwhile, we also show that our model works effectively with different modern deep neural network architectures: ResNet-50 [[15](#bib.bib15)] and Inception-v3 [[36](#bib.bib36)].
Experimental setup:
Based on the ImageNet 2012 dataset [[10](#bib.bib10)] (1.3M images of 1000 classes), we generate an open-set noisy dataset by randomly taking 200 classes of images as clean data, which are then mixed with uniformly and randomly selected images from other 800 classes. Finally, we obtain a noisy dataset of ∼290k images with 200 classes and 20% noise rate in each class. ResNet-50 and Inception-v3 networks are implemented in Keras [[7](#bib.bib7)]. We train the networks with batch size 128 and initial learning rate 0.01, which is reduced by 1/10 at the 30-th, 60-th and 80-th epoch. The training ends at the 100-th epoch. Several commonly used data augmentations are applied, i.e., 224×224 pixel random crops, horizontal random flips and scale data augmentation as in [[12](#bib.bib12)]. All images are normalized by the per-color mean and standard deviation. We test the models using the ImageNet validation set of the 200 clean classes along with a single center crop (224×224).
Results:
The baselines compared here are the same as that in Section [4.1](#S4.SS1 "4.1 Exploratory experiments on CIFAR-10 ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"), and we report the Top-1 and Top-5 classification accuracy. The results can be found in Table [6](#S4.T6 "Table 6 ‣ 4.3 Experiments on real-world noisy dataset ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"). Again, our model outperforms other baselines with significant margins for ResNet-50 as well as Inception-v3 architectures. Meanwhile, we notice that some baseline models such as CNN-CRF and Forward also demonstrate certain improvements compared to Cross-entropy. This may because that some ImageNet classes are fine-grained and visually similar such that some open-set noisy images are “closed-set” to some extent. In fact, this is in line with complex real-world situations, where a clear boundary between closed-set noisy labels and open-set noisy labels is often hard to draw. The superiority of our model on such datasets implies its advantages against real-world noisy data, where closed-set and open-set label noise may coexist.
###
4.3 Experiments on real-world noisy dataset
Finally, we assess our model on real-world noisy dataset, where noise type (closed-set or open-set) and noise rate are unknown. This is to demonstrate that our model can effectively make use of noisy data (web-search data) to learn accurate representation for real-world scenarios, *e.g*., no clean labeled data are available for a new domain.
Experimental setup:
We use Google image search engine to obtain a web-search image set of ∼1.2M images (∼0.8k per class) using the query list from ImageNet [[10](#bib.bib10)], SUN [[42](#bib.bib42)] and NEIL [[6](#bib.bib6)] datasets, similar as in [[5](#bib.bib5)], but we do not remove images from the searched results. For conceptually clear queries, Google returns relatively clean results, however, for “ambiguous” queries, the results are very similar to open-set label noise, *e.g*., ‘jasmine’ query returns a plant or a cartoon character. The experimental setup and training are same as Section [4.2](#S4.SS2 "4.2 Experiments on ImageNet ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"). For testing, however, this is a challenging task as there are no clean test data available. There are several ways to evaluate the learned features. Here we adopt a well defined classification task to achieve that. Specifically, we treat the learned network as a feature extractor by fixing all its layers except the softmax layer. We then train a softmax layer for classification using the CIFAR-100 training set. Because the optimization of the classification layer (*i.e*., multi-class logistic regression) is convex, the classification accuracy can be used to evaluate the learned features.
Results:
The classification accuracy on CIFAR-100 test set is reported in Table [7](#S4.T7 "Table 7 ‣ 4.3 Experiments on real-world noisy dataset ‣ 4 Experiments ‣ Iterative Learning with Open-set Noisy Labels"). As shown, our model achieves the best performance compared to baselines for both ResNet-50 and Inception-v3. This indicates that our model can robustly train CNNs from real-world noisy dataset. Note that the noisy data are from web, thus our model can also be extended to webly supervised learning [[5](#bib.bib5)].
In summary, empirical evidence has demonstrated that our model is a promising framework for learning from datasets with noisy labels (open-set and closed-set). Moreover, it can also be used for webly supervised learning.
| Method | ResNet-50 | Inception-v3 |
| --- | --- | --- |
| Top-1 | Top-5 | Top-1 | Top-5 |
| Cross-entropy | 58.51 | 75.62 | 60.73 | 76.75 |
| Backward | 59.32 | 75.61 | 61.27 | 76.74 |
| Forward | 64.17 | 79.43 | 65.48 | 80.68 |
| Bootstrapping | 59.05 | 75.00 | 61.50 | 76.13 |
| CNN-CRF | 66.54 | 82.37 | 67.23 | 84.12 |
| Ours | 70.29 | 86.04 | 71.43 | 87.87 |
Table 6: Accuracies (%) of different models on the 200-class ImageNet with 20% open-set noise. The best results are in bold.
| Method | ResNet-50 | Inception-v3 |
| --- | --- | --- |
| Cross-entropy | 57.32 | 53.82 |
| Backward | 58.75 | 54.02 |
| Forward | 61.65 | 58.28 |
| Bootstrapping | 57.62 | 54.49 |
| CNN-CRF | 63.94 | 60.47 |
| Ours | 67.90 | 64.21 |
Table 7: Accuracies (%) of different models trained on real-world noisy data (web-search data) and tested on CIFAR-100 test set. The best results are in bold.
5 Conclusions
--------------
In this paper, we identified and investigated the open-set noisy label problem – a more complex noisy label scenario that commonly occurs in real-world datasets. We proposed an iterative learning framework to address the problem with three powerful modules: iterative noisy label detection, discriminative feature learning, and reweighting. These modules are designed to benefit from each other and to be jointly improved over iterations. We empirically show that our model not only outperforms the state-of-the-arts for open-set label noise, but also effective for closed-set label noise, on datasets of various scales.
Acknowledgment.
This work is supported by National Natural Science Foundation of China (No. 61771273), NSFC U1609220, NSF IIS-1639792 EAGER, ONR N00014-15-1-2340, Intel ISTC, Amazon AWS and China Scholarship Council (CSC). |
1bbcfaa5-bee9-4b73-b562-e2d0fd260771 | trentmkelly/LessWrong-43k | LessWrong | New study on choice blindness in moral positions
Change blindness is the phenomenon whereby people fail to notice changes in scenery and whatnot if they're not directed to pay attention to it. There are countless videos online demonstrating this effect (one of my favorites here, by Richard Wiseman).
One of the most audacious and famous experiments is known informally as "the door study": an experimenter asks a passerby for directions, but is interrupted by a pair of construction workers carrying an unhinged door, concealing another person whom replaces the experimenter as the door passes. Incredibly, the person giving directions rarely notices they are now talking to a completely different person. This effect was reproduced by Derren Brown on British TV (here's an amateur re-enactment).
Subsequently a pair of Swedish researchers familiar with some sleight-of-hand magic conceived a new twist on this line of research, arguably even more audacious: have participants make a choice and quietly swap that choice with something else. People not only fail to notice the change, but confabulate reasons why they had preferred the counterfeit choice (video here). They called their new paradigm "Choice Blindness".
Just recently the same Swedish researchers published a new study that is even more shocking. Rather than demonstrating choice blindness by having participants choose between two photographs, they demonstrated the same effect with moral propositions. Participants completed a survey asking them to agree or disagree with statements such as "large scale governmental surveillance of e-mail and Internet traffic ought to be forbidden as a means to combat international crime and terrorism". When they reviewed their copy of the survey their responses had been covertly changed, but 69% failed to notice at least one of two changes, and when asked to explain their answers 53% argued in favor of what they falsely believed was their original choice, when they had previously indicated the opposite moral position (study here, vide |
cf170c25-79e5-4e03-8428-b3d29de67f56 | trentmkelly/LessWrong-43k | LessWrong | The Power to Teach Concepts Better
This is Part VIII of the Specificity Sequence
When you teach someone a concept, you’re building a structure in their mind by connecting up some of their mental concepts in a certain way. But you have to go in through their ears. It’s kind of like building this ship-in-a-bottle LEGO set.
In this post, we’ll visualize what’s happening in a learner’s brain and see how a teacher can wield their specificity powers to teach concepts better.
Mind-Hanging A Concept
Reading a startup’s pitch begins as a learning exercise: learning what the startup does. In How to Apply to Y Combinator, Paul Graham writes:
> We have to read about 100 [applications] a day. That means a YC partner who reads your application will on average have already read 50 that day and have 50 more to go. Yours has to stand out. So you have to be exceptionally clear and concise. Whatever you have to say, give it to us right in the first sentence, in the simplest possible terms. […]
> The first question I look at is, “What is your company going to make?” This isn’t the question I care most about, but I look at it first because I need something to hang the application on in my mind.
It’s worth unpacking and visualizing this part of PG’s advice, because we’ll see that the power to mind-hang the concept you’re trying to communicate is closely related to the power of specificity. Stay tuned for that.
First, one more snippet from PG:
> The best answers are the most matter of fact. It’s a mistake to use marketing-speak to make your idea sound more exciting. We’re immune to marketing-speak; to us it’s just noise. So don’t begin your answer with something like “We are going to transform the relationship between individuals and information.” That sounds impressive, but it conveys nothing. It could be a description of any technology company. Are you going to build a search engine? Database software? A router? I have no idea. […]
> One good trick for describing a project concisely is to explain it as a varian |
e09943e6-4a29-40cc-a7f2-0a4f2138ba88 | trentmkelly/LessWrong-43k | LessWrong | Bi-Weekly Rational Feed
===Highly Recommended Articles:
Bring Up Genius by Viliam (lesswrong) - An "80/20" translation. Positive motivation. Extreme resistance from the Hungarian government and press. Polgar's five principles. Biting criticism of the school system. Learning in early childhood. Is Genius a gift or curse? Celebrity. Detailed plan for daily instruction. Importance of diversity. Why chess? Teach the chess with love, playfully. Emancipation of women. Polgar's happy family.
The Shouting Class by Noah Smith - The majority of comments come from a tiny minority of commentators. Social media is giving a bullhorn to the people who constantly complain. Negativity is contagious. The level of discord in society is getting genuinely dangerous. The French Revolution. The author criticizes shouters on the Left and Right.
How Givewell Uses Cost Effectiveness Analyses by The GiveWell Blog - GiveWell doesn't take its estimates literally, unless one charity is measured as 2-3x as cost-effective GiveWell is unsure if a difference exists. Cost-effective is however the most important factor in GiveWell's recommendations. GiveWell goes into detail about how it deals with great uncertainty and suboptimal data.
Mode Collapse And The Norm One Principle by tristanm (lesswrong) - Generative Adversarial Networks. Applying the lessons of Machine Learning to discourse. How to make progress when the critical side of discourse is very powerful. "My claim is that any contribution to a discussion should satisfy the "Norm One Principle." In other words, it should have a well-defined direction, and the quantity of change should be feasible to implement."
The Face Of The Ice by Sarah Constantin (Otium) - Mountaineering. Survival Mindset vs Sexual-Selection Mindset. War and the Wilderness. Technical Skill.
Bayes: A Kinda Sorta Masterpost by Nostalgebraist - A long and very well thought-out criticism of Bayesianism. Explanation of Bayesian methodology. Comparison with classical statistics. Arguments for Baye |
62213d70-51b1-4a0e-917a-826bfc8740bf | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Paper+Summary: OMNIGROK: GROKKING BEYOND ALGORITHMIC DATA
There is a new paper by Liu et al. that claims to have understood the key mechanism underlying grokking (potentially even generalization more broadly).
**They argue:**
1. Grokking can be explained via the norm of the weights. They claim that there is a constant level of the weight norm that is optimal for generalization.
2. If there is an optimal level of the weight norm, the weight norm of your model after initialization can be either too low, too high or optimal. They claim that grokking is a phenomenon where we initialize the model with a large weight norm and it then slowly walks toward the optimal weight norm and then generalizes.
3. They also claim that you can get the same results as grokking but much faster if you set the weight norm correctly at every step.
4. They set the norm "correctly" by rescaling the weights after each unconstrained optimization step (so after every weight update loss.backward()?!)
**Implications:**
1. I think they have found a very important insight of grokking and finding generalizing circuits more broadly.
2. I'm still a bit skeptical of some of the claims and results. On some level "just fix your weight norm and the model generalizes" sounds too simple to be true for all tasks.
3. I think this result could have big implications but I'm not yet sure whether they are positive or negative. On the one hand, finding generalizing circuits seems to solve some of the problems associated with bad out-of-distribution generalization. On the other hand, it likely speeds up capabilities.
I'm very unsure about this paper but intuitively it feels important. Thoughts? |
1fa3edec-a7da-4f93-ab9d-a58b84ccfd30 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Some work on connecting UDT and Reinforcement Learning
|
07ca7343-41cf-4bc6-a740-9ae5a3fc2b87 | trentmkelly/LessWrong-43k | LessWrong | The Personal Implications of AGI Realism
Superintelligence Is On The Horizon
It’s widely accepted that powerful general AI, and soon after, superintelligence, may eventually be created.[1] There’s no fundamental law keeping humanity at the top of the intelligence hierarchy. While there are physical limits to intelligence, we can only speculate about where they lie. It’s reasonable to assume that even if we hit an S-curve in progress, that plateau will be far beyond anything even 15 John von Neumann clones could imagine.
Gwern was one of the first to recognise the "scaling hypothesis"; others followed later. While debate continues over whether scaling alone will lead to AI systems capable of self-improvement, it seems likely that scaling, combined with algorithmic progress and hardware advancements, will continue to drive progress for the foreseeable future. Dwarkesh Patel estimates a "70% chance scaling + algorithmic progress + hardware advances will get us to AGI by 2040". These odds are too high to ignore. Even if there are delays, superintelligence is still coming.
Some argue it's likely to be built by the end of this decade; others think it might take longer. But almost no one doubts that AGI will emerge this century, barring a global catastrophe. Even skeptics like Yann LeCun predict AGI could be reached in “years, if not a decade.” As Stuart Russell noted, estimates have shifted from “30-50 years” to “3-5 years.”
Leopold Aschenbrenner calls this shift "AGI realism." In this post, we focus on one key implication of this view—leaving aside geopolitical concerns:
> We are rapidly building machines smarter than the smartest humans. This is not another cool Silicon Valley boom; this isn’t some random community of coders writing an innocent open source software package; this isn’t fun and games. Superintelligence is going to be wild; it will be the most powerful weapon mankind has ever built. And for any of us involved, it’ll be the most important thing we ever do.
Of course, this could be wrong. |
45fc6dd0-bb01-4a39-830b-914f19458c72 | StampyAI/alignment-research-dataset/blogs | Blogs | A canonical and efficient byte-encoding for ints
A canonical and efficient byte-encoding for ints
------------------------------------------------
the internet has a bunch of efficient byte-encodings for fixed-length integers, from [LEB128](https://en.wikipedia.org/wiki/LEB128) to arguably [UTF-8](https://en.wikipedia.org/wiki/UTF-8); but none of them (except the trivial encoding of always encoding an integer as its fixed byte sizeof) seem to be canonical.
what i mean by canonical is that every integer has only one possible representation, and every representation means a single integer.
after some work, i've devised a mildly convoluted but, as far as i can tell, truly canonical encoding for int16, int32, and int64.
the formula i rely on is that 2ⁿ = 1 + (2¹ + … + 2ⁿ¯¹) = 2ⁱ + (2ⁱ + … 2ⁿ¯¹); so, to cover the entire, say, 2³² space of int32, i need to fit neatly into bytes variable bit patterns of length i through 31 (and have two patterns for i).
here are the bit patterns for int16 and int32 (int64, being larger, is at the bottom of the page).
each line mentions the number of variable bits you can check that the sum of two to the power of each of those account for every possible value; for example, for int16, 2⁷ + (2⁷ + … + 2¹⁵) = 2¹⁶
```
int16:
7: 0.......
* 7: 10000000 0.......
8: 10000001 ........
9: 1000001. ........
10: 100001.. ........
11: 10001... ........
12: 1001.... ........
13: 101..... ........
14: 11...... ........
* 15: 10000000 1....... ........
int32:
6: 00......
* 6: 01000000 00......
* 7: 10000000 0.......
8: 01000001 ........
9: 0100001. ........
10: 010001.. ........
11: 01001... ........
12: 0101.... ........
13: 011..... ........
* 14: 01000000 01...... ........
* 15: 10000000 1....... ........
16: 10000001 ........ ........
17: 1000001. ........ ........
18: 100001.. ........ ........
19: 10001... ........ ........
20: 1001.... ........ ........
21: 101..... ........ ........
* 22: 01000000 10...... ........(×2)
* 23: 11000000 0....... ........(×2)
24: 11000001 ........ ........(×2)
25: 1100001. ........ ........(×2)
26: 110001.. ........ ........(×2)
27: 11001... ........ ........(×2)
28: 1101.... ........ ........(×2)
29: 111..... ........ ........(×2)
* 30: 01000000 11...... ........(×3)
* 31: 11000000 1....... ........(×3)
```
with K = 1 for int64, K = 2 for int32, K = 3 for int64:
when the lower 8-K bits of the first byte are 0 (a pattern indicated with a \* next to the line), then the upper K bits of the first byte and the following 1 to 3 bits of the second byte (depending on the value of K) determine the amount of extra number-encoding bytes, while the remaining bits the second byte are the initial bits of the number.
when the lower 8-K bits are not 0, then the K upper bits indicate the number of number-encoding bytes after the first byte, and the bits after the first 1 in the lower 8-K bits are the initial bits of the number.
my explanation is probably not very clear, but the pattern should be visible if you look at it enough; just keep in mind that lines tagged with a \* work different from those not tagged that way, and that the first K bits are a special tag.
to efficiently encode a signed integer (optimizing for values near 0), encode:
* any positive or null number n as 2×n
* any strictly negative number n as 2×-n+1
below is the bit patterns for int64:
```
int64:
5: 000.....
* 5: 00100000 000.....
* 6: 01000000 00......
* 7: 10000000 0.......
8: 00100001 ........
9: 0010001. ........
10: 001001.. ........
11: 00101... ........
12: 0011.... ........
* 13: 00100000 001..... ........
* 14: 01000000 01...... ........
* 15: 10000000 1....... ........
16: 01000001 ........ ........
17: 0100001. ........ ........
18: 010001.. ........ ........
19: 01001... ........ ........
20: 0101.... ........ ........
* 21: 00100000 010..... ........(×2)
* 22: 01000000 10...... ........(×2)
* 23: 10100000 0....... ........(×2)
24: 01100001 ........ ........(×2)
25: 0110001. ........ ........(×2)
26: 011001.. ........ ........(×2)
27: 01101... ........ ........(×2)
28: 0111.... ........ ........(×2)
* 29: 00100000 011..... ........(×3)
* 30: 01000000 11...... ........(×3)
* 31: 10100000 1....... ........(×3)
32: 10000001 ........ ........(×3)
33: 1000001. ........ ........(×3)
34: 100001.. ........ ........(×3)
35: 10001... ........ ........(×3)
36: 1001.... ........ ........(×3)
* 37: 00100000 100..... ........(×4)
* 38: 01100000 00...... ........(×4)
* 39: 11000000 0....... ........(×4)
40: 10100001 ........ ........(×4)
41: 1010001. ........ ........(×4)
42: 101001.. ........ ........(×4)
43: 10101... ........ ........(×4)
44: 1011.... ........ ........(×4)
* 45: 00100000 101..... ........(×5)
* 46: 01100000 01...... ........(×5)
* 47: 11000000 1....... ........(×5)
48: 11000001 ........ ........(×5)
49: 1100001. ........ ........(×5)
50: 110001.. ........ ........(×5)
51: 11001... ........ ........(×5)
52: 1101.... ........ ........(×5)
* 53: 00100000 110..... ........(×6)
* 54: 01100000 10...... ........(×6)
* 55: 11100000 0....... ........(×6)
56: 11100001 ........ ........(×6)
57: 1110001. ........ ........(×6)
58: 111001.. ........ ........(×6)
59: 11101... ........ ........(×6)
60: 1111.... ........ ........(×6)
* 61: 00100000 111..... ........(×7)
* 62: 01100000 11...... ........(×7)
* 63: 11100000 1....... ........(×7)
``` |
e02c8904-2815-4bde-b078-59addc7b70ef | trentmkelly/LessWrong-43k | LessWrong | Meetup : Sydney Rationality Dojo - March
Discussion article for the meetup : Sydney Rationality Dojo - March
WHEN: 06 March 2016 04:00:00PM (+1100)
WHERE: 10 Shepherd Street, Chippendale
After a great session working on our values in February, we'll be following up to ensure we're working on goals that fulfill them, and that excite us. If you didn't make it last time, don't worry, there'll be plenty of opportunity to catch up! The usual group dinner will happen afterwards.
Discussion article for the meetup : Sydney Rationality Dojo - March |
d61abd5d-33ba-49ea-8ab7-44cff67d09fa | trentmkelly/LessWrong-43k | LessWrong | Misleading the witness
Related: Trust in Math
I was reading John Allen Paulos' A Mathematician Plays the Stock Market, in which Paulos relates a version of the well-known "missing dollar" riddle. I had heard it once before, but only vaguely remembered it. If you don't remember it, here it is:
> Three people stay in a hotel overnight. The innkeeper tells them that the price for three rooms is $30, so each pays $10.
>
> After the guests go to their rooms, the innkeeper realizes that there is a special discount for groups, and that the guests' total should have only been $25.
>
> The innkeeper gives a bellhop $5 with the instructions to return it to the guests.
>
> The bellhop, not wanting to get change, gives each guest $1 and keeps $2.
>
> Later, the bellhop thinks "Wait - something isn't right. Each guest paid $10. I gave them each back $1, so they each paid $9. $9 times 3 is $27. I kept $2. $27 + $2 is $29. Where did the missing dollar go?"
I remembered that the solution involves trickery, but it still took me a minute or two to figure out where it is. At first, I started mentally keeping track of the dollars in the riddle, trying to see where one got dropped so their sum would be 30.
Then I figured it out. The story should end:
> Later, the bellhop thinks "Wait - something isn't right. Each guest paid $10. I gave them each back $1, so they each paid $9. $9 times 3 is $27. The cost for their rooms was $25. $27 - $25 = $2, so they collectively overpaid by $2, which is the amount I kept. Why am I such a jerk?"
I told my fiance the riddle, and asked her where the missing dollar went. She went through the same process as I did, looking for a place in the story where $1 could go missing.
It's remarkable to me how blatantly deceptive the riddle is. The riddler states or implies at the end of the story that the dollars paid by the guests and the dollars kept by the bellhop should be summed, and that that sum should be $30. In fact, there's no reason to sum the dollars paid by the g |
d00f2f3d-f525-41e0-8d23-93284034a063 | trentmkelly/LessWrong-43k | LessWrong | Open thread, March 13 - March 19, 2017
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should start on Monday, and end on Sunday.
4. Unflag the two options "Notify me of new top level comments on this article" and "Make this post available under..." before submitting. |
906c1ba6-a780-4939-b05c-b5f2ba4066d8 | trentmkelly/LessWrong-43k | LessWrong | Cryonics-adjacent question
(This question involves my somewhat bizarre "mind backup" project, that was inspired by Rudy Rucker's "Lifebox" idea.)
Unfortunately, I made some HUGE blunders last year involving my life savings and the market (bonds, though I almost wish it was crypto). It's painful to admit this involved laziness on my part as well as bad judgment.
As a result, my finances are such I can't see a way to pay for both cryonics AND retire.
My question is: could I arrange to mail a hard drive (or future hyper-DVDs) to a company like Alcor after my death, containing all of my "mind backup" data. It's basically hundreds of thousands of disjointed diary entries and diagrams and flowcharts and sketches and lists and preserved records and pics and receipts and instructions and whatnot.
They would then store this media instead of my frozen corpse (which may be disposed of in a pauper's grave or ideally used for medical research).
Obviously, they should charge less to store a small box of disks for five hundred years at room temperature, than an upside-down frozen head in a canister of liquid nitrogen that needs constant topping up.
Once future software has become advanced enough, this data could then be used to make a crude "copy" of my mind; and hey presto, back from the dead.
It's not much, but better than the absolute certainty of oblivion, which is the alternative of doing nothing.
I'm not the only individual this question applies to.
Many users of this site already have huge collections of digital data, which will become MUCH larger before they die.
Some might not want it all to disappear after they're gone. |
d81654e8-6114-4935-80bd-0de06a3893fd | trentmkelly/LessWrong-43k | LessWrong | Weekly LW Meetups: Austin, Madison, NYC
This summary was posted to LW Main on July 27th, and has been moved to discussion.
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96581ec0-ac84-4232-830e-11a391c24531 | StampyAI/alignment-research-dataset/arxiv | Arxiv | Combining experts’ causal judgments.
1 Introduction
---------------
Consider a policymaker who is trying to decide which intervention,
that is, which actions,
should be implemented in order to bring about a desired outcome,
such as preventing violent behavior in prisons or reducing famine mortality in
some country.
The policymaker has access to various experts who can
advise her on which interventions to consider. Some experts may be (in the policymaker’s view)
more reliable than others; they may also have different areas of
expertise;
or may have perceived alternative factors in their analysis.
The goal of the policymaker is to choose the best intervention, taking into account
the experts’ advice.
There has been a great deal of work on combining experts’
probabilistic judgments. (Genest and Zidek ([1986](#bib.bib8))
provide a somewhat dated but still useful overview;
Dawid ([1987](#bib.bib5)) and Fenton et al. ([2016](#bib.bib6)), among others,
give a Bayesian analysis.)
We are interested in combining experts’
judgments in order to decide on the best intervention. Hence,
we need more than
probabilities. We need to have a causal understanding of the
situation. Thus, we assume that the experts provide the policymaker
with *causal models*.
In general, these models may
involve different variables (since the experts may be focusing on
different aspects of the problem). Even if two models both include
variables C1subscript𝐶1C\_{1}italic\_C start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and C2subscript𝐶2C\_{2}italic\_C start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, they may disagree on the relationships
between them. For example, one expert may believe that C2subscript𝐶2C\_{2}italic\_C start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is
independent of C1subscript𝐶1C\_{1}italic\_C start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT while another may believe that C1subscript𝐶1C\_{1}italic\_C start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT causally
depends on C2subscript𝐶2C\_{2}italic\_C start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
Yet somehow the policymaker
wants to use the information in these causal models to reach her
decision.
Despite the clear need for causal reasoning, and the examples in the
literature and in practice where experts work with
causal models (e.g., Chockler et al. ([2015](#bib.bib2)); Sampson et al. ([2013](#bib.bib24))),
there is surprisingly little work on combining causal judgments.
Indeed, the only work that we are aware of
that preceded our work
is that of
Bradley, Dietrich, and List ([2014](#bib.bib1)) (BDL from now on),
who prove an impossibility result.
Specifically, they describe certain arguably reasonable desiderata,
and show that there is no way of merging causal models so as
to satisfy all their desiderata. They then discuss various weakenings
of their assumptions to see the extent to which the impossibility can
be avoided, none of which seem that satisfactory.
Following the conference version of our paper,
Zennaro and Ivanovska [2018](#bib.bib31) examined the problem of
merging causal models where the merged model must satisfy a
fairness requirement (although the individual experts’ models may not
be fair). They proposed a way of combining models based on ideas of BDL.
Friedenberg and Halpern [2018](#bib.bib7) also considered the same
problem of combining causal model of experts, but allowed for the
possibility that experts disagree on the causal structure of
variables due to having different focus areas.
There is also much work on the closely related problem
of *causal discovery*: constructing a single causal model
from a data set.
A variety of techniques have been used to find the model that
best describes how the data is generated (see,
e.g., Claassen and Heskes ([2010](#bib.bib3), [2012](#bib.bib4)); Hyttinen et al. ([2014](#bib.bib16)); Tillman and Spirtes ([2011](#bib.bib26)); Triantafillou and Tsamardinos ([2015](#bib.bib27));
Triantafillou and Tsamardinos ([2015](#bib.bib27)) provide a good overview of work in the
area).
Of course, if we have the data that the experts used to generate their
models, then we should apply the more refined techniques of the work
on causal discovery. However, while the causals model constructed by
experts are presumably based on data, the data itself is typically no
longer available. Rather, the models represent the distillation
of years of experience, obtained by querying the experts.
In this paper, we present an approach to combining experts’ causal models
when sufficient data for discovering the overall causal model is not available.
The key step in combining experts’ causal models lies in defining when
two causal models are *compatible*. Causal models can be
merged only if they are compatible.
We start with a notion of *strong* compatibility, where the conditions say,
among other things, that if both M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
involve variables C1subscript𝐶1C\_{1}italic\_C start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and C2subscript𝐶2C\_{2}italic\_C start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, then they must agree on the causal
relationship between C1subscript𝐶1C\_{1}italic\_C start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and C2subscript𝐶2C\_{2}italic\_C start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. But that is not enough.
Suppose that in both models C1subscript𝐶1C\_{1}italic\_C start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT depends on C2subscript𝐶2C\_{2}italic\_C start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, C3subscript𝐶3C\_{3}italic\_C start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT, and C4subscript𝐶4C\_{4}italic\_C start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT.
Then in a precise sense, the two models must agree on *how* the
dependence works, despite describing the world using possibly different
sets of variables. Roughly speaking, this is the case when,
for every variable C𝐶Citalic\_C that the two models have in common, we can
designate one of the models as being “dominant” with respect to C𝐶Citalic\_C, and
use that model to determine the relationships for C𝐶Citalic\_C.
When M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are compatible, we are able to
construct a merged model M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT that can be viewed as
satisfying all but one of BDL’s desiderata (and we argue that the one
it does not satisfy is unreasonable).
In a precise sense, all conclusions that hold
in either of the models M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT also hold in the merged model (see
Theorem [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")(e) and Theorem [2](#Thmthm2 "Theorem 2 ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")(e)).
In this way, the merged model takes advantage of the
information supplied by all the experts (at least, to the extent that
the experts’ models are compatible), and can go beyond what we can do with
either of the individual models (e.g., considering interventions that
simultaneously act on variables that are in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT but not in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
variables that are in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT but not in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT).
The set of constraints that need to be satisfied for models to be
compatible is quite restrictive; as we show on
real-life examples,
models are often not compatible in this strong sense.
We thus define two successively more general notions of compatibility,
and demonstrate, by means of examples, how this allows for the merging
a wider class of models, and reasoning about interventions under less
strict conditions.
But even with these more general notions, some
experts’ models may still be incompatible due to disagreements about some
parts of the model, even though possible interventions to be considered do not affect
those parts of the model.
We therefore introduce a notion of causal model decomposition to
allow policymakers to
“localize” the incompatibility between models,and
merge the parts of the models that are compatible.
Having set out the formal foundation for merging causal models, we
show how probabilities can be assigned to different reasonable ways
of combining experts’ causal models based on the perceived reliability
of the experts who proposed them, using relatively standard techniques.
The policymaker will then have a probability on causal
models that she can use to decide which interventions to
implement. Specifically, we can use the probability on causal models to
compute the probability that an intervention is efficacious.
Combining that with the cost of implementing the intervention, the
policymaker can
compute the most effective intervention. As we shall see, although we
work with the same causal structures used to define causality,
interventions are different from (and actually simpler to analyze
than) causes.
We draw on various examples from the literature (including real-world
scenarios involving complex sociological phenomena) to illustrate
our approach, including
crime-prevention scenarios Sampson et al. ([2013](#bib.bib24)), radicalization in prisons
Wikström and Bouhana ([2017](#bib.bib29)), and child abuse Marinetto ([2011](#bib.bib21)).
These examples reinforce our belief that our
approach provides a
useful formal framework that
can be applied to the determination of appropriate interventions for
policymaking.
The rest of the paper is organized as follows. Section [2](#S2 "2 Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")
provides some background material on causal
models. We formally define our notion of intervention and compare it
to causality in Section [3](#S3 "3 Interventions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
We discuss our concept of compatibility and how causal
models can be merged in Section [4](#S4 "4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). We discuss how the
notions of interventions and of compatible models can be used by the
policymakers
to choose optimal interventions in Section [5](#S5 "5 Combining Experts’ Opinions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). Finally,
we summarize our results and outline future directions in
Section [6](#S6 "6 Conclusions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
2 Causal Models
----------------
In this section, we review the definition of causal
models introduced by Halpern and Pearl ([2005](#bib.bib13)).
The material in this section is largely taken from Halpern ([2016a](#bib.bib11)).
We assume that the world is described in terms of
variables and their values.
Some variables may have a causal influence on others. This
influence is modeled by a set of structural equations.
It is conceptually useful to split the variables into two
sets: the exogenous variables, whose values are
determined by
factors outside the model, and the
endogenous variables, whose values are ultimately determined by
the exogenous variables.
For example, in a voting scenario, we could have endogenous variables
that describe what the voters actually do (i.e., which candidate they
vote for), exogenous variables
that describe the factors
that determine how the voters vote, and a
variable describing the outcome (who wins).
The structural equations
describe how these values are
determined (majority rules; a candidate
wins if A𝐴Aitalic\_A and at least two of B𝐵Bitalic\_B, C𝐶Citalic\_C, D𝐷Ditalic\_D, and E𝐸Eitalic\_E vote for him;
etc.).
Formally, a *causal model* M𝑀Mitalic\_M
is a pair (𝒮,ℱ)𝒮ℱ({\cal S},{\cal F})( caligraphic\_S , caligraphic\_F ), where 𝒮𝒮{\cal S}caligraphic\_S is a *signature*, which explicitly
lists the endogenous and exogenous variables and characterizes
their possible values, and ℱℱ{\cal F}caligraphic\_F defines a set of *(modifiable)
structural equations*, relating the values of the variables.
A signature 𝒮𝒮{\cal S}caligraphic\_S is a tuple (𝒰,𝒱,ℛ)𝒰𝒱ℛ({\cal U},{\cal V},{\cal R})( caligraphic\_U , caligraphic\_V , caligraphic\_R ), where 𝒰𝒰{\cal U}caligraphic\_U is a set of
exogenous variables, 𝒱𝒱{\cal V}caligraphic\_V is a set
of endogenous variables, and ℛℛ{\cal R}caligraphic\_R associates with every variable Y∈𝒰∪𝒱𝑌𝒰𝒱Y\in{\cal U}\cup{\cal V}italic\_Y ∈ caligraphic\_U ∪ caligraphic\_V a nonempty set ℛ(Y)ℛ𝑌{\cal R}(Y)caligraphic\_R ( italic\_Y ) of possible values for
Y𝑌Yitalic\_Y (that is, the set of values over which Y𝑌Yitalic\_Y ranges).
For simplicity, we assume here that 𝒱𝒱{\cal V}caligraphic\_V is finite, as is ℛ(Y)ℛ𝑌{\cal R}(Y)caligraphic\_R ( italic\_Y ) for
every endogenous variable Y∈𝒱𝑌𝒱Y\in{\cal V}italic\_Y ∈ caligraphic\_V.
ℱℱ{\cal F}caligraphic\_F associates with each endogenous variable X∈𝒱𝑋𝒱X\in{\cal V}italic\_X ∈ caligraphic\_V a
function denoted FXsubscript𝐹𝑋F\_{X}italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT
(i.e., FX=ℱ(X)subscript𝐹𝑋ℱ𝑋F\_{X}={\cal F}(X)italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT = caligraphic\_F ( italic\_X ))
such that FX:(×U∈𝒰ℛ(U))×(×Y∈𝒱−{X}ℛ(Y))→ℛ(X)F\_{X}:(\times\_{U\in{\cal U}}{\cal R}(U))\times(\times\_{Y\in{\cal V}-\{X\}}{\cal R}(Y))\rightarrow{\cal R}(X)italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT : ( × start\_POSTSUBSCRIPT italic\_U ∈ caligraphic\_U end\_POSTSUBSCRIPT caligraphic\_R ( italic\_U ) ) × ( × start\_POSTSUBSCRIPT italic\_Y ∈ caligraphic\_V - { italic\_X } end\_POSTSUBSCRIPT caligraphic\_R ( italic\_Y ) ) → caligraphic\_R ( italic\_X ).
This mathematical notation just makes precise the fact that
FXsubscript𝐹𝑋F\_{X}italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT determines the value of X𝑋Xitalic\_X,
given the values of all the other variables in 𝒰∪𝒱𝒰𝒱{\cal U}\cup{\cal V}caligraphic\_U ∪ caligraphic\_V.
If there is one exogenous variable U𝑈Uitalic\_U and three endogenous
variables, X𝑋Xitalic\_X, Y𝑌Yitalic\_Y, and Z𝑍Zitalic\_Z, then FXsubscript𝐹𝑋F\_{X}italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT defines the values of X𝑋Xitalic\_X in
terms of the values of Y𝑌Yitalic\_Y, Z𝑍Zitalic\_Z, and U𝑈Uitalic\_U. For example, we might have
FX(u,y,z)=u+ysubscript𝐹𝑋𝑢𝑦𝑧𝑢𝑦F\_{X}(u,y,z)=u+yitalic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT ( italic\_u , italic\_y , italic\_z ) = italic\_u + italic\_y, which is usually written as
X=U+Y𝑋𝑈𝑌X=U+Yitalic\_X = italic\_U + italic\_Y. Thus, if Y=3𝑌3Y=3italic\_Y = 3 and U=2𝑈2U=2italic\_U = 2, then
X=5𝑋5X=5italic\_X = 5, regardless of how Z𝑍Zitalic\_Z is set.222The fact that X𝑋Xitalic\_X is assigned U+Y𝑈𝑌U+Yitalic\_U + italic\_Y (i.e., the value
of X𝑋Xitalic\_X is the sum of the values of U𝑈Uitalic\_U and Y𝑌Yitalic\_Y) does not imply
that Y𝑌Yitalic\_Y is assigned X−U𝑋𝑈X-Uitalic\_X - italic\_U; that is, FY(U,X,Z)=X−Usubscript𝐹𝑌𝑈𝑋𝑍𝑋𝑈F\_{Y}(U,X,Z)=X-Uitalic\_F start\_POSTSUBSCRIPT italic\_Y end\_POSTSUBSCRIPT ( italic\_U , italic\_X , italic\_Z ) = italic\_X - italic\_U does not
necessarily hold.
The structural equations define what happens in the presence of external
interventions.
Setting the value of some variable X𝑋Xitalic\_X to x𝑥xitalic\_x in a causal
model M=(𝒮,ℱ)𝑀𝒮ℱM=({\cal S},{\cal F})italic\_M = ( caligraphic\_S , caligraphic\_F ) results in a new causal model, denoted MX←xsubscript𝑀←𝑋𝑥M\_{X\leftarrow x}italic\_M start\_POSTSUBSCRIPT italic\_X ← italic\_x end\_POSTSUBSCRIPT, which is identical to M𝑀Mitalic\_M, except that the
equation for X𝑋Xitalic\_X in ℱℱ{\cal F}caligraphic\_F is replaced by X=x𝑋𝑥X=xitalic\_X = italic\_x.
The dependencies between variables in a causal model M𝑀Mitalic\_M
can be described using a causal network (or *causal graph*),
whose nodes are labeled by the endogenous and exogenous variables in
M=((𝒰,𝒱,ℛ),ℱ)𝑀𝒰𝒱ℛℱM=(({\cal U},{\cal V},{\cal R}),{\cal F})italic\_M = ( ( caligraphic\_U , caligraphic\_V , caligraphic\_R ) , caligraphic\_F ), with one node for each variable in 𝒰∪𝒱𝒰𝒱{\cal U}\cup{\cal V}caligraphic\_U ∪ caligraphic\_V. The roots of the graph are (labeled by)
the exogenous variables. There is a directed edge from variable X𝑋Xitalic\_X
to Y𝑌Yitalic\_Y if Y𝑌Yitalic\_Y *depends on* X𝑋Xitalic\_X; this is the case
if there is some setting of all the variables in
𝒰∪𝒱𝒰𝒱{\cal U}\cup{\cal V}caligraphic\_U ∪ caligraphic\_V other than X𝑋Xitalic\_X and Y𝑌Yitalic\_Y such that varying the value of
X𝑋Xitalic\_X in that setting results in a variation in the value of Y𝑌Yitalic\_Y; that
is, there is
a setting z→→𝑧\vec{z}over→ start\_ARG italic\_z end\_ARG of the variables other than X𝑋Xitalic\_X and Y𝑌Yitalic\_Y and values
x𝑥xitalic\_x and x′superscript𝑥′x^{\prime}italic\_x start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of X𝑋Xitalic\_X such that
FY(x,z→)≠FY(x′,z→)subscript𝐹𝑌𝑥→𝑧subscript𝐹𝑌superscript𝑥′→𝑧F\_{Y}(x,\vec{z})\neq F\_{Y}(x^{\prime},\vec{z})italic\_F start\_POSTSUBSCRIPT italic\_Y end\_POSTSUBSCRIPT ( italic\_x , over→ start\_ARG italic\_z end\_ARG ) ≠ italic\_F start\_POSTSUBSCRIPT italic\_Y end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_z end\_ARG ).
A causal model M𝑀Mitalic\_M is *recursive* (or *acyclic*)
if its causal graph is acyclic.333Halpern [2016a](#bib.bib11) calls this *strongly
recursive*, and takes a recursive model to be one where, for each
context u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG, the dependency graph is acyclic (but it may be a
different acyclic graph for context, so that in one context A𝐴Aitalic\_A may
be an ancestor of B𝐵Bitalic\_B, while in another, B𝐵Bitalic\_B may be an ancestor of
A𝐴Aitalic\_A).
It should be clear that if M𝑀Mitalic\_M is an acyclic causal model,
then given a *context*, that is, a setting u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG for the
exogenous variables in 𝒰𝒰{\cal U}caligraphic\_U, the values of all the other variables are
determined (i.e., there is a unique solution to all the equations).
We can determine these values by starting at the top of the graph and
working our way down.
In this paper, following the literature, we restrict to acyclic models.
The following example, due to Lewis ([2000](#bib.bib19)), describes a simple
causal scenario.
######
Example 2.1.
Suzy and Billy both pick up rocks and throw them at a bottle. Suzy’s rock gets there first, shattering
the bottle. Since both throws are perfectly accurate, Billy’s would have shattered the bottle had it not
been preempted by Suzy’s throw.
Consider a model having
an exogenous variable U𝑈Uitalic\_U that encapsulates whatever background factors
cause Suzy and Billy to decide to throw the rock
(the details of U𝑈Uitalic\_U
do not matter, since we are interested only in the context where U𝑈Uitalic\_U’s
value is such that both Suzy and Billy throw). Thus, U𝑈Uitalic\_U has four
possible values, depending on which of Suzy and Billy throw. We also
have three binary variables: ST for Suzy
throws, BT for Billy throws, and BS for bottle shatters.
𝑆𝑇=1𝑆𝑇1\mbox{{\it ST}}=1ST = 1 means “Suzy throws”; 𝑆𝑇=0𝑆𝑇0\mbox{{\it ST}}=0ST = 0 means that she does not. We
interpret 𝐵𝑇=1𝐵𝑇1\mbox{{\it BT}}=1BT = 1, 𝐵𝑇=0𝐵𝑇0\mbox{{\it BT}}=0BT = 0, 𝐵𝑆=1𝐵𝑆1\mbox{{\it BS}}=1BS = 1, and 𝐵𝑆=0𝐵𝑆0\mbox{{\it BS}}=0BS = 0 similarly.
The values of ST and BT are determined by the
context.
The value of BS
is determined by the equation F𝐵𝑆(u→,𝑆𝑇,𝐵𝑇)=𝑆𝑇∨𝐵𝑇subscript𝐹𝐵𝑆→𝑢𝑆𝑇𝐵𝑇𝑆𝑇𝐵𝑇F\_{{\it BS}}(\vec{u},\mbox{{\it ST}},\mbox{{\it BT}})=\mbox{{\it ST}}\vee\mbox{{\it BT}}italic\_F start\_POSTSUBSCRIPT italic\_BS end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_u end\_ARG , ST , BT ) = ST ∨ BT.
The causal graph corresponding to this model is depicted in
Figure [1](#S2.F1 "Figure 1 ‣ Example 2.1. ‣ 2 Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").444We could also model this problem using time-indexed variables that
explicitly model the temporal order of events; see Halpern ([2016a](#bib.bib11)); Halpern and Pearl ([2005](#bib.bib13)).
The issues raised by this example could be made using either approach.
BSSTBTU𝑈Uitalic\_U
Figure 1: The causal graph for the rock-throwing example.
In many papers in the literature (e.g., Bradley et al. ([2014](#bib.bib1)); Sampson et al. ([2013](#bib.bib24)))
a causal model is defined simply by a causal graph indicating the
dependencies, perhaps with an indication of whether a change
has a positive or negative effect; that is, edges are annotated with
+++ or −--, so that an edge from A𝐴Aitalic\_A to B𝐵Bitalic\_B annotated with +++ means
that an increase in A𝐴Aitalic\_A results in an increase in B𝐵Bitalic\_B, while
if it is annotated with a −--, then an increase in A𝐴Aitalic\_A results in a
decrease in B𝐵Bitalic\_B (where what constitutes an increase or decrease is
determined by the model). Examples of these are shown in Section [4](#S4 "4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
Our models are more expressive, since the
equations typically provide much more detailed information regarding
the dependence between variables (as shown in Example [2.1](#S2.Thmtheorem1 "Example 2.1. ‣ 2 Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")); the causal graphs capture only part
of this information. Of course, this extra information makes
merging models more difficult (although, as the results of
BDL show, the difficulties in
merging models already arise with purely qualitative graphs).
To define interventions carefully, it is useful to have a language in
which we can make statements about interventions.
Given a signature 𝒮=(𝒰,𝒱,ℛ)𝒮𝒰𝒱ℛ{\cal S}=({\cal U},{\cal V},{\cal R})caligraphic\_S = ( caligraphic\_U , caligraphic\_V , caligraphic\_R ), a *primitive event* is a
formula of the form X=x𝑋𝑥X=xitalic\_X = italic\_x, for X∈𝒱𝑋𝒱X\in{\cal V}italic\_X ∈ caligraphic\_V and x∈ℛ(X)𝑥ℛ𝑋x\in{\cal R}(X)italic\_x ∈ caligraphic\_R ( italic\_X ).
A causal formula (over 𝒮𝒮{\cal S}caligraphic\_S) is one of the form
[Y1←y1,…,Yk←yk]φdelimited-[]formulae-sequence←subscript𝑌1subscript𝑦1…←subscript𝑌𝑘subscript𝑦𝑘𝜑[Y\_{1}\leftarrow y\_{1},\ldots,Y\_{k}\leftarrow y\_{k}]\varphi[ italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ] italic\_φ,
where
φ𝜑\varphiitalic\_φ is a Boolean
combination of primitive events,
Y1,…,Yksubscript𝑌1…subscript𝑌𝑘Y\_{1},\ldots,Y\_{k}italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT are distinct variables in 𝒰∪𝒱𝒰𝒱{\cal U}\cup{\cal V}caligraphic\_U ∪ caligraphic\_V, and
yi∈ℛ(Yi)subscript𝑦𝑖ℛsubscript𝑌𝑖y\_{i}\in{\cal R}(Y\_{i})italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∈ caligraphic\_R ( italic\_Y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ).555In earlier work Halpern ([2016a](#bib.bib11)); Halpern and Pearl ([2005](#bib.bib13)), each Yisubscript𝑌𝑖Y\_{i}italic\_Y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT was taken to
be an endogenous variable. For technical reasons (explained in
Section [4](#S4 "4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")), we also allow Y𝑌Yitalic\_Y to be exogenous.
Such a formula is abbreviated
as [Y→←y→]φdelimited-[]←→𝑌→𝑦𝜑[\vec{Y}\leftarrow\vec{y}]\varphi[ over→ start\_ARG italic\_Y end\_ARG ← over→ start\_ARG italic\_y end\_ARG ] italic\_φ.
The special
case where k=0𝑘0k=0italic\_k = 0
is abbreviated as
φ𝜑\varphiitalic\_φ.
Intuitively,
[Y1←y1,…,Yk←yk]φdelimited-[]formulae-sequence←subscript𝑌1subscript𝑦1…←subscript𝑌𝑘subscript𝑦𝑘𝜑[Y\_{1}\leftarrow y\_{1},\ldots,Y\_{k}\leftarrow y\_{k}]\varphi[ italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ] italic\_φ says that
φ𝜑\varphiitalic\_φ would hold if
Yisubscript𝑌𝑖Y\_{i}italic\_Y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT were set to yisubscript𝑦𝑖y\_{i}italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, for i=1,…,k𝑖1…𝑘i=1,\ldots,kitalic\_i = 1 , … , italic\_k.
We call a pair (M,u→)𝑀→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ) consisting of a causal model M𝑀Mitalic\_M and a
context u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG a *(causal) setting*.
A causal formula ψ𝜓\psiitalic\_ψ is true or false in a setting.
We write (M,u→)⊧ψmodels𝑀→𝑢𝜓(M,\vec{u})\models\psi( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_ψ if
the causal formula ψ𝜓\psiitalic\_ψ is true in
the setting (M,u→)𝑀→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ).
The ⊧models\models⊧ relation is defined inductively.
(M,u→)⊧X=xmodels𝑀→𝑢𝑋𝑥(M,\vec{u})\models X=x( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_X = italic\_x if
the variable X𝑋Xitalic\_X has value x𝑥xitalic\_x
in the unique (since we are dealing with acyclic models) solution
to the equations in
M𝑀Mitalic\_M in context u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG (that is, the
unique vector of values for the exogenous variables that simultaneously satisfies all
equations
in M𝑀Mitalic\_M
with the variables in 𝒰𝒰{\cal U}caligraphic\_U set to u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG).
If k≥1𝑘1k\geq 1italic\_k ≥ 1 and Yksubscript𝑌𝑘Y\_{k}italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT is an endogenous
variable, then
| | | |
| --- | --- | --- |
| | (M,u→)⊧[Y1←y1,…,Yk←yk]φ iff (MYk←yk,u→)⊧[Y1←y1,…,Yk−1←yk−1]φ.models𝑀→𝑢delimited-[]formulae-sequence←subscript𝑌1subscript𝑦1…←subscript𝑌𝑘subscript𝑦𝑘𝜑 iff missing-subexpressionmodelssubscript𝑀←subscript𝑌𝑘subscript𝑦𝑘→𝑢delimited-[]formulae-sequence←subscript𝑌1subscript𝑦1…←subscript𝑌𝑘1subscript𝑦𝑘1𝜑missing-subexpression\begin{array}[]{ll}(M,\vec{u})\models[Y\_{1}\leftarrow y\_{1},\ldots,Y\_{k}\leftarrow y\_{k}]\varphi\mbox{ iff }\\
(M\_{Y\_{k}\leftarrow y\_{k}},\vec{u})\models[Y\_{1}\leftarrow y\_{1},\ldots,Y\_{k-1}\leftarrow y\_{k-1}]\varphi.\end{array}start\_ARRAY start\_ROW start\_CELL ( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ [ italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ] italic\_φ iff end\_CELL start\_CELL end\_CELL end\_ROW start\_ROW start\_CELL ( italic\_M start\_POSTSUBSCRIPT italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ [ italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_k - 1 end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT italic\_k - 1 end\_POSTSUBSCRIPT ] italic\_φ . end\_CELL start\_CELL end\_CELL end\_ROW end\_ARRAY | |
If Yksubscript𝑌𝑘Y\_{k}italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT is an exogenous variable, then
| | | |
| --- | --- | --- |
| | (M,u→)⊧[Y1←y1,…,Yk←yk]φ iff (M,u→[Yk/yk])⊧[Y1←y1,…,Yk−1←yk−1]φ,models𝑀→𝑢delimited-[]formulae-sequence←subscript𝑌1subscript𝑦1…←subscript𝑌𝑘subscript𝑦𝑘𝜑 iff missing-subexpressionmodels𝑀→𝑢delimited-[]subscript𝑌𝑘subscript𝑦𝑘delimited-[]formulae-sequence←subscript𝑌1subscript𝑦1…←subscript𝑌𝑘1subscript𝑦𝑘1𝜑missing-subexpression\begin{array}[]{ll}(M,\vec{u})\models[Y\_{1}\leftarrow y\_{1},\ldots,Y\_{k}\leftarrow y\_{k}]\varphi\mbox{ iff }\\
(M,\vec{u}[Y\_{k}/y\_{k}])\models[Y\_{1}\leftarrow y\_{1},\ldots,Y\_{k-1}\leftarrow y\_{k-1}]\varphi,\end{array}start\_ARRAY start\_ROW start\_CELL ( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ [ italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ] italic\_φ iff end\_CELL start\_CELL end\_CELL end\_ROW start\_ROW start\_CELL ( italic\_M , over→ start\_ARG italic\_u end\_ARG [ italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT / italic\_y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ] ) ⊧ [ italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_k - 1 end\_POSTSUBSCRIPT ← italic\_y start\_POSTSUBSCRIPT italic\_k - 1 end\_POSTSUBSCRIPT ] italic\_φ , end\_CELL start\_CELL end\_CELL end\_ROW end\_ARRAY | |
where u→[Yk/yk]→𝑢delimited-[]subscript𝑌𝑘subscript𝑦𝑘\vec{u}[Y\_{k}/y\_{k}]over→ start\_ARG italic\_u end\_ARG [ italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT / italic\_y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ] is the result of replacing the value of Yksubscript𝑌𝑘Y\_{k}italic\_Y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT
in u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG by yksubscript𝑦𝑘y\_{k}italic\_y start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT.
3 Interventions
----------------
In this section, we define (causal) interventions, and compare
the notion of intervention to that of cause.
######
Definition 3.1.
[Intervention]
X→=x→normal-→𝑋normal-→𝑥\vec{X}=\vec{x}over→ start\_ARG italic\_X end\_ARG = over→ start\_ARG italic\_x end\_ARG is an *intervention leading to
¬φ𝜑\neg{\varphi}¬ italic\_φ
in (M,u→)𝑀normal-→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG )* if the following
three conditions hold:
I1.
(M,u→)⊧φmodels𝑀→𝑢𝜑(M,\vec{u})\models\varphi( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_φ.
I2.
(M,u→)⊧[X→←x→]¬φ.models𝑀→𝑢delimited-[]←→𝑋→𝑥𝜑(M,\vec{u})\models[\vec{X}\leftarrow\vec{x}]\neg\varphi.( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ¬ italic\_φ .
I3.
X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG is minimal; there is no strict subset X→′superscript→𝑋′\vec{X}^{\prime}over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG
and values x→′superscript→𝑥′\vec{x}^{\prime}over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT such that X→′=x→′superscript→𝑋′superscript→𝑥′\vec{X}^{\prime}=\vec{x}^{\prime}over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT satisfies I2.
I1 says φ𝜑\varphiitalic\_φ must be true in the current setting (M,u→)𝑀→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ), while
I2 says that performing the intervention results in
φ𝜑\varphiitalic\_φ no longer being true. I3 is a minimality condition. From
a policymaker’s perspective, I2 is the key condition. It says that by
making the appropriate changes, we can bring about a change in φ𝜑\varphiitalic\_φ.
Our definition of an intervention leading to ¬φ𝜑\neg{{\varphi}}¬ italic\_φ
slightly generalizes others in the literature.
Pearl ([2009](#bib.bib23)) assumes that the causal model is first
analyzed, and then a new intervention variable IVsubscript𝐼𝑉I\_{V}italic\_I start\_POSTSUBSCRIPT italic\_V end\_POSTSUBSCRIPT is added for each
variable V𝑉Vitalic\_V on which we want to intervene.
If IV=1subscript𝐼𝑉1I\_{V}=1italic\_I start\_POSTSUBSCRIPT italic\_V end\_POSTSUBSCRIPT = 1, then the appropriate intervention
on V𝑉Vitalic\_V takes place, independent of the values of the other parents of
V𝑉Vitalic\_V; if IV=0subscript𝐼𝑉0I\_{V}=0italic\_I start\_POSTSUBSCRIPT italic\_V end\_POSTSUBSCRIPT = 0, then IVsubscript𝐼𝑉I\_{V}italic\_I start\_POSTSUBSCRIPT italic\_V end\_POSTSUBSCRIPT has no effect, and the behavior of V𝑉Vitalic\_V
is determined by its parents, just as it was in the original model.
Lu and Druzdzel ([2002](#bib.bib20)), Korb et al. ([2004](#bib.bib18)),
and Woodward ([2003](#bib.bib30)) take similar approaches.
We do not require special intervention variables; we just allow
interventions directly on the variables in the model.
But we can certainly assume as a special case that for
each variable V𝑉Vitalic\_V in the model there is a special intervention
variable IVsubscript𝐼𝑉I\_{V}italic\_I start\_POSTSUBSCRIPT italic\_V end\_POSTSUBSCRIPT that works just like Pearl’s intervention variables,
and thus recover the other approaches considered in the literature.
Our definition also focuses on the outcome of the intervention, not
just the intervention itself.
In any case, it
should be
clear that all these definitions are trying to capture exactly the
same intuitions, and differ only in minor ways.
Although there seems to be relatively little disagreement about how to
capture intervention, the same cannot be said for causality.
Even among definitions that involve counterfactuals and structural
equations
Glymour and Wimberly ([2007](#bib.bib9)); Halpern ([2015](#bib.bib10)); Halpern and Pearl ([2005](#bib.bib13)); Hitchcock ([2001](#bib.bib14), [2007](#bib.bib15)); Woodward ([2003](#bib.bib30)),
there are a number of subtle variations.
Fortunately for us,
the definition of intervention does not depend on
how causality is defined.
While we do not get into the details of causality here, it is
instructive to compare the definitions of causality and intervention.
For definiteness, we focus on the definition of causality given by
Halpern ([2015](#bib.bib10)). It has
conditions
AC1–3 that are analogous of I1–3.
Specifically, AC1 says X→=x→→𝑋→𝑥\vec{X}=\vec{x}over→ start\_ARG italic\_X end\_ARG = over→ start\_ARG italic\_x end\_ARG is a cause of φ𝜑\varphiitalic\_φ
in (M,u→)𝑀→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ) if (M,u→)⊧(X→=x→)∧φmodels𝑀→𝑢→𝑋→𝑥𝜑(M,\vec{u})\models(\vec{X}=\vec{x})\land\varphi( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ ( over→ start\_ARG italic\_X end\_ARG = over→ start\_ARG italic\_x end\_ARG ) ∧ italic\_φ and
AC3 is a minimality condition. AC2 is a more complicated condition; it
says that there exist values x→′superscript→𝑥′\vec{x}^{\prime}over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT for the variables in
X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG, a (possibly empty) subset W→→𝑊\vec{W}over→ start\_ARG italic\_W end\_ARG of variables, and
values w→→𝑤\vec{w}over→ start\_ARG italic\_w end\_ARG for the variables in W→→𝑊\vec{W}over→ start\_ARG italic\_W end\_ARG such
that (M,u→)⊧W→=w→models𝑀→𝑢→𝑊→𝑤(M,\vec{u})\models\vec{W}=\vec{w}( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ over→ start\_ARG italic\_W end\_ARG = over→ start\_ARG italic\_w end\_ARG and
(M,u→)⊧[X→←x→,W→←w→]¬φmodels𝑀→𝑢delimited-[]formulae-sequence←→𝑋→𝑥←→𝑊→𝑤𝜑(M,\vec{u})\models[\vec{X}\leftarrow\vec{x},\vec{W}\leftarrow\vec{w}]\neg\varphi( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG , over→ start\_ARG italic\_W end\_ARG ← over→ start\_ARG italic\_w end\_ARG ] ¬ italic\_φ. We do not attempt to explain or motivate AC2 here,
since our focus is not causality.
Consider Example [2.1](#S2.Thmtheorem1 "Example 2.1. ‣ 2 Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") again.
Changing the value of Suzy’s throw by itself is not an intervention
leading to the bottle not being shattered.
Even if we prevent Suzy from throwing, the bottle will still shatter
because of Billy’s throw.
That is,
although 𝑆𝑇=1𝑆𝑇1\mbox{{\it ST}}=1ST = 1 is a cause of the bottle shattering,
𝑆𝑇=0𝑆𝑇0\mbox{{\it ST}}=0ST = 0 is not an intervention leading to the bottle
not being shattered;
intervening on ST alone does not change the outcome.
On the
other hand, 𝑆𝑇=0∧𝐵𝑇=0𝑆𝑇0𝐵𝑇0\mbox{{\it ST}}=0\land\mbox{{\it BT}}=0ST = 0 ∧ BT = 0 is an intervention leading to the bottle
not being shattered,
but 𝑆𝑇=1∧𝐵𝑇=1𝑆𝑇1𝐵𝑇1\mbox{{\it ST}}=1\land\mbox{{\it BT}}=1ST = 1 ∧ BT = 1 is not a cause of the bottle shattering; it violates
the minimality condition AC3.
It is almost immediate from the definitions that we have the following
relationship between interventions and causes:
######
Proposition 3.2.
If X→=x→normal-→𝑋normal-→𝑥\vec{X}=\vec{x}over→ start\_ARG italic\_X end\_ARG = over→ start\_ARG italic\_x end\_ARG is an intervention
leading to ¬φ𝜑\neg\varphi¬ italic\_φ in
(M,u→)𝑀normal-→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ), then there is some subset X→′superscriptnormal-→𝑋normal-′\vec{X}^{\prime}over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of X→normal-→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG
such that X→′=x→′superscriptnormal-→𝑋normal-′superscriptnormal-→𝑥normal-′\vec{X}^{\prime}=\vec{x}^{\prime}over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT is a cause of φ𝜑\varphiitalic\_φ in (M,u→)𝑀normal-→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ),
where
x→′superscriptnormal-→𝑥normal-′\vec{x}^{\prime}over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT is such that
(M,u→)⊧X→′=x→′models𝑀normal-→𝑢superscriptnormal-→𝑋normal-′superscriptnormal-→𝑥normal-′(M,\vec{u})\models\vec{X}^{\prime}=\vec{x}^{\prime}( italic\_M , over→ start\_ARG italic\_u end\_ARG ) ⊧ over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT.
Conversely, if X→=x→normal-→𝑋normal-→𝑥\vec{X}=\vec{x}over→ start\_ARG italic\_X end\_ARG = over→ start\_ARG italic\_x end\_ARG is a cause of φ𝜑\varphiitalic\_φ in
(M,u→)𝑀normal-→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ) then there is a superset X→′superscriptnormal-→𝑋normal-′\vec{X}^{\prime}over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of X→normal-→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG and
values x→′superscriptnormal-→𝑥normal-′\vec{x}^{\prime}over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT such that X→′=x→′superscriptnormal-→𝑋normal-′superscriptnormal-→𝑥normal-′\vec{X}^{\prime}=\vec{x}^{\prime}over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT is an intervention
leading to ¬φ𝜑\neg\varphi¬ italic\_φ.
Halpern ([2015](#bib.bib10)) proved that (for his latest definition) the
complexity of determining
whether X→=x→→𝑋→𝑥\vec{X}=\vec{x}over→ start\_ARG italic\_X end\_ARG = over→ start\_ARG italic\_x end\_ARG is a cause
of φ𝜑\varphiitalic\_φ in (M,u→)𝑀→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ) is DP-complete,
where DP consists of those languages L𝐿Litalic\_L for which there exist a
language L1subscript𝐿1L\_{1}italic\_L start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT in NP and a language L2subscript𝐿2L\_{2}italic\_L start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT in co-NP such that L=L1∩L2𝐿subscript𝐿1subscript𝐿2L=L\_{1}\cap L\_{2}italic\_L = italic\_L start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ italic\_L start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT Papadimitriou and Yannakakis ([1982](#bib.bib22)).
It is well known that DP is at least as hard as NP and co-NP (and
most likely strictly harder).
The following theorem shows that the
problem of determining whether
X→=x→→𝑋→𝑥\vec{X}=\vec{x}over→ start\_ARG italic\_X end\_ARG = over→ start\_ARG italic\_x end\_ARG is an intervention is in a lower complexity class.
######
Theorem 3.3.
Given a causal model M𝑀Mitalic\_M, a context u→normal-→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG, and a Boolean formula
φ𝜑\varphiitalic\_φ, the problem of
determining whether
X→=x→normal-→𝑋normal-→𝑥\vec{X}=\vec{x}over→ start\_ARG italic\_X end\_ARG = over→ start\_ARG italic\_x end\_ARG is an intervention
leading to ¬φ𝜑\neg{\varphi}¬ italic\_φ
in (M,u→)𝑀normal-→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ) is co-NP-complete.
Proof. First, we prove membership in co-NP. It is easy to see that checking
conditions I1 and I2 of Definition [3.1](#S3.Thmtheorem1 "Definition 3.1. ‣ 3 Interventions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")
can be done in polynomial time by simply evaluating φ𝜑\varphiitalic\_φ first in (M,u→)𝑀→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ) and then in the modified context
where the values of X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG are set to x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG.
Checking whether I3 holds is in co-NP, because the complementary condition is
in NP; indeed,
we simply have to guess a subset X→′superscript→𝑋′\vec{X}^{\prime}over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG
and values x→′superscript→𝑥′\vec{x}^{\prime}over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and verify that I1 and I2 hold for X→′=x→′superscript→𝑋′superscript→𝑥′\vec{X}^{\prime}=\vec{x}^{\prime}over→ start\_ARG italic\_X end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = over→ start\_ARG italic\_x end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and φ𝜑\varphiitalic\_φ, which, as we observed, can be done in polynomial time.
For co-NP-hardness, we provide a reduction from UNSAT, which is
the language of all unsatisfiable Boolean formulas,
to the intervention problem.
Given a formula ψ𝜓\psiitalic\_ψ that mentions the set V→→𝑉\vec{V}over→ start\_ARG italic\_V end\_ARG of variables,
we construct a causal
model Mψsubscript𝑀𝜓M\_{\psi}italic\_M start\_POSTSUBSCRIPT italic\_ψ end\_POSTSUBSCRIPT, context
u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG, and formula φ𝜑\varphiitalic\_φ such that V→=1→𝑉1\vec{V}=1over→ start\_ARG italic\_V end\_ARG = 1 is an intervention
5joe42
leading to ¬φ𝜑\neg\varphi¬ italic\_φ in (M,u→)𝑀→𝑢(M,\vec{u})( italic\_M , over→ start\_ARG italic\_u end\_ARG ) iff ψ𝜓\psiitalic\_ψ is unsatisfiable.
The set of endogenous variables in M𝑀Mitalic\_M is V→∪{V′,Y}→𝑉superscript𝑉′𝑌\vec{V}\cup\{V^{\prime},Y\}over→ start\_ARG italic\_V end\_ARG ∪ { italic\_V start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_Y }, where V′superscript𝑉′V^{\prime}italic\_V start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and Y𝑌Yitalic\_Y are fresh variables not in V→→𝑉\vec{V}over→ start\_ARG italic\_V end\_ARG.
Let W→=V→∪{V′}→𝑊→𝑉superscript𝑉′\vec{W}=\vec{V}\cup\{V^{\prime}\}over→ start\_ARG italic\_W end\_ARG = over→ start\_ARG italic\_V end\_ARG ∪ { italic\_V start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT }.
There is a single exogenous variable U𝑈Uitalic\_U that determines the value
of the variables in W→→𝑊\vec{W}over→ start\_ARG italic\_W end\_ARG: we have the equation V=U𝑉𝑈V=Uitalic\_V = italic\_U for each
variable V∈W→𝑉→𝑊V\in\vec{W}italic\_V ∈ over→ start\_ARG italic\_W end\_ARG.
The equation for Y𝑌Yitalic\_Y is
Y=∨V∈W→(V=0)𝑌subscript𝑉→𝑊𝑉0Y=\lor\_{V\in\vec{W}\,}(V=0)italic\_Y = ∨ start\_POSTSUBSCRIPT italic\_V ∈ over→ start\_ARG italic\_W end\_ARG end\_POSTSUBSCRIPT ( italic\_V = 0 ) (so Y=1𝑌1Y=1italic\_Y = 1 if at least one variable in
W→→𝑊\vec{W}over→ start\_ARG italic\_W end\_ARG is 0).
Let φ𝜑\varphiitalic\_φ be ¬ψ∧(Y=1)𝜓𝑌1\neg\psi\land(Y=1)¬ italic\_ψ ∧ ( italic\_Y = 1 ). We claim that W→=1→→𝑊→1\vec{W}=\vec{1}over→ start\_ARG italic\_W end\_ARG = over→ start\_ARG 1 end\_ARG is an
intervention leading to ¬φ𝜑\neg{\varphi}¬ italic\_φ in (Mψ,0)subscript𝑀𝜓0(M\_{\psi},0)( italic\_M start\_POSTSUBSCRIPT italic\_ψ end\_POSTSUBSCRIPT , 0 ) iff ψ∈𝜓absent\psi\initalic\_ψ ∈ UNSAT.
Suppose that ψ∈𝜓absent\psi\initalic\_ψ ∈ UNSAT. Then, it is easy to see that
(M,0)⊧φmodels𝑀0𝜑(M,0)\models\varphi( italic\_M , 0 ) ⊧ italic\_φ (since ¬ψ𝜓\neg\psi¬ italic\_ψ is valid)
and (M,0)⊧[W→←1→]¬φmodels𝑀0delimited-[]←→𝑊→1𝜑(M,0)\models[\vec{W}\leftarrow\vec{1}]\neg\varphi( italic\_M , 0 ) ⊧ [ over→ start\_ARG italic\_W end\_ARG ← over→ start\_ARG 1 end\_ARG ] ¬ italic\_φ (since
(M,0)⊧[W→←1→](Y=0)models𝑀0delimited-[]←→𝑊→1𝑌0(M,0)\models[\vec{W}\leftarrow\vec{1}](Y=0)( italic\_M , 0 ) ⊧ [ over→ start\_ARG italic\_W end\_ARG ← over→ start\_ARG 1 end\_ARG ] ( italic\_Y = 0 )).
To see that I3 holds, suppose by way of contradiction that
W→′←w→′←superscript→𝑊′superscript→𝑤′\vec{W}^{\prime}\leftarrow\vec{w}^{\prime}over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ← over→ start\_ARG italic\_w end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT satisfies I1 and I2 for some strict subset
W→′superscript→𝑊′\vec{W}^{\prime}over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of W→→𝑊\vec{W}over→ start\_ARG italic\_W end\_ARG. In particular, we must have
(M,0)⊧[W→′←w→′]¬φmodels𝑀0delimited-[]←superscript→𝑊′superscript→𝑤′𝜑(M,0)\models[\vec{W}^{\prime}\leftarrow\vec{w}^{\prime}]\neg\varphi( italic\_M , 0 ) ⊧ [ over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ← over→ start\_ARG italic\_w end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ] ¬ italic\_φ. We clearly have
(M,0)⊧[W→′←w→′](Y=1)models𝑀0delimited-[]←superscript→𝑊′superscript→𝑤′𝑌1(M,0)\models[\vec{W}^{\prime}\leftarrow\vec{w}^{\prime}](Y=1)( italic\_M , 0 ) ⊧ [ over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ← over→ start\_ARG italic\_w end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ] ( italic\_Y = 1 ), so we must have (M,0)⊧[W→′←w→′]ψmodels𝑀0delimited-[]←superscript→𝑊′superscript→𝑤′𝜓(M,0)\models[\vec{W}^{\prime}\leftarrow\vec{w}^{\prime}]\psi( italic\_M , 0 ) ⊧ [ over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ← over→ start\_ARG italic\_w end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ] italic\_ψ, contradicting the assumption
that ψ∈𝜓absent\psi\initalic\_ψ ∈ UNSAT. Thus, W→←1→←→𝑊→1\vec{W}\leftarrow\vec{1}over→ start\_ARG italic\_W end\_ARG ← over→ start\_ARG 1 end\_ARG is an
intervention leading to ¬φ𝜑\neg{\varphi}¬ italic\_φ, as desired.
For the converse, suppose that W→←1→←→𝑊→1\vec{W}\leftarrow\vec{1}over→ start\_ARG italic\_W end\_ARG ← over→ start\_ARG 1 end\_ARG is an
intervention leading to ¬φ𝜑\neg{\varphi}¬ italic\_φ. Then we must have
(M,0)⊧[W→′←w→′]¬ψmodels𝑀0delimited-[]←superscript→𝑊′superscript→𝑤′𝜓(M,0)\models[\vec{W}^{\prime}\leftarrow\vec{w}^{\prime}]\neg\psi( italic\_M , 0 ) ⊧ [ over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ← over→ start\_ARG italic\_w end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ] ¬ italic\_ψ for all strict
subsets W→′superscript→𝑊′\vec{W}^{\prime}over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of W→→𝑊\vec{W}over→ start\_ARG italic\_W end\_ARG and all settings w→′superscript→𝑤′\vec{w}^{\prime}over→ start\_ARG italic\_w end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of the
variables in W→′superscript→𝑊′\vec{W}^{\prime}over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. Since, in particular, this is true for all
subsets W→′superscript→𝑊′\vec{W}^{\prime}over→ start\_ARG italic\_W end\_ARG start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of W→→𝑊\vec{W}over→ start\_ARG italic\_W end\_ARG that do not involve V′superscript𝑉′V^{\prime}italic\_V start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, it follows
that ¬ψ𝜓\neg\psi¬ italic\_ψ is true for all truth assignments, so ψ∈𝜓absent\psi\initalic\_ψ ∈
UNSAT.
In practice, however, we rarely expect to face the co-NP complexity.
For reasons of cost or practicality, we would expect a policymaker to
consider interventions on at most k𝑘kitalic\_k variables, for some small k𝑘kitalic\_k.
The straightforward algorithm that, for a given k𝑘kitalic\_k,
checks all sets of
variables of the model M𝑀Mitalic\_M of size at most k𝑘kitalic\_k runs in time O(|M|k)𝑂superscript𝑀𝑘O(|M|^{k})italic\_O ( | italic\_M | start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ).
4 Merging Compatible Causal Models
-----------------------------------
This section presents our definition for compatibility of expert opinions.
We consider each expert’s opinion to be represented by a causal model
and, for simplicity, that each expert expresses her opinion with certainty.
(We can easily extend our approach to allow the experts to have some
uncertainty about the correct model; see
the end of Section [5](#S5 "5 Combining Experts’ Opinions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").)
We start with a strong notion of compatibility,
and then consider generalizations of this notion that are more widely
applicable.
###
4.1 Full compatibility
To specify what it means for a set of models to be compatible, we first
define what it means for the causal model
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT to contain at least as much information about variable C𝐶Citalic\_C as
the causal model M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, denoted M1⪰CM2subscriptsucceeds-or-equals𝐶subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
Intuitively, M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT contains at least as much information about C𝐶Citalic\_C as M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT if
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT say the same things about the causal structure of C𝐶Citalic\_C
as far as the variables that M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT share, but M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT contains
(possibly) more information about C𝐶Citalic\_C, because, for example, there are
additional variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT that affect C𝐶Citalic\_C.
We capture this property formally below. We say
that B𝐵Bitalic\_B is an *immediate M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT-ancestor of Y𝑌Yitalic\_Y in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT*
if B∈𝒰2∪𝒱2𝐵subscript𝒰2subscript𝒱2B\in{\cal U}\_{2}\cup{\cal V}\_{2}italic\_B ∈ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, B𝐵Bitalic\_B is an ancestor of Y𝑌Yitalic\_Y in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, and
there is a path from B𝐵Bitalic\_B to Y𝑌Yitalic\_Y in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT that has no nodes in 𝒰2∪𝒱2subscript𝒰2subscript𝒱2{\cal U}\_{2}\cup{\cal V}\_{2}caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT other than B𝐵Bitalic\_B and Y𝑌Yitalic\_Y (if Y∈𝒰2∪𝒱2𝑌subscript𝒰2subscript𝒱2Y\in{\cal U}\_{2}\cup{\cal V}\_{2}italic\_Y ∈ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT).
That is, Y𝑌Yitalic\_Y is the first node in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT after B𝐵Bitalic\_B on a path from B𝐵Bitalic\_B to Y𝑌Yitalic\_Y in
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
######
Definition 4.1.
[Strong domination]
Let M1=((𝒰1,𝒱1,ℛ1),ℱ1)subscript𝑀1subscript𝒰1subscript𝒱1subscriptℛ1subscriptℱ1M\_{1}=(({\cal U}\_{1},{\cal V}\_{1},{\cal R}\_{1}),{\cal F}\_{1})italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , caligraphic\_R start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) and M2=((𝒰2,M\_{2}=(({\cal U}\_{2},italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , 𝒱2,subscript𝒱2{\cal V}\_{2},caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , ℛ2),ℱ2){\cal R}\_{2}),{\cal F}\_{2})caligraphic\_R start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ).
Let 𝑃𝑎𝑟M(C)subscript𝑃𝑎𝑟𝑀𝐶\mathit{Par}\_{M}(C)italic\_Par start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT ( italic\_C ) denote
the variables that are parents of C𝐶Citalic\_C in
M𝑀Mitalic\_M.
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT *strongly dominates M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT with respect to C𝐶Citalic\_C*, denoted
M1⪰CM2subscriptsucceeds-or-equals𝐶subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, if the
following conditions hold:
MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT.
The parents of C𝐶Citalic\_C in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, 𝑃𝑎𝑟M2(C)subscript𝑃𝑎𝑟subscript𝑀2𝐶\mathit{Par}\_{M\_{2}}(C)italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ), are the
immediate M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT-ancestors of C𝐶Citalic\_C in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
MI2M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT.
Every path from an exogenous variable to
C𝐶Citalic\_C in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT goes through a variable in 𝑃𝑎𝑟M2(C)subscript𝑃𝑎𝑟subscript𝑀2𝐶\mathit{Par}\_{M\_{2}}(C)italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ).
MI3M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT.
Let X=((𝒰1∪𝒱1)∩(𝒰2∪𝒱2))−{C}𝑋subscript𝒰1subscript𝒱1subscript𝒰2subscript𝒱2𝐶X=(({\cal U}\_{1}\cup{\cal V}\_{1})\cap({\cal U}\_{2}\cup{\cal V}\_{2}))-\{C\}italic\_X = ( ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ∩ ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ) - { italic\_C }. Then for all settings
x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG of the variables in X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG, all values c𝑐citalic\_c of C𝐶Citalic\_C, all
contexts u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT for M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, and all contexts u→2subscript→𝑢2\vec{u}\_{2}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT for
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
we have
| | | |
| --- | --- | --- |
| | (M1,u→1)⊧[X→←x→](C=c) iff (M2,u→2)⊧[X→←x→](C=c).modelssubscript𝑀1subscript→𝑢1delimited-[]←→𝑋→𝑥𝐶𝑐 iff subscript𝑀2subscript→𝑢2modelsdelimited-[]←→𝑋→𝑥𝐶𝑐\begin{array}[]{l}(M\_{1},\vec{u}\_{1})\models[\vec{X}\leftarrow\vec{x}](C=c)\mbox{ iff }(M\_{2},\vec{u}\_{2})\models[\vec{X}\leftarrow\vec{x}](C=c).\end{array}start\_ARRAY start\_ROW start\_CELL ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) iff ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) . end\_CELL end\_ROW end\_ARRAY | |
If MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT holds and, for example, B𝐵Bitalic\_B is a parent
of C𝐶Citalic\_C in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
then there may be a variable B′superscript𝐵′B^{\prime}italic\_B start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT on the path from B𝐵Bitalic\_B to C𝐶Citalic\_C in
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. Thus, M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT has in a sense more detailed information than
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT about the causal paths leading to C𝐶Citalic\_C.
MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT is not by itself
enough to say that M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT agree on
the causal relations for C𝐶Citalic\_C. This is guaranteed by
MI2M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT and MI3M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT. MI2M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT says that the
variables in 𝑃𝑎𝑟M2(C)subscript𝑃𝑎𝑟subscript𝑀2𝐶\mathit{Par}\_{M\_{2}}(C)italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ) screen off
C𝐶Citalic\_C from the exogenous variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. (Clearly the variables
in 𝑃𝑎𝑟M2(C)subscript𝑃𝑎𝑟subscript𝑀2𝐶\mathit{Par}\_{M\_{2}}(C)italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ) also screen off C𝐶Citalic\_C from the exogenous variables in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.)
It follows that if (M1,u→1)⊧[𝑃𝑎𝑟M2(C)←x→](C=c)modelssubscript𝑀1subscript→𝑢1delimited-[]←subscript𝑃𝑎𝑟subscript𝑀2𝐶→𝑥𝐶𝑐(M\_{1},\vec{u}\_{1})\models[\mathit{Par}\_{M\_{2}}(C)\leftarrow\vec{x}](C=c)( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ⊧ [ italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ) ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) for some context u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, then
(M1,u→)⊧[𝑃𝑎𝑟M2(C)←x→](C=c)modelssubscript𝑀1→𝑢delimited-[]←subscript𝑃𝑎𝑟subscript𝑀2𝐶→𝑥𝐶𝑐(M\_{1},\vec{u})\models[\mathit{Par}\_{M\_{2}}(C)\leftarrow\vec{x}](C=c)( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ [ italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ) ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) for all contexts
u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, and similarly for M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. In light of this
observation, it follows that
MI3M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT assures us that C𝐶Citalic\_C satisfies the same
causal relations in both models.
We write M1⋡CM2subscriptnot-succeeds-or-equals𝐶subscript𝑀1subscript𝑀2M\_{1}\not\succeq\_{C}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⋡ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT if any of the conditions
above does not hold.
Two technical comments regarding Definition [4.1](#S4.Thmtheorem1 "Definition 4.1. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."):
First, note that in MI3 we used the fact that we allow the X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG in
formulas of the form [X→←x→]φdelimited-[]←→𝑋→𝑥𝜑[\vec{X}\leftarrow\vec{x}]{\varphi}[ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] italic\_φ to include exogenous
variables, since some of the parents of C𝐶Citalic\_C may be exogenous.
Second, despite the suggestive notation, ⪰Csubscriptsucceeds-or-equals𝐶\succeq\_{C}⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT is not a partial
order. In particular, it is not hard to construct examples showing
that it is not transitive. However, ⪰Csubscriptsucceeds-or-equals𝐶\succeq\_{C}⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT
is a partial order on compatible models (see the proof of
Proposition [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")), which is the only context in which
we are interested in transitivity, so the abuse of notation is
somewhat justified.
Note that we have a relation ⪰Csubscriptsucceeds-or-equals𝐶\succeq\_{C}⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT for each variable C𝐶Citalic\_C
that appears in both M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
Model M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT may be more informative than M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT with respect to C𝐶Citalic\_C
whereas M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT may be more informative than M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT with respect to another
variable C′superscript𝐶′C^{\prime}italic\_C start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT.
Roughly speaking, M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are *fully
compatible* if for each variable C∈𝒱1∩𝒱2𝐶subscript𝒱1subscript𝒱2C\in{\cal V}\_{1}\cap{\cal V}\_{2}italic\_C ∈ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, either M1⪰CM2subscriptsucceeds-or-equals𝐶subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT or M2⪰CM1subscriptsucceeds-or-equals𝐶subscript𝑀2subscript𝑀1M\_{2}\succeq\_{C}M\_{1}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
We then merge M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT by taking
the equations for C𝐶Citalic\_C to be determined
by the model that has more information about C𝐶Citalic\_C.
Consider another example demonstrating the notion of strong dominance,
taken from Bradley et al. ([2014](#bib.bib1)).
######
Example 4.2.
Bradley et al. ([2014](#bib.bib1))
An aid agency consults two experts about causes of famine in a region.
Both experts agree that the amount of rainfall (R𝑅Ritalic\_R) affects
crop yield (Y𝑌Yitalic\_Y). Specifically, a shortage of rainfall leads to poor
crop yield.
Expert 2222 says that political conflict (P𝑃Pitalic\_P) can also directly
affect famine. Expert 1, on the other hand, says that P𝑃Pitalic\_P affects
F𝐹Fitalic\_F only via Y𝑌Yitalic\_Y.
The experts’ causal graphs are depicted in Figure [2](#S4.F2 "Figure 2 ‣ Example 4.2. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), where
the graph on the left, M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, describes expert 1’s model, while
the graph on the right, M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, describes expert 2’s model.
These graphs already appear in
the work of BDL.
In these graphs (as in many other causal
graphs in the literature),
the exogenous variables are omitted; all the
variables are taken to be endogenous.
R𝑅Ritalic\_RP𝑃Pitalic\_PY𝑌Yitalic\_YF𝐹Fitalic\_FM1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPTR𝑅Ritalic\_RP𝑃Pitalic\_PY𝑌Yitalic\_YF𝐹Fitalic\_FM2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
Figure 2: Two expert’s models of famine.
Neither MI1M1,M2,Fsubscript𝑀1subscript𝑀2𝐹{}\_{M\_{1},M\_{2},F}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_F end\_FLOATSUBSCRIPT nor MI1M2,M1,Fsubscript𝑀2subscript𝑀1𝐹{}\_{M\_{2},M\_{1},F}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_F end\_FLOATSUBSCRIPT holds, since P𝑃Pitalic\_P is not
an M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT-immediate ancestor of F𝐹Fitalic\_F in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. Similarly,
neither MI1M1,M2,Ysubscript𝑀1subscript𝑀2𝑌{}\_{M\_{1},M\_{2},Y}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_Y end\_FLOATSUBSCRIPT nor MI1M2,M1,Ysubscript𝑀2subscript𝑀1𝑌{}\_{M\_{2},M\_{1},Y}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_Y end\_FLOATSUBSCRIPT holds, since P𝑃Pitalic\_P is
not an M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT-immediate
ancestor of Y𝑌Yitalic\_Y in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT (indeed, it is not an ancestor at all).
MI2M1,M2,Fsubscript𝑀1subscript𝑀2𝐹{}\_{M\_{1},M\_{2},F}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_F end\_FLOATSUBSCRIPT holds since every path in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT from an exogenous
variable to F𝐹Fitalic\_F goes through a variable
that is a parent of F𝐹Fitalic\_F in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT (namely, Y𝑌Yitalic\_Y);
MI2M2,M1,Fsubscript𝑀2subscript𝑀1𝐹{}\_{M\_{2},M\_{1},F}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_F end\_FLOATSUBSCRIPT does not hold (there is a path in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT to F𝐹Fitalic\_F
via P𝑃Pitalic\_P that does not go through a parent of F𝐹Fitalic\_F in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT).
Although we are not given the equations, we also know that
MI3M1,M2,Fsubscript𝑀1subscript𝑀2𝐹{}\_{M\_{1},M\_{2},F}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_F end\_FLOATSUBSCRIPT does not hold. Since P𝑃Pitalic\_P is a parent of
F𝐹Fitalic\_F in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT according to expert 2, there must be a setting y𝑦yitalic\_y of Y𝑌Yitalic\_Y
such that the value of F𝐹Fitalic\_F changes depending on the value of P𝑃Pitalic\_P if Y=y𝑌𝑦Y=yitalic\_Y = italic\_y.
This cannot be the case in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, since Y𝑌Yitalic\_Y screens off P𝑃Pitalic\_P from
F𝐹Fitalic\_F. It easily follows that taking X→=(P,Y)→𝑋𝑃𝑌\vec{X}=(P,Y)over→ start\_ARG italic\_X end\_ARG = ( italic\_P , italic\_Y ) we get a
counterexample to MI3M1,M2,Fsubscript𝑀1subscript𝑀2𝐹{}\_{M\_{1},M\_{2},F}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_F end\_FLOATSUBSCRIPT.
Therefore, we have neither M1⪰FM2subscriptsucceeds-or-equals𝐹subscript𝑀1subscript𝑀2M\_{1}\succeq\_{F}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT nor
M2⪰FM1subscriptsucceeds-or-equals𝐹subscript𝑀2subscript𝑀1M\_{2}\succeq\_{F}M\_{1}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
While the definition of dominance given above is
useful, it does not cover all cases where a policymaker may want to merge
models. Consider the following example, taken from the work of
Sampson et al. ([2013](#bib.bib24)).
######
Example 4.3.
Two experts have provided causal models regarding the
causes of domestic violence.
According to the first expert, an appropriate arrest policy (AP)
may affect
both an offender’s belief that his partner would report
any abuse to police (PLS)
and the amount of
domestic violence (DV).
The amount of domestic violence also affects the likelihood of
a victim calling to report abuse (C), which in turn
affects the likelihood of there being
a random arrest (A). (Decisions on whether to arrest the offender in
cases of domestic violence were randomized.)
According to the second expert,
DV affects A directly, while A affects the amount of
repeated violence (RV) through both formal sanction (FS) and
informal sanction on socially embedded individuals (IS).
Sampson et al. ([2013](#bib.bib24)) use the causal
graphs shown in Figure [3](#S4.F3 "Figure 3 ‣ Example 4.3. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), which are annotated with the
direction of the influence
(the only information provided by the experts) to describe the expert’s
opinions.
APDV+++−--CAM1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT+++−--PLS−--−--RVM2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPTDV+++FS−--+++IS−--A
Figure 3: Experts’ models of domestic violence.
For the two common variables DV and A, MI1M1,M2,DVsubscript𝑀1subscript𝑀2𝐷𝑉{}\_{M\_{1},M\_{2},DV}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_D italic\_V end\_FLOATSUBSCRIPT
and MI1M1,M2,Asubscript𝑀1subscript𝑀2𝐴{}\_{M\_{1},M\_{2},A}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_A end\_FLOATSUBSCRIPT both hold.
If the only variables that have exogenous parents are AP in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and
DV in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and the set of parents of AP in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT is a subset
of the set of parents of DV in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, then MI2M1,M2,DVsubscript𝑀1subscript𝑀2𝐷𝑉{}\_{M\_{1},M\_{2},DV}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_D italic\_V end\_FLOATSUBSCRIPT holds.
Sampson et al. seem to be implicitly assuming this, and that MI3 holds, so they
merge M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT to get the
causal graph shown in Figure [4](#S4.F4 "Figure 4 ‣ Example 4.3. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
−--−--PLS−--DV+++−--+++CAFSIS++++++−--−--RV
Figure 4: The result of merging experts’ model of domestic violence.
Sampson et al. do not provide structural equations. Moreover, for
edges that do not have a +++ or −-- annotation, such as the edge from
DV to A𝐴Aitalic\_A in Figure [3](#S4.F3 "Figure 3 ‣ Example 4.3. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), we do not even know qualitatively
what the impact of interventions is.
Presumably, the lack of annotation represents the expert’s uncertainty.
We can capture this uncertainty by viewing the
expert as having a
probability on two models that disagree on the direction of DV’s
influence on A𝐴Aitalic\_A
(and thus are incompatible because they disagree
on the equations). We discuss in Section [5](#S5 "5 Combining Experts’ Opinions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") how
such uncertainty can be handled.
Suppose that some parent of AP
(or AP itself)
in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT is not a
parent of DV in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. Then,
in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, it may be possible to change the value of DV by intervening
on AP, while keeping the values of all the exogenous variables that
are
parents of DV in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT fixed.
This will seem like an
inexplicable change in the value of DV from the perspective of the
second expert.
If the second expert had been aware of such
possible changes, she
surely would have added additional variables to M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT to capture this
situation. One explanation of the fact that no changes were
observed is that the second expert was
working in a setting where the values of all variables that she cannot
affect by an intervention are determined by some default setting of
exogenous variables of which she is not aware (or not modeling).
Given models M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
we now want to define a notion of weak domination relative to a
default setting v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT of the exogenous variables in 𝒰1∪𝒰2subscript𝒰1subscript𝒰2{\cal U}\_{1}\cup{\cal U}\_{2}caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. We say that contexts
u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT for M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and u→2subscript→𝑢2\vec{u}\_{2}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT for
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are *compatible with v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT* if u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and
u→2subscript→𝑢2\vec{u}\_{2}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT agree on the variables
in 𝒰1∩𝒰2subscript𝒰1subscript𝒰2{\cal U}\_{1}\cap{\cal U}\_{2}caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT agrees with v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT on the
variables in 𝒰1−𝒰2subscript𝒰1subscript𝒰2{\cal U}\_{1}-{\cal U}\_{2}caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT - caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and u→2subscript→𝑢2\vec{u}\_{2}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT agrees with
v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT on the variables in 𝒰2−𝒰1subscript𝒰2subscript𝒰1{\cal U}\_{2}-{\cal U}\_{1}caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT - caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
######
Definition 4.4.
[Domination relative to a
default setting]
Let v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT be a default setting for the variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT *dominates* M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT with respect to C𝐶Citalic\_C
relative to v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, denoted M1⪰Cv→\*M2superscriptsubscriptsucceeds-or-equals𝐶superscriptnormal-→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, if
MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT holds,
and, in addition,
the following condition (which can be viewed as a replacement for
MI2M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT and MI3M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT) holds:
MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscriptnormal-→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT
Let X→=(𝒰1∪𝒱1)∩(𝒰2∪𝒱2)−{C}→𝑋subscript𝒰1subscript𝒱1subscript𝒰2subscript𝒱2𝐶\vec{X}=({\cal U}\_{1}\cup{\cal V}\_{1})\cap({\cal U}\_{2}\cup{\cal V}\_{2})-\{C\}over→ start\_ARG italic\_X end\_ARG = ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ∩ ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) - { italic\_C }. Then for all settings
x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG of the variables in X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG, all values c𝑐citalic\_c of C𝐶Citalic\_C, and
all contexts u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT for M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and u→2subscript→𝑢2\vec{u}\_{2}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT for
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
that are compatible with v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, we have
| | | |
| --- | --- | --- |
| | (M1,u→1)⊧[X→←x→](C=c) iff (M2,u→2)⊧[X→←x→](C=c).modelssubscript𝑀1subscript→𝑢1delimited-[]←→𝑋→𝑥𝐶𝑐 iff subscript𝑀2subscript→𝑢2modelsdelimited-[]←→𝑋→𝑥𝐶𝑐\begin{array}[]{l}(M\_{1},\vec{u}\_{1})\models[\vec{X}\leftarrow\vec{x}](C=c)\mbox{ iff }(M\_{2},\vec{u}\_{2})\models[\vec{X}\leftarrow\vec{x}](C=c).\end{array}start\_ARRAY start\_ROW start\_CELL ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) iff ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) . end\_CELL end\_ROW end\_ARRAY | |
It is easy to see that ⪰Csubscriptsucceeds-or-equals𝐶\succeq\_{C}⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT is a special case of
⪰Cv→\*superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣\succeq\_{C}^{\vec{v}^{\*}}⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT:
######
Lemma 4.5.
If M1⪰CM2subscriptsucceeds-or-equals𝐶subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, then, for all
default settings v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT of the variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, we have
M1⪰Cv→\*M2superscriptsubscriptsucceeds-or-equals𝐶superscriptnormal-→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
Proof. Suppose that M1⪰Cℳ2subscriptsucceeds-or-equals𝐶subscript𝑀1subscriptℳ2M\_{1}\succeq\_{C}{\cal M}\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT caligraphic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. Fix default values
v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. Clearly MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT is a special case
of MI2. Thus, M1⪰Cv→\*ℳ2superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣subscript𝑀1subscriptℳ2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}{\cal M}\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT caligraphic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
In light of Lemma [4.5](#S4.Thmtheorem5 "Lemma 4.5. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), we give all the definitions
in the remainder of the paper using ⪰Cv→\*superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣\succeq\_{C}^{\vec{v}^{\*}}⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT.
All the technical results hold if we
replace ⪰Cv→\*superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣\succeq\_{C}^{\vec{v}^{\*}}⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT by ⪰Csubscriptsucceeds-or-equals𝐶\succeq\_{C}⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT throughout.
######
Definition 4.6.
[Full compatibility of causal
models]
If M1=((𝒰1,𝒱1,ℛ1),ℱ1)subscript𝑀1subscript𝒰1subscript𝒱1subscriptℛ1subscriptℱ1M\_{1}=(({\cal U}\_{1},{\cal V}\_{1},{\cal R}\_{1}),{\cal F}\_{1})italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , caligraphic\_R start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) and M2=((𝒰2,𝒱2,M\_{2}=(({\cal U}\_{2},{\cal V}\_{2},italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , ℛ2),ℱ2){\cal R}\_{2}),{\cal F}\_{2})caligraphic\_R start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ),
then M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are *fully compatible with respect to
default setting v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT*
if (1) for all variables C∈(𝒰1∪𝒱1)∩(𝒰2∪𝒱2)𝐶subscript𝒰1subscript𝒱1subscript𝒰2subscript𝒱2C\in({\cal U}\_{1}\cup{\cal V}\_{1})\cap({\cal U}\_{2}\cup{\cal V}\_{2})italic\_C ∈ ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ∩ ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ), we have ℛ1(C)=ℛ2(C)subscriptℛ1𝐶subscriptℛ2𝐶{\cal R}\_{1}(C)={\cal R}\_{2}(C)caligraphic\_R start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_C ) = caligraphic\_R start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C ) and (2) for all variables
C∈(𝒱1∩𝒱2)∪(𝒱1∩𝒰2)∪(𝒱2∩𝒰1)𝐶subscript𝒱1subscript𝒱2subscript𝒱1subscript𝒰2subscript𝒱2subscript𝒰1C\in({\cal V}\_{1}\cap{\cal V}\_{2})\cup({\cal V}\_{1}\cap{\cal U}\_{2})\cup({\cal V}\_{2}\cap{\cal U}\_{1})italic\_C ∈ ( caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ∪ ( caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ∪ ( caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∩ caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ),
either M1⪰Cv→\*M2superscriptsubscriptsucceeds-or-equals𝐶superscriptnormal-→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT or M2⪰Cv→\*M1superscriptsubscriptsucceeds-or-equals𝐶superscriptnormal-→𝑣subscript𝑀2subscript𝑀1M\_{2}\succeq\_{C}^{\vec{v}^{\*}}M\_{1}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
We say that M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are *fully compatible* if 𝒰1=𝒰2subscript𝒰1subscript𝒰2{\cal U}\_{1}={\cal U}\_{2}caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT (so we can ignore the default setting).
######
Definition 4.7.
[Merging fully compatible
models]
If M1=((𝒰1,𝒱1,ℛ1),ℱ1)subscript𝑀1subscript𝒰1subscript𝒱1subscriptℛ1subscriptℱ1M\_{1}=(({\cal U}\_{1},{\cal V}\_{1},{\cal R}\_{1}),{\cal F}\_{1})italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , caligraphic\_R start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) and M2=((𝒰2,𝒱2,ℛ2),ℱ2)subscript𝑀2subscript𝒰2subscript𝒱2subscriptℛ2subscriptℱ2M\_{2}=(({\cal U}\_{2},{\cal V}\_{2},{\cal R}\_{2}),{\cal F}\_{2})italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , caligraphic\_R start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) are
fully compatible with respect to v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, then the merged model
M1⊕v→\*M2superscriptdirect-sumsuperscriptnormal-→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is the causal
model
((𝒰,𝒱,ℛ),ℱ)𝒰𝒱ℛℱ(({\cal U},{\cal V},{\cal R}),{\cal F})( ( caligraphic\_U , caligraphic\_V , caligraphic\_R ) , caligraphic\_F ), where
* 1.
𝒰=𝒰1∪𝒰2−(𝒱1∪𝒱2)𝒰subscript𝒰1subscript𝒰2subscript𝒱1subscript𝒱2{\cal U}={\cal U}\_{1}\cup{\cal U}\_{2}-({\cal V}\_{1}\cup{\cal V}\_{2})caligraphic\_U = caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT - ( caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT );
* 2.
𝒱=𝒱1∪𝒱2𝒱subscript𝒱1subscript𝒱2{\cal V}={\cal V}\_{1}\cup{\cal V}\_{2}caligraphic\_V = caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT;
* 3.
if C∈𝒰1∪𝒱1𝐶subscript𝒰1subscript𝒱1C\in{\cal U}\_{1}\cup{\cal V}\_{1}italic\_C ∈ caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, then ℛ(C)=ℛ1(C)ℛ𝐶subscriptℛ1𝐶{\cal R}(C)={\cal R}\_{1}(C)caligraphic\_R ( italic\_C ) = caligraphic\_R start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_C ), and if C∈𝒰2∪𝒱2𝐶subscript𝒰2subscript𝒱2C\in{\cal U}\_{2}\cup{\cal V}\_{2}italic\_C ∈ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
then ℛ(C)=ℛ2(C)ℛ𝐶subscriptℛ2𝐶{\cal R}(C)={\cal R}\_{2}(C)caligraphic\_R ( italic\_C ) = caligraphic\_R start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C );
* 4.
if C∈𝒱1−𝒱2𝐶subscript𝒱1subscript𝒱2C\in{\cal V}\_{1}-{\cal V}\_{2}italic\_C ∈ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT - caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT or if both C∈𝒱1∩𝒱2𝐶subscript𝒱1subscript𝒱2C\in{\cal V}\_{1}\cap{\cal V}\_{2}italic\_C ∈ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
M1⪰Cv→\*M2superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
then ℱ(C)=ℱ1(C)ℱ𝐶subscriptℱ1𝐶{\cal F}(C)={\cal F}\_{1}(C)caligraphic\_F ( italic\_C ) = caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_C );
if C∈𝒱2−𝒱1𝐶subscript𝒱2subscript𝒱1C\in{\cal V}\_{2}-{\cal V}\_{1}italic\_C ∈ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT - caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT or if both C∈𝒱1∩𝒱2𝐶subscript𝒱1subscript𝒱2C\in{\cal V}\_{1}\cap{\cal V}\_{2}italic\_C ∈ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
M2⪰Cv→\*M1superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣subscript𝑀2subscript𝑀1M\_{2}\succeq\_{C}^{\vec{v}^{\*}}M\_{1}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT,
then ℱ(C)=ℱ2(C)ℱ𝐶subscriptℱ2𝐶{\cal F}(C)={\cal F}\_{2}(C)caligraphic\_F ( italic\_C ) = caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C ).666We are abusing notation here and viewing ℱi(C)subscriptℱ𝑖𝐶{\cal F}\_{i}(C)caligraphic\_F start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( italic\_C ) as a
function from the
values of the parents of C𝐶Citalic\_C in Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT to the value of C𝐶Citalic\_C, as opposed to a
function from all the values of all variables other than C𝐶Citalic\_C to the value
of C𝐶Citalic\_C.
We write M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT if 𝒰1=𝒰2subscript𝒰1subscript𝒰2{\cal U}\_{1}={\cal U}\_{2}caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
Note that we assume that when experts use the same variable, they are
referring to the same phenomenon within the same domain. Our approach does
not deal with the possibility of two experts using the same variable
name to refer to different phenomenon.
Returning to Example [4.3](#S4.Thmtheorem3 "Example 4.3. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), assume that either
MI2M1,M2,DVsubscript𝑀1subscript𝑀2𝐷𝑉{}\_{M\_{1},M\_{2},DV}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_D italic\_V end\_FLOATSUBSCRIPT, MI2M1,M2,Asubscript𝑀1subscript𝑀2𝐴{}\_{M\_{1},M\_{2},A}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_A end\_FLOATSUBSCRIPT,
MI3M1,M2,DVsubscript𝑀1subscript𝑀2𝐷𝑉{}\_{M\_{1},M\_{2},DV}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_D italic\_V end\_FLOATSUBSCRIPT, and MI3M1,M2,Asubscript𝑀1subscript𝑀2𝐴{}\_{M\_{1},M\_{2},A}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_A end\_FLOATSUBSCRIPT all hold, or
there is a default setting v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT such that
Then M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT has the causal graph described in
Figure [4](#S4.F4 "Figure 4 ‣ Example 4.3. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."); that is, even though
Sampson et al. ([2013](#bib.bib24)) do not have a formal theory for
merging models, they actually merge models in just the way
that we are suggesting.
Let M1∼Cv→\*M2superscriptsubscriptsimilar-to𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\sim\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∼ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT be an abbreviation for
M1⪰Cv→\*M2superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
M2⪰Cv→\*M1superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣subscript𝑀2subscript𝑀1M\_{2}\succeq\_{C}^{\vec{v}^{\*}}M\_{1}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
We also write M1≻Cv→\*M2superscriptsubscriptsucceeds𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succ\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ≻ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT if
M1⪰Cv→\*M2superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
M2⋡Cv→\*M1superscriptsubscriptnot-succeeds-or-equals𝐶superscript→𝑣subscript𝑀2subscript𝑀1M\_{2}\not\succeq\_{C}^{\vec{v}^{\*}}M\_{1}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⋡ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
The next theorem
provides evidence that Definition [4.6](#S4.Thmtheorem6 "Definition 4.6. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")
is reasonable
and captures our intuitions. Part (b) says that it is well defined,
so that in the clauses in the definition where there might be
potential conflict, such as in the definition of ℱ(C)ℱ𝐶{\cal F}(C)caligraphic\_F ( italic\_C ) when C∈𝒱1∩𝒱2𝐶subscript𝒱1subscript𝒱2C\in{\cal V}\_{1}\cap{\cal V}\_{2}italic\_C ∈ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and M1∼Cv→\*M2superscriptsubscriptsimilar-to𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\sim\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∼ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
there is in fact no conflict;
part (a) is a technical result
needed to prove part (b).
Part (c) says that the merged model
is guaranteed to be acyclic.
Part (d) says that causal paths in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT are preserved in M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, while
part (e) says that at least as far
as formulas involving the variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT go, M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
and M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT agree, provided that the default values are used for the
exogenous variables not in 𝒰1∩𝒰2subscript𝒰1subscript𝒰2{\cal U}\_{1}\cap{\cal U}\_{2}caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
Parts (d) and (e) can be viewed as saying that the essential
causal structure of M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is preserved in M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. All conclusions that can be drawn in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT individually can be drawn in M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
(In the language of Halpern ([2016b](#bib.bib12)), part (e) says that M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is essentially a conservative extension of M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.)
But it is important to note that M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT lets us go beyond
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, since we can, for example, consider interventions
that simultaneously affect variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT that are not in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
variables in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT that are not in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
Finally, parts (f) and (g) say
that ⊕v→\*superscriptdirect-sumsuperscript→𝑣\oplus^{\vec{v}^{\*}}⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT is commutative and associative over its domain.
###### Theorem 1
Suppose that M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and M3subscript𝑀3M\_{3}italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT are pairwise fully
compatible with respect to
default setting v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Then
the following conditions hold.
* (a)
If M1∼Cv→\*M2superscriptsubscriptsimilar-to𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\sim\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∼ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT then
(i) 𝑃𝑎𝑟M1(C)=𝑃𝑎𝑟M2(C)subscript𝑃𝑎𝑟subscript𝑀1𝐶subscript𝑃𝑎𝑟subscript𝑀2𝐶\mathit{Par}\_{M\_{1}}(C)=\mathit{Par}\_{M\_{2}}(C)italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ) = italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ) and (ii) ℱ1(C)=ℱ2(C)subscriptℱ1𝐶subscriptℱ2𝐶{\cal F}\_{1}(C)={\cal F}\_{2}(C)caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_C ) = caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C );
* (b)
M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is well defined.
* (c)
M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is acyclic.
* (d)
If A𝐴Aitalic\_A and B𝐵Bitalic\_B are variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, then A𝐴Aitalic\_A is an
ancestor of B𝐵Bitalic\_B in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT iff A𝐴Aitalic\_A is an ancestor of B𝐵Bitalic\_B in M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
* (e)
If φ𝜑\varphiitalic\_φ is a formula that mentions only the endogenous
variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG is a context for M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT is a context for M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT,
and u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG and u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT are compatible with v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT,
then
(M1,u→1)⊧φmodelssubscript𝑀1subscript→𝑢1𝜑(M\_{1},\vec{u}\_{1})\models\varphi( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ⊧ italic\_φ iff (M1⊕M2,u→)⊧φmodelsdirect-sumsubscript𝑀1subscript𝑀2→𝑢𝜑(M\_{1}\oplus M\_{2},\vec{u})\models\varphi( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_φ.
* (f)
M1⊕v→\*M2=M2⊕v→\*M1superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2superscriptdirect-sumsuperscript→𝑣subscript𝑀2subscript𝑀1M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}=M\_{2}\oplus^{\vec{v}^{\*}}M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
* (g)
If M3subscript𝑀3M\_{3}italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT is fully compatible with M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT with respect to v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT and M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
is fully compatible with M2⊕v→\*M3superscriptdirect-sumsuperscript→𝑣subscript𝑀2subscript𝑀3M\_{2}\oplus^{\vec{v}^{\*}}M\_{3}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT with respect to v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, then
M1⊕v→\*(M2⊕v→\*M3)=(M1⊕v→\*M2)⊕v→\*M3superscriptdirect-sumsuperscript→𝑣subscript𝑀1superscriptdirect-sumsuperscript→𝑣subscript𝑀2subscript𝑀3superscriptdirect-sumsuperscript→𝑣superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2subscript𝑀3M\_{1}\oplus^{\vec{v}^{\*}}(M\_{2}\oplus^{\vec{v}^{\*}}M\_{3})=(M\_{1}\oplus^{\vec{v}^{\*}}M\_{2})\oplus^{\vec{v}^{\*}}M\_{3}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ) = ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT.
The proof of Theorem [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")
is rather involved; the details can be found in [A](#A1 "Appendix A Proof of Proposition 1 ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
We define what it means for a collection ℳ={M1,…,Mn}ℳsubscript𝑀1…subscript𝑀𝑛{\cal M}=\{M\_{1},\ldots,M\_{n}\}caligraphic\_M = { italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT } of causal
models to be *mutually compatible with respect to default setting
v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT (for all the exogenous variables that are not common to
M1,…,Mnsubscript𝑀1normal-…subscript𝑀𝑛M\_{1},\ldots,M\_{n}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT)* by induction on the
cardinality of ℳℳ{\cal M}caligraphic\_M.
If |ℳ|=1ℳ1|{\cal M}|=1| caligraphic\_M | = 1, then mutual compatibility trivially
holds. If |ℳ|=2ℳ2|{\cal M}|=2| caligraphic\_M | = 2, then the models in ℳℳ{\cal M}caligraphic\_M are mutually compatible
with respect to v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT if
they are fully compatible with respect to v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT according to
Definition [4.6](#S4.Thmtheorem6 "Definition 4.6. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). If |ℳ|=nℳ𝑛|{\cal M}|=n| caligraphic\_M | = italic\_n,
then the models in ℳℳ{\cal M}caligraphic\_M are mutually compatible with respect to
v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT if the models in every
subset of ℳℳ{\cal M}caligraphic\_M of cardinality n−1𝑛1n-1italic\_n - 1 are mutually compatible with
respect to v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, and for
each model M∈ℳ𝑀ℳM\in{\cal M}italic\_M ∈ caligraphic\_M, M𝑀Mitalic\_M is fully compatible with
⊕M′≠Mv→\*M′subscriptsuperscriptdirect-sumsuperscript→𝑣superscript𝑀′𝑀superscript𝑀′\oplus^{\vec{v}^{\*}}\_{M^{\prime}\neq M}M^{\prime}⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_M start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_M end\_POSTSUBSCRIPT italic\_M start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT with respect to v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
By Theorem [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), if M1,…,Mnsubscript𝑀1…subscript𝑀𝑛M\_{1},\ldots,M\_{n}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT are mutually compatible with respect to v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, then the causal model
M1⊕v→\*⋯⊕v→\*Mnsuperscriptdirect-sumsuperscript→𝑣superscriptdirect-sumsuperscript→𝑣subscript𝑀1⋯subscript𝑀𝑛M\_{1}\oplus^{\vec{v}^{\*}}\cdots\oplus^{\vec{v}^{\*}}M\_{n}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT ⋯ ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT is well defined;
we do not have to worry about parenthesization, nor the order in which
the settings are combined. Thus, the model ⊕M′≠Mv→\*M′subscriptsuperscriptdirect-sumsuperscript→𝑣superscript𝑀′𝑀superscript𝑀′\oplus^{\vec{v}^{\*}}\_{M^{\prime}\neq M}M^{\prime}⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_M start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_M end\_POSTSUBSCRIPT italic\_M start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT
considered in the definition is also well defined.
Theorem [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")(e) also tells us that M1⊕v→\*⋯⊕v→\*Mnsuperscriptdirect-sumsuperscript→𝑣superscriptdirect-sumsuperscript→𝑣subscript𝑀1⋯subscript𝑀𝑛M\_{1}\oplus^{\vec{v}^{\*}}\cdots\oplus^{\vec{v}^{\*}}M\_{n}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT ⋯ ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT
contains, in a precise sense, at least as much information as each
model Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT individually.
Thus, by merging mutually compatible models, we are maximizing our
use of information.
This approach to merging models is our main contribution.
Using it, we
show in Section [5](#S5 "5 Combining Experts’ Opinions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") how
experts’ models can be combined to reason about interventions.
We now discuss the extent to which our approach to merging models
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT satisfies BDL’s desiderata.
Recall that BDL considered
only causal networks, not
causal models in our sense; they also assume that all models mention
the same set of variables. They consider four desiderata. We
briefly describe them and their status in our setting.
Since BDL do not consider contexts explicity, we assume for simplicity
in this discussion that the context is the same for all models, and
talk only about ⊕direct-sum\oplus⊕ rather than ⊕v→\*superscriptdirect-sumsuperscript→𝑣\oplus^{\vec{v}^{\*}}⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT.
* 1.
*Universal Domain*: the rule for combining models accepts
all possible inputs and can output any logically possible model.
This clearly holds for us.
* 2.
*Acyclicity*: the result of merging M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is
acyclic. This follows from Theorem [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")(c),
provided that M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is defined.
* 3.
*Unbiasedness*: if M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is defined, and M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT mention the same variables, then
whether B𝐵Bitalic\_B is a parent of C𝐶Citalic\_C in M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT depends only on whether B𝐵Bitalic\_B is a parent of C𝐶Citalic\_C in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
and in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. This property holds trivially for us, since if B𝐵Bitalic\_B
and C𝐶Citalic\_C are in
both M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is defined, then B𝐵Bitalic\_B is a
parent of C𝐶Citalic\_C in M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT iff B𝐵Bitalic\_B is a parent of C𝐶Citalic\_C in both
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
(The version of this requirement given by BDL does not say “if M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is defined”,
since they assume that arbitrary models can be merged.) BDL also
have a *neutrality* requirement as part of
unbiasedness. Unfortunately, an aggregation rule that says that B𝐵Bitalic\_B is
a parent of C𝐶Citalic\_C in M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT iff B𝐵Bitalic\_B is a parent of C𝐶Citalic\_C in
both M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
(which seems quite reasonable to us)
is not neutral in their sense, so we do not
satisfy neutrality
(nor, in light of the observation above, do we consider it a
reasonble requirement to satisfy).
* 4.
*Non-dictatorship:* no single expert determines the
aggregation. This clearly holds for us.
We conclude this section with a characterization of the complexity of
determining whether two causal models are fully compatible.
Determining whether M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are fully
compatible requires checking whether the conditions of
Definition [4.6](#S4.Thmtheorem6 "Definition 4.6. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") hold.
This amounts to checking the conditions MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT and
MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT for all variables
C∈(𝒰1∪𝒱1)∩(𝒰2∪𝒱2)𝐶subscript𝒰1subscript𝒱1subscript𝒰2subscript𝒱2C\in({\cal U}\_{1}\cup{\cal V}\_{1})\cap({\cal U}\_{2}\cup{\cal V}\_{2})italic\_C ∈ ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ∩ ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ).
How hard this is to do depends in part on how the models are presented.
If the models are presented *explicitly*, which means that, for
each variable C𝐶Citalic\_C, the
equation for C𝐶Citalic\_C is described as a (huge) table, giving the value of
C𝐶Citalic\_C for each possible setting of all the other variables, the problem
is *polynomial* in the sizes of the input models.
However, the size of the model will be exponential in the number of
variables.
In this case, checking whether
MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT holds
for all C𝐶Citalic\_C
amounts to
checking whether
the parents of C𝐶Citalic\_C in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are the immediate M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT-ancestors of C𝐶Citalic\_C
in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
To solve this, we need to determine, for each pair of endogenous
variables X𝑋Xitalic\_X and Y𝑌Yitalic\_Y in Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for i=1,2𝑖12i=1,2italic\_i = 1 , 2, whether X𝑋Xitalic\_X depends on Y𝑌Yitalic\_Y.
With this information, we can construct the causal graphs
for M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and then quickly determine whether
MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT holds.
If the model is given explicitly, then
determining wheter X𝑋Xitalic\_X depends on Y𝑌Yitalic\_Y amounts to finding two
rows in the table of values of FXsubscript𝐹𝑋F\_{X}italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT that differ only in the value of
Y𝑌Yitalic\_Y and
in the outcome. As the number of pairs of rows is quadratic in the
size of the table, this is polynomial in the size of the input.
Thus, we can determine if MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT holds in
polynomial time.
Checking if MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT holds
amounts to checking whether
| | | |
| --- | --- | --- |
| | (M1,u→1)⊧[X→←x→](C=c),modelssubscript𝑀1subscript→𝑢1delimited-[]←→𝑋→𝑥𝐶𝑐(M\_{1},\vec{u}\_{1})\models[\vec{X}\leftarrow\vec{x}](C=c),( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) , | |
iff
| | | |
| --- | --- | --- |
| | (M2,u→2)⊧[X→←x→](C=c).modelssubscript𝑀2subscript→𝑢2delimited-[]←→𝑋→𝑥𝐶𝑐(M\_{2},\vec{u}\_{2})\models[\vec{X}\leftarrow\vec{x}](C=c).( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) . | |
For a specific setting u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG and choice of X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG and
x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG, we can easily compute the value of C𝐶Citalic\_C in a context
u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG if X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG is set to x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG (even if the model is not
given explicitly). Since the number of possible settings is smaller
than the size of an explicitly presented model, we can also determine
whether MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT holds
in polynomial time if the model is presented explicitly.
On the other hand, if the models are presented in a more compact way,
with the structural equations, and hence the (descriptions of the)
models, being of size polynomial in the number of
variables in the
model, checking full compatibility is in a higher complexity class, as
we now show.
######
Proposition 4.8.
Given two causal models M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT of size polynomial in the
number of variables,
determining whether they are fully compatible
with respect to a default setting v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT
is in 𝑃||NP\mbox{P}^{{\rm NP}}\_{||}P start\_POSTSUPERSCRIPT roman\_NP end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT | | end\_POSTSUBSCRIPTand is
co-NP-hard
in the sizes of M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
Proof. We prove a slightly stronger claim: that checking
MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT is co-NP-complete in
the sizes of M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and that checking
MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT is in P||NP\mbox{P}^{{\rm NP}}\_{||}P start\_POSTSUPERSCRIPT roman\_NP end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT | | end\_POSTSUBSCRIPT.
The complexity class
P||NP\mbox{P}^{{\rm NP}}\_{||}P start\_POSTSUPERSCRIPT roman\_NP end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT | | end\_POSTSUBSCRIPTconsists of all decision problems that can
be solved in polynomial time with parallel (i.e., non-adaptive)
queries to an NP oracle (see Wag90; JT95; Joh90).
We start by showing that checking that MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT holds is
in co-NP by showing that the
complementary problem,
namely demonstrating that MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT does not hold,
is in NP.
To do this, we guess a witness:
a setting x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG for the
common variables X→→𝑋\vec{X}over→ start\_ARG italic\_X end\_ARG of M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT other than C𝐶Citalic\_C, a value
c𝑐citalic\_c of C𝐶Citalic\_C, and contexts u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and u→2subscript→𝑢2\vec{u}\_{2}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT for M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
respectively, such that
u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and u→2subscript→𝑢2\vec{u}\_{2}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are compatible with v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT,
| | | |
| --- | --- | --- |
| | (M1,u→1)⊧[X→←x→](C=c),modelssubscript𝑀1subscript→𝑢1delimited-[]←→𝑋→𝑥𝐶𝑐(M\_{1},\vec{u}\_{1})\models[\vec{X}\leftarrow\vec{x}](C=c),( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ⊧ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C = italic\_c ) , | |
but
| | | |
| --- | --- | --- |
| | (M2,u→2)⊧̸[X→←x→](C≠c)not-modelssubscript𝑀2subscript→𝑢2delimited-[]←→𝑋→𝑥𝐶𝑐(M\_{2},\vec{u}\_{2})\not\models[\vec{X}\leftarrow\vec{x}](C\neq c)( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ⊧̸ [ over→ start\_ARG italic\_X end\_ARG ← over→ start\_ARG italic\_x end\_ARG ] ( italic\_C ≠ italic\_c ) | |
(or vice versa). A witness can be verified in polynomial time in the
size of the model, as it amounts to assigning
values to all variables in the models and checking the value of C𝐶Citalic\_C.
The proof that the problem is co-NP hard is
by reduction from the known co-NP-complete
problem Tautology: determining whether a Boolean formula
φ𝜑\varphiitalic\_φ is a tautology.
Let φ𝜑\varphiitalic\_φ be a Boolean formula over the variables {Y1,…,Yn}subscript𝑌1…subscript𝑌𝑛\{Y\_{1},\ldots,Y\_{n}\}{ italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT }. We construct a causal model M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT as follows:
1. 1.
𝒰1={U1,…,Un}subscript𝒰1subscript𝑈1…subscript𝑈𝑛{\cal U}\_{1}=\{U\_{1},\ldots,U\_{n}\}caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = { italic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_U start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT };
2. 2.
𝒱1={Y1,…,Yn,C}subscript𝒱1subscript𝑌1…subscript𝑌𝑛𝐶{\cal V}\_{1}=\{Y\_{1},\ldots,Y\_{n},C\}caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = { italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT , italic\_C };
3. 3.
ℛ1(X)={0,1}subscriptℛ1𝑋01{\cal R}\_{1}(X)=\{0,1\}caligraphic\_R start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_X ) = { 0 , 1 } for all X∈𝒱1𝑋subscript𝒱1X\in{\cal V}\_{1}italic\_X ∈ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT;
4. 4.
the equations are Yi=Uisubscript𝑌𝑖subscript𝑈𝑖Y\_{i}=U\_{i}italic\_Y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = italic\_U start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for i=1,…,n𝑖
1…𝑛i=1,\ldots,nitalic\_i = 1 , … , italic\_n and C=φ𝐶𝜑C={\varphi}italic\_C = italic\_φ.
We note that since the set of exogenous variables is the same in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
and in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, we don’t need to
define the default setting v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
In other words, the variables {Y1,…,Yn}subscript𝑌1…subscript𝑌𝑛\{Y\_{1},\ldots,Y\_{n}\}{ italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT } are binary
variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, and the value of C𝐶Citalic\_C is determined by φ𝜑\varphiitalic\_φ.
The second causal model M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is constructed as follows:
1. 1.
𝒰2={U1,…,Un}subscript𝒰2subscript𝑈1…subscript𝑈𝑛{\cal U}\_{2}=\{U\_{1},\ldots,U\_{n}\}caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = { italic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_U start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT };
2. 2.
𝒱2={Y1,…,Yn,C}subscript𝒱2subscript𝑌1…subscript𝑌𝑛𝐶{\cal V}\_{2}=\{Y\_{1},\ldots,Y\_{n},C\}caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = { italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT , italic\_C };
3. 3.
ℛ2(X)={0,1}subscriptℛ2𝑋01{\cal R}\_{2}(X)=\{0,1\}caligraphic\_R start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_X ) = { 0 , 1 } for all X∈𝒱2𝑋subscript𝒱2X\in{\cal V}\_{2}italic\_X ∈ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT;
4. 4.
the equations are Yi=Uisubscript𝑌𝑖subscript𝑈𝑖Y\_{i}=U\_{i}italic\_Y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = italic\_U start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for i=1,…,n𝑖
1…𝑛i=1,\ldots,nitalic\_i = 1 , … , italic\_n, and C=1𝐶1C=1italic\_C = 1.
MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT holds iff φ𝜑\varphiitalic\_φ is a tautology.
Indeed, if φ𝜑{\varphi}italic\_φ is a tautology, then MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT holds
trivially. On
the other hand, if φ𝜑{\varphi}italic\_φ is not
a tautology, then it is easy to see that MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscript→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT does
not hold, since there is some setting of the variables Y1,…,Ynsubscript𝑌1…subscript𝑌𝑛Y\_{1},\ldots,Y\_{n}italic\_Y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_Y start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT that makes C=0𝐶0C=0italic\_C = 0.
To prove membership of MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT in P||NP\mbox{P}^{{\rm NP}}\_{||}P start\_POSTSUPERSCRIPT roman\_NP end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT | | end\_POSTSUBSCRIPT, we describe a
polynomial-time algorithm
for deciding MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT that makes parallel queries
to an NP oracle.
We define an oracle ODep(M,X,Y)superscript𝑂𝐷𝑒𝑝𝑀𝑋𝑌O^{Dep}(M,X,Y)italic\_O start\_POSTSUPERSCRIPT italic\_D italic\_e italic\_p end\_POSTSUPERSCRIPT ( italic\_M , italic\_X , italic\_Y ) as follows: for a causal model
M𝑀Mitalic\_M and two variables X𝑋Xitalic\_X and Y𝑌Yitalic\_Y of M𝑀Mitalic\_M, it answers
“yes” if FXsubscript𝐹𝑋F\_{X}italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT depends on
the variable Y𝑌Yitalic\_Y in M𝑀Mitalic\_M and “no” otherwise. It is easy to see that
ODep(M,X,Y)superscript𝑂𝐷𝑒𝑝𝑀𝑋𝑌O^{Dep}(M,X,Y)italic\_O start\_POSTSUPERSCRIPT italic\_D italic\_e italic\_p end\_POSTSUPERSCRIPT ( italic\_M , italic\_X , italic\_Y ) is in NP, since a witness for the
positive answer is
a pair of
assignments to the variables of FXsubscript𝐹𝑋F\_{X}italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT that differ only in the value of
Y𝑌Yitalic\_Y and in the result. A witness is clearly verifiable in polynomial
time; we simply instantiate FXsubscript𝐹𝑋F\_{X}italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT on these two assignments and
verify that the results are different.
(We have implicitly assumed here that the equation FXsubscript𝐹𝑋F\_{X}italic\_F start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT can be
computed in polynomial time, as it is a part of M𝑀Mitalic\_M.)
By querying the oracle
ODep(Mi,X,Y)superscript𝑂𝐷𝑒𝑝subscript𝑀𝑖𝑋𝑌O^{Dep}(M\_{i},X,Y)italic\_O start\_POSTSUPERSCRIPT italic\_D italic\_e italic\_p end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_X , italic\_Y ) for all endogenous variables X𝑋Xitalic\_X and
Y𝑌Yitalic\_Y in Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT,
for i=1,2𝑖12i=1,2italic\_i = 1 , 2, we can determine the causal graphs of M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
and thus whether MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT holds.
The number of queries is at most quadratic in the sizes of M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, hence the algorithm terminates in polynomial time.
###
4.2 Partial compatibility
While the notion of dominance used in Definition [4.6](#S4.Thmtheorem6 "Definition 4.6. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") is
useful, it still does not cover many cases of interest.
We briefly describe an example here on causal models for the emergence
of radicalization in US prisons. The
material is taken from
Useem and Clayton ([2009](#bib.bib28)).
Although Useem and Clayton do not provide causal models,
we construct these based on the description provided.
We do not provide a detailed explanation of all the variables
and their dependencies here
(details are
provided in the full paper); for our purposes, it suffices to focus on
the structure of these models.
######
Example 4.9.
Consider the two causal models in Figure [5](#S4.F5 "Figure 5 ‣ Example 4.9. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT represents expert 1’s opinion about emerging
radicalization (R𝑅Ritalic\_R)
in
the State Correctional
Institution Camp Hill in Pennsylvania. M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT represents
expert 2’s opinion about the causes of emergence in the Texas Department
of Corrections and Rehabilitation. Both experts agree on
the structural equations for R𝑅Ritalic\_R. However they differ
on the structural equations for PD, CB and AM.
The authors point to
three main factors upon which the emergence of radicalization
settings in both prison settings is dependent: “order in prisons” (PD),
“a boundary between the prison and potentially radicalizing
communities” (CB), and “having missionary leadership within the prison
organizations” (AM). The also both share
the same outcome —emerging radicalization (R𝑅Ritalic\_R).
As can be observed from the descriptions provided, some variables
and their dependency relations are specific to
a prison.
In M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, PD is attributed to
corruption (CG) and lax management (LM) in the prison’s staff
together with prisons being allowed to roam freely (FM).
CB is viewed as a result of religious leaders within the
facilities being permitted to provide religious services freely
(IL) and by prisoners showing membership within a prison community
(CM) which in turn is signalled by prisoners being allowed to wear
distinguished street clothing (SC).
Prison authorities’ exercising of internal punishments, such as
administrative segregation (AS), away from external oversight, and
IL are considered to directly contribute to AM.
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT instead considers PD to be linked to the rapid growth in
inmate numbers (RG), inmates being allowed to assist authorities in
maintaining order (AA), and inmates feeling
significantly deprived (D) within the prisons—the latter as a
result of being forced to engage in unpaid work (W) and having
limited contact with visitors (C).
For the common variables CB and AM, assuming default values for CG, LM
and FM in SCICH and for AA, D and RG in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
we can show that M1⪰𝐶𝐵v→\*M2superscriptsubscriptsucceeds-or-equals𝐶𝐵superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{\mbox{{\scriptsize{{\it CB}}}}}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT CB end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
M1⪰𝐴𝑀v→\*M2superscriptsubscriptsucceeds-or-equals𝐴𝑀superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{\mbox{{\scriptsize{{\it AM}}}}}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT AM end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
However, neither model dominates the other with respect to PD;
neither MI1M1,M2,𝑃𝐷subscript𝑀1subscript𝑀2𝑃𝐷{}\_{M\_{1},M\_{2},\mbox{{\scriptsize{\it PD}}}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , PD end\_FLOATSUBSCRIPT nor MI1M2,M1,𝑃𝐷subscript𝑀2subscript𝑀1𝑃𝐷{}\_{M\_{2},M\_{1},\mbox{{\scriptsize{\it PD}}}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , PD end\_FLOATSUBSCRIPT
holds. Therefore the models are not fully compatible according to Definition
[4.6](#S4.Thmtheorem6 "Definition 4.6. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
ASSCCMILFMCGLMPDCBAMR𝑅Ritalic\_RM1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPTCMASCWAADRGPDCBAMR𝑅Ritalic\_RM2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
Figure 5: Schematic representation of the two prison
models.
Although the models are not fully compatible according to our definition,
the incompatibility is “localized” to the variable PD, a point we
make precise in Section [4.3](#S4.SS3 "4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
Moreover,
it is not even clear that there is disagreement with regard to PD;
the experts could just be focusing on different variables.
(The issue of what the experts are focusing on and how this might
affect the issue of combining experts’ models is discussed by
Friedenberg and Halpern ([2018](#bib.bib7)).)
In a richer model, PD might have
six parents. The trouble is, knowing the two models does not give
us any insight into what
the equations for PD would be in the richer model.
######
Definition 4.10 (Weak domination).
Let v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT be a default setting for the variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
We say that
*M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT weakly dominates M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT with respect to a node C𝐶Citalic\_C relative to
v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT*, written M1⪰w,Cv→\*superscriptsubscriptsucceeds-or-equals𝑤𝐶superscriptnormal-→𝑣subscript𝑀1absentM\_{1}\succeq\_{w,C}^{\vec{v}^{\*}}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_w , italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT if
MI4M1,M2,C,v→\*subscript𝑀1subscript𝑀2𝐶superscriptnormal-→𝑣{}\_{M\_{1},M\_{2},C,\vec{v}^{\*}}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C , over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_FLOATSUBSCRIPT and the following weakening of
MI1M1,M2,Csubscript𝑀1subscript𝑀2𝐶{}\_{M\_{1},M\_{2},C}start\_FLOATSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_FLOATSUBSCRIPT hold:
MI1M1,M2,C′subscriptsuperscriptabsentnormal-′subscript𝑀1subscript𝑀2𝐶{}^{\prime}\_{M\_{1},M\_{2},C}start\_FLOATSUPERSCRIPT ′ end\_FLOATSUPERSCRIPT start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_C end\_POSTSUBSCRIPT
If A𝐴Aitalic\_A is a node in both M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT then A𝐴Aitalic\_A
is an immediate M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT-ancestor of C𝐶Citalic\_C in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT iff A𝐴Aitalic\_A is a parent of
C𝐶Citalic\_C in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
Note that in Example [4.2](#S4.Thmtheorem2 "Example 4.2. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), neither M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT nor M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT weakly
dominates the other with respect to F𝐹Fitalic\_F: P𝑃Pitalic\_P is a parent of F𝐹Fitalic\_F in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and is
not an immediate M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ancestor of F𝐹Fitalic\_F in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, so M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT does not
weakly dominate M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT with respect to F𝐹Fitalic\_F, while P𝑃Pitalic\_P is an immediate
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ancestor of F𝐹Fitalic\_F in
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and is not a parent of F𝐹Fitalic\_F in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, so M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT does not weakly
dominate M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT either.
Also note that MI1 implies MI1′′{}^{\prime}start\_FLOATSUPERSCRIPT ′ end\_FLOATSUPERSCRIPT; MI1′′{}^{\prime}start\_FLOATSUPERSCRIPT ′ end\_FLOATSUPERSCRIPT is a strictly weaker
condition than MI1, since it allows M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT to weakly dominate M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
with respect to C𝐶Citalic\_C if C𝐶Citalic\_C has parents in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT that are not in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
at all.
######
Definition 4.11.
[Partial compatibility]
If M1=((𝒰1,𝒱1,ℛ1),ℱ1)subscript𝑀1subscript𝒰1subscript𝒱1subscriptℛ1subscriptℱ1M\_{1}=(({\cal U}\_{1},{\cal V}\_{1},{\cal R}\_{1}),{\cal F}\_{1})italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , caligraphic\_R start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) and M2=((𝒰2,𝒱2,ℛ2),ℱ2)subscript𝑀2subscript𝒰2subscript𝒱2subscriptℛ2subscriptℱ2M\_{2}=(({\cal U}\_{2},{\cal V}\_{2},{\cal R}\_{2}),{\cal F}\_{2})italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , caligraphic\_R start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ),
then
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are *partially compatible with respect to default
setting v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT* iff
(1) for all variables C∈(𝒰1∪𝒱1)∩(𝒰2∪𝒱2)𝐶subscript𝒰1subscript𝒱1subscript𝒰2subscript𝒱2C\in({\cal U}\_{1}\cup{\cal V}\_{1})\cap({\cal U}\_{2}\cup{\cal V}\_{2})italic\_C ∈ ( caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ∩ ( caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∪ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ), we have ℛ1(C)=ℛ2(C)subscriptℛ1𝐶subscriptℛ2𝐶{\cal R}\_{1}(C)={\cal R}\_{2}(C)caligraphic\_R start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_C ) = caligraphic\_R start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C ) and (2) for all variables
C∈(𝒱1∩𝒱2)∪(𝒱1∩𝒰2)∪(𝒱2∩𝒰1)𝐶subscript𝒱1subscript𝒱2subscript𝒱1subscript𝒰2subscript𝒱2subscript𝒰1C\in({\cal V}\_{1}\cap{\cal V}\_{2})\cup({\cal V}\_{1}\cap{\cal U}\_{2})\cup({\cal V}\_{2}\cap{\cal U}\_{1})italic\_C ∈ ( caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ∪ ( caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ∪ ( caligraphic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ∩ caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ),
either M1⪰w,Cv→\*M2superscriptsubscriptsucceeds-or-equals𝑤𝐶superscriptnormal-→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{w,C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_w , italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT or
M2⪰w,Cv→\*M1superscriptsubscriptsucceeds-or-equals𝑤𝐶superscriptnormal-→𝑣subscript𝑀2subscript𝑀1M\_{2}\succeq\_{w,C}^{\vec{v}^{\*}}M\_{1}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_w , italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. If M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are partially
compatible with respect to v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, then M1⊕v→\*M2=((𝒰,𝒱,ℛ),ℱ)superscriptdirect-sumsuperscriptnormal-→𝑣subscript𝑀1subscript𝑀2𝒰𝒱ℛℱM\_{1}\oplus^{\vec{v}^{\*}}M\_{2}=(({\cal U},{\cal V},{\cal R}),{\cal F})italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = ( ( caligraphic\_U , caligraphic\_V , caligraphic\_R ) , caligraphic\_F )
M1⊕M2=((𝒰,𝒱,ℛ),ℱ)direct-sumsubscript𝑀1subscript𝑀2𝒰𝒱ℛℱM\_{1}\oplus M\_{2}=(({\cal U},{\cal V},{\cal R}),{\cal F})italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = ( ( caligraphic\_U , caligraphic\_V , caligraphic\_R ) , caligraphic\_F )
is defined as follows:
* 1.
𝒰𝒰{\cal U}caligraphic\_U, 𝒱𝒱{\cal V}caligraphic\_V, ℛℛ{\cal R}caligraphic\_R are defined just as in Definition [4.6](#S4.Thmtheorem6 "Definition 4.6. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
* 2.
For ℱℱ{\cal F}caligraphic\_F,
if C∈V1−V2𝐶subscript𝑉1subscript𝑉2C\in V\_{1}-V\_{2}italic\_C ∈ italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT - italic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT then ℱ(C)=ℱ1(C)ℱ𝐶subscriptℱ1𝐶{\cal F}(C)={\cal F}\_{1}(C)caligraphic\_F ( italic\_C ) = caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_C ).
and if C∈V2−V1𝐶subscript𝑉2subscript𝑉1C\in V\_{2}-V\_{1}italic\_C ∈ italic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT - italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT then ℱ(C)=ℱ2(C)ℱ𝐶subscriptℱ2𝐶{\cal F}(C)={\cal F}\_{2}(C)caligraphic\_F ( italic\_C ) = caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C ).
If C∈V1∩V2𝐶subscript𝑉1subscript𝑉2C\in V\_{1}\cap V\_{2}italic\_C ∈ italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ italic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
M1⪰Cv→\*M2superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, then let
P→1subscript→𝑃1\vec{P}\_{1}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT consist of the parents of C𝐶Citalic\_C in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and let
P→2subscript→𝑃2\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT consist of the parents of C𝐶Citalic\_C in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT that are not in
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. Then the parents of C𝐶Citalic\_C in M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are the nodes
P→1∪P→2subscript→𝑃1subscript→𝑃2\vec{P}\_{1}\cup\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. Let v→2subscript→𝑣2\vec{v}\_{2}over→ start\_ARG italic\_v end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT consist be the values of
the variables in P→2subscript→𝑃2\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT when the exogenous variables in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
have their default value in v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Given an arbitrary setting x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG of the variables in P→1subscript→𝑃1\vec{P}\_{1}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT,
we define ℱ(C)(x→,v→2)=ℱ2(C)(x→)ℱ𝐶→𝑥subscript→𝑣2subscriptℱ2𝐶→𝑥{\cal F}(C)(\vec{x},\vec{v}\_{2})={\cal F}\_{2}(C)(\vec{x})caligraphic\_F ( italic\_C ) ( over→ start\_ARG italic\_x end\_ARG , over→ start\_ARG italic\_v end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C ) ( over→ start\_ARG italic\_x end\_ARG ).
Symmetrically, if C∈V2−V1𝐶subscript𝑉2subscript𝑉1C\in V\_{2}-V\_{1}italic\_C ∈ italic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT - italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT or both C∈V1∩V2𝐶subscript𝑉1subscript𝑉2C\in V\_{1}\cap V\_{2}italic\_C ∈ italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ italic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
M1⪰w,Cv→\*M2superscriptsubscriptsucceeds-or-equals
𝑤𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{w,C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_w , italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, then let
P→1subscript→𝑃1\vec{P}\_{1}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT consist of the parents of C𝐶Citalic\_C in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and let
P→2subscript→𝑃2\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT consist of the parents of C𝐶Citalic\_C in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT that are not in
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. Then the parents of C𝐶Citalic\_C in M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are the nodes
P→1∪P→2subscript→𝑃1subscript→𝑃2\vec{P}\_{1}\cup\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. Let v→2subscript→𝑣2\vec{v}\_{2}over→ start\_ARG italic\_v end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT consist be the values of
the variables in P→2subscript→𝑃2\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT when the exogenous variables in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
have their default value in v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Given an arbitrary setting x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG of the variables in P→1subscript→𝑃1\vec{P}\_{1}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT,
we define ℱ(C)(x→,v→2)=ℱ2(C)(x→)ℱ𝐶→𝑥subscript→𝑣2subscriptℱ2𝐶→𝑥{\cal F}(C)(\vec{x},\vec{v}\_{2})={\cal F}\_{2}(C)(\vec{x})caligraphic\_F ( italic\_C ) ( over→ start\_ARG italic\_x end\_ARG , over→ start\_ARG italic\_v end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C ) ( over→ start\_ARG italic\_x end\_ARG ).
If C∈V1∩V2𝐶subscript𝑉1subscript𝑉2C\in V\_{1}\cap V\_{2}italic\_C ∈ italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ italic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
M1⪰w,Cv→\*M2superscriptsubscriptsucceeds-or-equals
𝑤𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{w,C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_w , italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, then let
P→1subscript→𝑃1\vec{P}\_{1}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT consist of the parents of C𝐶Citalic\_C in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and let
P→2subscript→𝑃2\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT consist of the parents of C𝐶Citalic\_C in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT that are not in
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. Then the parents of C𝐶Citalic\_C in M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are the nodes
P→1∪P→2subscript→𝑃1subscript→𝑃2\vec{P}\_{1}\cup\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∪ over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. Let v→2subscript→𝑣2\vec{v}\_{2}over→ start\_ARG italic\_v end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT consist be the values of
the variables in P→2subscript→𝑃2\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT when the exogenous variables in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
have their default value in v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Given an arbitrary setting x→→𝑥\vec{x}over→ start\_ARG italic\_x end\_ARG of the variables in P→1subscript→𝑃1\vec{P}\_{1}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT,
we define ℱ(C)(x→,v→2)=ℱ2(C)(x→)ℱ𝐶→𝑥subscript→𝑣2subscriptℱ2𝐶→𝑥{\cal F}(C)(\vec{x},\vec{v}\_{2})={\cal F}\_{2}(C)(\vec{x})caligraphic\_F ( italic\_C ) ( over→ start\_ARG italic\_x end\_ARG , over→ start\_ARG italic\_v end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C ) ( over→ start\_ARG italic\_x end\_ARG ).
We have a symmetric definition of ℱ(C)ℱ𝐶{\cal F}(C)caligraphic\_F ( italic\_C ) if
C∈V1∩V2𝐶subscript𝑉1subscript𝑉2C\in V\_{1}\cap V\_{2}italic\_C ∈ italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ italic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and M2⪰w,Cv→\*M1superscriptsubscriptsucceeds-or-equals
𝑤𝐶superscript→𝑣subscript𝑀2subscript𝑀1M\_{2}\succeq\_{w,C}^{\vec{v}^{\*}}M\_{1}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_w , italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
Again, we write M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT if 𝒰1=𝒰2subscript𝒰1subscript𝒰2{\cal U}\_{1}={\cal U}\_{2}caligraphic\_U start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = caligraphic\_U start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
This definition does not define ℱ(C)ℱ𝐶{\cal F}(C)caligraphic\_F ( italic\_C ) for all possible values of the
parents of C𝐶Citalic\_C.
For example,
if C∈V1∩V2𝐶subscript𝑉1subscript𝑉2C\in V\_{1}\cap V\_{2}italic\_C ∈ italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∩ italic\_V start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and M1⪰Cv→\*M2superscriptsubscriptsucceeds-or-equals𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\succeq\_{C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, we
have not defined ℱ(C)(x→,y→)ℱ𝐶→𝑥→𝑦{\cal F}(C)(\vec{x},\vec{y})caligraphic\_F ( italic\_C ) ( over→ start\_ARG italic\_x end\_ARG , over→ start\_ARG italic\_y end\_ARG ) if y→→𝑦\vec{y}over→ start\_ARG italic\_y end\_ARG is a setting of
the variables in P→2subscript→𝑃2\vec{P}\_{2}over→ start\_ARG italic\_P end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT other than v→2subscript→𝑣2\vec{v}\_{2}over→ start\_ARG italic\_v end\_ARG start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. Intuitively,
this is because the experts have not given us the information to
determine ℱ(C)ℱ𝐶{\cal F}(C)caligraphic\_F ( italic\_C ) in these cases.
We can think if M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT as a *partial* causal model.
Intuitively, we cannot define
⊧models\models⊧ in M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT since we will not be able to define
value of (M1⊕M2,u→)⊧C=cmodelsdirect-sumsubscript𝑀1subscript𝑀2→𝑢𝐶𝑐(M\_{1}\oplus M\_{2},\vec{u})\models C=c( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_C = italic\_c for some setting u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG.
Say that causal model M\*=((𝒰\*,𝒱\*,ℛ\*),ℱ\*)superscript𝑀superscript𝒰superscript𝒱superscriptℛsuperscriptℱM^{\*}=(({\cal U}^{\*},{\cal V}^{\*},{\cal R}^{\*}),{\cal F}^{\*})italic\_M start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT = ( ( caligraphic\_U start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , caligraphic\_V start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , caligraphic\_R start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) , caligraphic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
*extends* M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT if (𝒰\*,𝒱\*,ℛ\*)=(𝒰,𝒱,ℛ)superscript𝒰superscript𝒱superscriptℛ𝒰𝒱ℛ({\cal U}^{\*},{\cal V}^{\*},{\cal R}^{\*})=({\cal U},{\cal V},{\cal R})( caligraphic\_U start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , caligraphic\_V start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , caligraphic\_R start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) = ( caligraphic\_U , caligraphic\_V , caligraphic\_R ) and
ℱ\*(C)=ℱ(C)superscriptℱ𝐶ℱ𝐶{\cal F}^{\*}(C)={\cal F}(C)caligraphic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ( italic\_C ) = caligraphic\_F ( italic\_C ) whenever ℱ(C)ℱ𝐶{\cal F}(C)caligraphic\_F ( italic\_C ) is defined. We now define a
3-valued version of ⊧models\models⊧ in M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT by taking
(M1⊕M2,u→)⊧φmodelsdirect-sumsubscript𝑀1subscript𝑀2→𝑢𝜑(M\_{1}\oplus M\_{2},\vec{u})\models\varphi( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_φ iff (M\*,u→)⊧φmodelssuperscript𝑀→𝑢𝜑(M^{\*},\vec{u})\models\varphi( italic\_M start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_φ for
all (complete) causal models M\*superscript𝑀M^{\*}italic\_M start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT extending M1⊕M2direct-sumsubscript𝑀1subscript𝑀2M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and
taking
(M1⊕M2,u→)⊧φmodelsdirect-sumsubscript𝑀1subscript𝑀2→𝑢𝜑(M\_{1}\oplus M\_{2},\vec{u})\models\varphi( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_φ to be *undefined* if neither
(M1⊕M2,u→)⊧φmodelsdirect-sumsubscript𝑀1subscript𝑀2→𝑢𝜑(M\_{1}\oplus M\_{2},\vec{u})\models\varphi( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_φ nor
(M1⊕M2,u→)⊧¬φmodelsdirect-sumsubscript𝑀1subscript𝑀2→𝑢𝜑(M\_{1}\oplus M\_{2},\vec{u})\models\neg\varphi( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ ¬ italic\_φ holds.
Note that, according to the definition above, the two models in
Figure [5](#S4.F5 "Figure 5 ‣ Example 4.9. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")
are partially compatible.
We can now prove a generalization of Theorem [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
###### Theorem 2
Suppose that M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT,
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and M3subscript𝑀3M\_{3}italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT are pairwise partially compatible with respect to
v→\*superscriptnormal-→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. Then
the following conditions hold.
* (a)
If M1∼w,Cv→\*M2superscriptsubscriptsimilar-to
𝑤𝐶superscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\sim\_{w,C}^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∼ start\_POSTSUBSCRIPT italic\_w , italic\_C end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT then
(i) 𝑃𝑎𝑟M1(C)=𝑃𝑎𝑟M2(C)subscript𝑃𝑎𝑟subscript𝑀1𝐶subscript𝑃𝑎𝑟subscript𝑀2𝐶\mathit{Par}\_{M\_{1}}(C)=\mathit{Par}\_{M\_{2}}(C)italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ) = italic\_Par start\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_C ) and (ii) ℱ1(C)=ℱ2(C)subscriptℱ1𝐶subscriptℱ2𝐶{\cal F}\_{1}(C)={\cal F}\_{2}(C)caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_C ) = caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_C ).
* (b)
M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is well defined.
* (c)
M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is acyclic.
* (d)
If A𝐴Aitalic\_A and B𝐵Bitalic\_B are variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, then A𝐴Aitalic\_A is an
ancestor of B𝐵Bitalic\_B in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT iff A𝐴Aitalic\_A is an ancestor of B𝐵Bitalic\_B in M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
* (e)
If φ𝜑\varphiitalic\_φ is a formula that mentions only the endogenous
variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG is a context for M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT,
u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT is a context for M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT,
and u→→𝑢\vec{u}over→ start\_ARG italic\_u end\_ARG and u→1subscript→𝑢1\vec{u}\_{1}over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT are compatible with v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, then
(M1,u→1)⊧φmodelssubscript𝑀1subscript→𝑢1𝜑(M\_{1},\vec{u}\_{1})\models\varphi( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ⊧ italic\_φ iff (M1⊕v→\*M2,u→)⊧φmodelssuperscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2→𝑢𝜑(M\_{1}\oplus^{\vec{v}^{\*}}M\_{2},\vec{u})\models\varphi( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , over→ start\_ARG italic\_u end\_ARG ) ⊧ italic\_φ.
* (f)
M1⊕v→\*M2=M2⊕v→\*M1superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2superscriptdirect-sumsuperscript→𝑣subscript𝑀2subscript𝑀1M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}=M\_{2}\oplus^{\vec{v}^{\*}}M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT = italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT.
* (g)
If M3subscript𝑀3M\_{3}italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT is partially compatible with M1⊕v→\*M2superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2M\_{1}\oplus^{\vec{v}^{\*}}M\_{2}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
is partially compatible with M2⊕v→\*M3superscriptdirect-sumsuperscript→𝑣subscript𝑀2subscript𝑀3M\_{2}\oplus^{\vec{v}^{\*}}M\_{3}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT, then
M1⊕v→\*(M2⊕v→\*M3)=(M1⊕v→\*M2)⊕v→\*M3superscriptdirect-sumsuperscript→𝑣subscript𝑀1superscriptdirect-sumsuperscript→𝑣subscript𝑀2subscript𝑀3superscriptdirect-sumsuperscript→𝑣superscriptdirect-sumsuperscript→𝑣subscript𝑀1subscript𝑀2subscript𝑀3M\_{1}\oplus^{\vec{v}^{\*}}(M\_{2}\oplus^{\vec{v}^{\*}}M\_{3})=(M\_{1}\oplus^{\vec{v}^{\*}}M\_{2})\oplus^{\vec{v}^{\*}}M\_{3}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ) = ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ⊕ start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT.
The proof is almost identical to that of
Theorem [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), so we omit the details here.
It is easy to see that the problem of determining whether M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
are partially compatible is of the same complexity as the problem of determining
full compatibility
by the following argument.
Clearly, if M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are fully compatible, they
are also
partially compatible, so the problem of determining partial
compatibility is at least as hard
as the problem of determining full compatibility. On the other hand,
by Definition [4.11](#S4.Thmtheorem11 "Definition 4.11. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."),
the scope of compatibility is defined by the set of the common
variables, hence given
two models M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, it is easy to determine the subset of
variables with respect to
which we need to check compatibility.
Therefore, checking partial compatibility is of the same complexity as checking full compatibility.
###
4.3 Decomposition of causal models
Even with the generalised notions of compatibility introduced above, it may
still be the case that two expert’s models are
not compatible. But we would expect that these models have submodels
that are compatible.
Finding such submodels has several advantages.
First, consider the situation where the
policymaker is given several different causal models that are not
fully compatible according to Definition [4.6](#S4.Thmtheorem6 "Definition 4.6. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). If we could decompose
the models, we might be able to “localize” the incompatibility, and
merge the parts of the models that are fully compatible. Doing so may suggest
effective interventions.
Another advantage of decomposing a model is that it allows the policymaker to reason about
each submodel in isolation. Since the problem of computing causes is DP-complete and the problem of computing interventions is co-NP-complete,
having a smaller model to reason about could have a
significant impact on the complexity of the problem.
In order to define the notion of decomposition,
we need some preliminary definitions.
######
Definition 4.12.
[Order-preserving partition]
A sequence ⟨𝒱1,…,𝒱k⟩subscript𝒱1normal-…subscript𝒱𝑘\langle{\cal V}\_{1},\ldots,{\cal V}\_{k}\rangle⟨ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , caligraphic\_V start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ⟩ of subsets of variables in 𝒱𝒱{\cal V}caligraphic\_V
variables in a causal model M𝑀Mitalic\_M is an
*order-preserving partition* if Vi∩Vj=∅subscript𝑉𝑖subscript𝑉𝑗V\_{i}\cap V\_{j}=\emptysetitalic\_V start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∩ italic\_V start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT = ∅ for i≠j𝑖𝑗i\neq jitalic\_i ≠ italic\_j, ∪i=1k𝒱i=𝒱superscriptsubscript𝑖1𝑘subscript𝒱𝑖𝒱\cup\_{i=1}^{k}{\cal V}\_{i}={\cal V}∪ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT caligraphic\_V start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = caligraphic\_V (so {V1,…,𝒱k}subscript𝑉1normal-…subscript𝒱𝑘\{V\_{1},\ldots,{\cal V}\_{k}\}{ italic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , caligraphic\_V start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT } is a
partition of 𝒱𝒱{\cal V}caligraphic\_V), and
for all i,j𝑖𝑗i,jitalic\_i , italic\_j with i<j𝑖𝑗i<jitalic\_i < italic\_j,
no variable in 𝒱jsubscript𝒱𝑗{\cal V}\_{j}caligraphic\_V start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT is an ancestor of a variable in 𝒱isubscript𝒱𝑖{\cal V}\_{i}caligraphic\_V start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT.
######
Definition 4.13.
[Decomposable causal models]
M=((𝒰,𝒱,ℛ),ℱ)𝑀𝒰𝒱ℛℱM=(({\cal U},{\cal V},{\cal R}),{\cal F})italic\_M = ( ( caligraphic\_U , caligraphic\_V , caligraphic\_R ) , caligraphic\_F ) is *decomposable* if there
exist k≥1𝑘1k\geq 1italic\_k ≥ 1 fully compatible causal models
{Mi=((𝒰i,𝒱i,ℛi),ℱi):1≤i≤k}conditional-setsubscript𝑀𝑖subscript𝒰𝑖subscript𝒱𝑖subscriptℛ𝑖subscriptℱ𝑖1𝑖𝑘\{M\_{i}=(({\cal U}\_{i},{\cal V}\_{i},{\cal R}\_{i}),{\cal F}\_{i}):1\leq i\leq k\}{ italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , caligraphic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) : 1 ≤ italic\_i ≤ italic\_k },
such that ⟨𝒱1,…,𝒱k⟩subscript𝒱1normal-…subscript𝒱𝑘\langle{\cal V}\_{1},\ldots,{\cal V}\_{k}\rangle⟨ caligraphic\_V start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , caligraphic\_V start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ⟩ is an order-preserving
partition of 𝒱𝒱{\cal V}caligraphic\_V, ℱi⊆ℱsubscriptℱ𝑖ℱ{\cal F}\_{i}\subseteq{\cal F}caligraphic\_F start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ⊆ caligraphic\_F is the set of structural equations
that assign the values to the variables in 𝒱isubscript𝒱𝑖{\cal V}\_{i}caligraphic\_V start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT,
and for each model Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, the set 𝒰isubscript𝒰𝑖{\cal U}\_{i}caligraphic\_U start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT consists of
all the endogenous and exogenous variables in M𝑀Mitalic\_M not in 𝒱isubscript𝒱𝑖{\cal V}\_{i}caligraphic\_V start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT
that participate in the structural equations for the variables of
𝒱isubscript𝒱𝑖{\cal V}\_{i}caligraphic\_V start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT.
M1,…,Mksubscript𝑀1normal-…subscript𝑀𝑘M\_{1},\ldots,M\_{k}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT is called a *decomposition* of M𝑀Mitalic\_M.
######
Lemma 4.14.
If M1,…,Mksubscript𝑀1normal-…subscript𝑀𝑘M\_{1},\ldots,M\_{k}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT is a decomposition of M𝑀Mitalic\_M, then
M1⊕⋯⊕Mk=Mdirect-sumsubscript𝑀1normal-⋯subscript𝑀𝑘𝑀M\_{1}\oplus\cdots\oplus M\_{k}=Mitalic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ ⋯ ⊕ italic\_M start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT = italic\_M.
Proof. The proof is immediate given the observation that we do not
change any of
the structural equations of M𝑀Mitalic\_M when decomposing it into submodels.
It is easy to see that, for a given model, there can be many ways to
decompose it into a set of submodels according to
Definition [4.13](#S4.Thmtheorem13 "Definition 4.13. ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). Moreover,
all models are decomposable by Definition [4.13](#S4.Thmtheorem13 "Definition 4.13. ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). Indeed,
any model M𝑀Mitalic\_M can be trivially decomposed to
|𝒱|𝒱|{\cal V}|| caligraphic\_V | submodels, each of which consists of exactly one endogenous
variable. Of course, such a decomposition is useless for practical
purposes; the decompositions we consider are those that help in
either analysing the model or reducing the complexity of computing
causes. In Example [4.15](#S4.Thmtheorem15 "Example 4.15. ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") below, we demonstrate a
nontrivial decomposition.
######
Example 4.15.
Consider the causal models in Figure [5](#S4.F5 "Figure 5 ‣ Example 4.9. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") from the prison
example (Example [4.9](#S4.Thmtheorem9 "Example 4.9. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")). We can
decompose M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
according to Definition
[4.13](#S4.Thmtheorem13 "Definition 4.13. ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") into M11subscript𝑀11M\_{11}italic\_M start\_POSTSUBSCRIPT 11 end\_POSTSUBSCRIPT, M12subscript𝑀12M\_{12}italic\_M start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT, M13subscript𝑀13M\_{13}italic\_M start\_POSTSUBSCRIPT 13 end\_POSTSUBSCRIPT, and
M14subscript𝑀14M\_{14}italic\_M start\_POSTSUBSCRIPT 14 end\_POSTSUBSCRIPT, where Mij=((𝒰ij,𝒱ij,M\_{ij}=(({\cal U}\_{ij},{\cal V}\_{ij},italic\_M start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT , ℛij),{\cal R}\_{ij}),caligraphic\_R start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT ) , ℱij){\cal F}\_{ij})caligraphic\_F start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT ),
𝒱11={𝐹𝑀,𝐶𝐺,𝐿𝑀,𝑃𝐷}subscript𝒱11𝐹𝑀𝐶𝐺𝐿𝑀𝑃𝐷{\cal V}\_{11}=\{\mbox{{\it FM}},\mbox{{\it CG}},\mbox{{\it LM}},\mbox{{\it PD}}\}caligraphic\_V start\_POSTSUBSCRIPT 11 end\_POSTSUBSCRIPT = { FM , CG , LM , PD }, 𝒰11subscript𝒰11{\cal U}\_{11}caligraphic\_U start\_POSTSUBSCRIPT 11 end\_POSTSUBSCRIPT consists of all
the exogenous variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT (which are not
explicitly given in Figure [5](#S4.F5 "Figure 5 ‣ Example 4.9. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")) that are ancestors of
the variables in 𝒱11subscript𝒱11{\cal V}\_{11}caligraphic\_V start\_POSTSUBSCRIPT 11 end\_POSTSUBSCRIPT, 𝒱12={𝑆𝐶,𝐶𝑀,𝐶𝐵,𝐼𝐿}subscript𝒱12𝑆𝐶𝐶𝑀𝐶𝐵𝐼𝐿{\cal V}\_{12}=\{\mbox{{\it SC}},\mbox{{\it CM}},\mbox{{\it CB}},\mbox{{\it IL}}\}caligraphic\_V start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT = { SC , CM , CB , IL },
𝒰12subscript𝒰12{\cal U}\_{12}caligraphic\_U start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT consists of all
the exogenous variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
that are ancestors of the variables in 𝒱12subscript𝒱12{\cal V}\_{12}caligraphic\_V start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT together FM,
𝒱13={𝐴𝑆,𝐴𝑀}subscript𝒱13𝐴𝑆𝐴𝑀{\cal V}\_{13}=\{\mbox{{\it AS}},\mbox{{\it AM}}\}caligraphic\_V start\_POSTSUBSCRIPT 13 end\_POSTSUBSCRIPT = { AS , AM }, 𝒰13subscript𝒰13{\cal U}\_{13}caligraphic\_U start\_POSTSUBSCRIPT 13 end\_POSTSUBSCRIPT consists of all
the exogenous variables in M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
that are ancestors of the variables in 𝒱13subscript𝒱13{\cal V}\_{13}caligraphic\_V start\_POSTSUBSCRIPT 13 end\_POSTSUBSCRIPT together IL,
𝒱14={R}subscript𝒱14𝑅{\cal V}\_{14}=\{R\}caligraphic\_V start\_POSTSUBSCRIPT 14 end\_POSTSUBSCRIPT = { italic\_R }, and 𝒰14={𝑃𝐷,𝐶𝐵,𝐴𝑀}subscript𝒰14𝑃𝐷𝐶𝐵𝐴𝑀{\cal U}\_{14}=\{\mbox{{\it PD}},\mbox{{\it CB}},\mbox{{\it AM}}\}caligraphic\_U start\_POSTSUBSCRIPT 14 end\_POSTSUBSCRIPT = { PD , CB , AM }.
Similarly we can decompose M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT into four submodels M21subscript𝑀21M\_{21}italic\_M start\_POSTSUBSCRIPT 21 end\_POSTSUBSCRIPT,
M22subscript𝑀22M\_{22}italic\_M start\_POSTSUBSCRIPT 22 end\_POSTSUBSCRIPT, M23subscript𝑀23M\_{23}italic\_M start\_POSTSUBSCRIPT 23 end\_POSTSUBSCRIPT, where
M24subscript𝑀24M\_{24}italic\_M start\_POSTSUBSCRIPT 24 end\_POSTSUBSCRIPT, where Mij=((𝒰ij,𝒱ij,ℛij),ℱij)subscript𝑀𝑖𝑗subscript𝒰𝑖𝑗subscript𝒱𝑖𝑗subscriptℛ𝑖𝑗subscriptℱ𝑖𝑗M\_{ij}=(({\cal U}\_{ij},{\cal V}\_{ij},{\cal R}\_{ij}),{\cal F}\_{ij})italic\_M start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT = ( ( caligraphic\_U start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT , caligraphic\_R start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT ) , caligraphic\_F start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT ),
𝒱21={𝐶,𝑊,𝐴𝐴,𝐷,𝑅𝐺,𝑃𝐷}subscript𝒱21𝐶𝑊𝐴𝐴𝐷𝑅𝐺𝑃𝐷{\cal V}\_{21}=\{\mbox{{\it C}},\mbox{{\it W}},\mbox{{\it AA}},\mbox{{\it D}},\mbox{{\it RG}},\mbox{{\it PD}}\}caligraphic\_V start\_POSTSUBSCRIPT 21 end\_POSTSUBSCRIPT = { C , W , AA , D , RG , PD }, 𝒰21subscript𝒰21{\cal U}\_{21}caligraphic\_U start\_POSTSUBSCRIPT 21 end\_POSTSUBSCRIPT consists of all
the exogenous variables in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
that are ancestors of the variables in 𝒱21subscript𝒱21{\cal V}\_{21}caligraphic\_V start\_POSTSUBSCRIPT 21 end\_POSTSUBSCRIPT. 𝒱22={𝐶𝑀,𝐶𝐵}subscript𝒱22𝐶𝑀𝐶𝐵{\cal V}\_{22}=\{\mbox{{\it CM}},\mbox{{\it CB}}\}caligraphic\_V start\_POSTSUBSCRIPT 22 end\_POSTSUBSCRIPT = { CM , CB },
𝒰22subscript𝒰22{\cal U}\_{22}caligraphic\_U start\_POSTSUBSCRIPT 22 end\_POSTSUBSCRIPT consists of all the exogenous variables in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
that are ancestors of the variables in 𝒱22subscript𝒱22{\cal V}\_{22}caligraphic\_V start\_POSTSUBSCRIPT 22 end\_POSTSUBSCRIPT,
𝒱23={𝐴𝑆,𝐴𝑀}subscript𝒱23𝐴𝑆𝐴𝑀{\cal V}\_{23}=\{\mbox{{\it AS}},\mbox{{\it AM}}\}caligraphic\_V start\_POSTSUBSCRIPT 23 end\_POSTSUBSCRIPT = { AS , AM }, 𝒰23subscript𝒰23{\cal U}\_{23}caligraphic\_U start\_POSTSUBSCRIPT 23 end\_POSTSUBSCRIPT consists of all the exogenous variables in M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
that are ancestors of the variables in 𝒱23subscript𝒱23{\cal V}\_{23}caligraphic\_V start\_POSTSUBSCRIPT 23 end\_POSTSUBSCRIPT,
𝒱24={R}subscript𝒱24𝑅{\cal V}\_{24}=\{R\}caligraphic\_V start\_POSTSUBSCRIPT 24 end\_POSTSUBSCRIPT = { italic\_R }, and 𝒰24={𝑃𝐷,𝐶𝐵,𝐴𝑀}subscript𝒰24𝑃𝐷𝐶𝐵𝐴𝑀{\cal U}\_{24}=\{\mbox{{\it PD}},\mbox{{\it CB}},\mbox{{\it AM}}\}caligraphic\_U start\_POSTSUBSCRIPT 24 end\_POSTSUBSCRIPT = { PD , CB , AM }.
Figures [6](#S4.F6 "Figure 6 ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") and Figures [7](#S4.F7 "Figure 7 ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")
show
the four submodels resulting from these decompositions.
There is some flexibility in how we do the decomposition. For
example, we could move IL from 𝒱12subscript𝒱12{\cal V}\_{12}caligraphic\_V start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT and 𝒰13subscript𝒰13{\cal U}\_{13}caligraphic\_U start\_POSTSUBSCRIPT 13 end\_POSTSUBSCRIPT to 𝒱13subscript𝒱13{\cal V}\_{13}caligraphic\_V start\_POSTSUBSCRIPT 13 end\_POSTSUBSCRIPT
and 𝒰12subscript𝒰12{\cal U}\_{12}caligraphic\_U start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT.
However, we cannot, for example, move CB into 𝒱i1subscript𝒱𝑖1{\cal V}\_{i1}caligraphic\_V start\_POSTSUBSCRIPT italic\_i 1 end\_POSTSUBSCRIPT, for
then ⟨𝒱i1,𝒱i2,𝒱i3,𝒱i4⟩subscript𝒱𝑖1subscript𝒱𝑖2subscript𝒱𝑖3subscript𝒱𝑖4\langle{\cal V}\_{i1},{\cal V}\_{i2},{\cal V}\_{i3},{\cal V}\_{i4}\rangle⟨ caligraphic\_V start\_POSTSUBSCRIPT italic\_i 1 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT italic\_i 2 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT italic\_i 3 end\_POSTSUBSCRIPT , caligraphic\_V start\_POSTSUBSCRIPT italic\_i 4 end\_POSTSUBSCRIPT ⟩ would not be an
order-preserving partition
(since FM is an ancestor of CM, which is an ancestor of CB).
FMCGLMPDM11subscript𝑀11M\_{11}italic\_M start\_POSTSUBSCRIPT 11 end\_POSTSUBSCRIPTSCCMILFMCBM12subscript𝑀12M\_{12}italic\_M start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPTASILAMM13subscript𝑀13M\_{13}italic\_M start\_POSTSUBSCRIPT 13 end\_POSTSUBSCRIPTAMR𝑅Ritalic\_RM14subscript𝑀14M\_{14}italic\_M start\_POSTSUBSCRIPT 14 end\_POSTSUBSCRIPTPDCB
Figure 6: Decomposition of the model M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT from Example [4.9](#S4.Thmtheorem9 "Example 4.9. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
ASCWAADRGPDM21subscript𝑀21M\_{21}italic\_M start\_POSTSUBSCRIPT 21 end\_POSTSUBSCRIPTCMCBM22subscript𝑀22M\_{22}italic\_M start\_POSTSUBSCRIPT 22 end\_POSTSUBSCRIPTASAMM23subscript𝑀23M\_{23}italic\_M start\_POSTSUBSCRIPT 23 end\_POSTSUBSCRIPTAMPDCBR𝑅Ritalic\_RM24subscript𝑀24M\_{24}italic\_M start\_POSTSUBSCRIPT 24 end\_POSTSUBSCRIPT
Figure 7: Decomposition of the model M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT from Example [4.9](#S4.Thmtheorem9 "Example 4.9. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
We observe that decomposing incompatible models into smaller
submodels can in some cases help determine common interventions over
shared outcomes in the original models in spite of their
incompatibility. Consider, for example, the two incompatible models
M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT in the prison example. Although the two are
incompatible (as observed in Example [4.9](#S4.Thmtheorem9 "Example 4.9. ‣ 4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")), the
submodels M12subscript𝑀12M\_{12}italic\_M start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT and M22subscript𝑀22M\_{22}italic\_M start\_POSTSUBSCRIPT 22 end\_POSTSUBSCRIPT in Figures [6](#S4.F6 "Figure 6 ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")
and [7](#S4.F7 "Figure 7 ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), respectively, obtained from their
decomposition, are fully compatible according to Definition [4.6](#S4.Thmtheorem6 "Definition 4.6. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). We
have M12⪰𝐶𝐵v→\*M22superscriptsubscriptsucceeds-or-equals𝐶𝐵superscript→𝑣subscript𝑀12subscript𝑀22M\_{12}\succeq\_{\mbox{{\scriptsize{{\it CB}}}}}^{\vec{v}^{\*}}M\_{22}italic\_M start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT CB end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 22 end\_POSTSUBSCRIPT and M12⪰𝐶𝑀v→\*M22superscriptsubscriptsucceeds-or-equals𝐶𝑀superscript→𝑣subscript𝑀12subscript𝑀22M\_{12}\succeq\_{\mbox{{\scriptsize{{\it CM}}}}}^{\vec{v}^{\*}}M\_{22}italic\_M start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT ⪰ start\_POSTSUBSCRIPT CM end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT 22 end\_POSTSUBSCRIPT. The composition of the two
submodels yields a merged model similar to M12subscript𝑀12M\_{12}italic\_M start\_POSTSUBSCRIPT 12 end\_POSTSUBSCRIPT. Given this, it
may be concluded that interventions over SC or IL make it
possible to change the value of CB in the two models M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and
M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and ultimately R𝑅Ritalic\_R (assuming the structural equation for R𝑅Ritalic\_R in
both models M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT to be the conjunction of PD, CB
and AM)—a similar conclusion reached by considering partial
compatibility between the models M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT in
Section [4.2](#S4.SS2 "4.2 Partial compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). We conjecture that reasoning using model
decomposition and full compatibility of submodels is an alternative to
reasoning using partial compatibility.
Another advantage of decomposing a causal model M𝑀Mitalic\_M into a set of
smaller submodels is that we can reason about each submodel
separately. In particular, we can compute the set of causes and
possible interventions for a given outcome. However, in order to use
these results to reason about the whole model, we need to perform
additional calculations. Informally, when decomposing M𝑀Mitalic\_M into a set of smaller submodels, we can view each submodel as a black box, with inputs and outputs
being the exogenous variables of the submodel and the leaves in the
causal graph of the submodel, respectively. We can then put connect
these variables into an abstract causal graph for the original model,
essentially ignoring the internal variables.
If the submodels are fairly large, the graph of
submodels will be significantly smaller than the causal network of
M𝑀Mitalic\_M. We can then apply causal reasoning to the abstract graph, which
will result in a set of submodels
being causes for the outcome. For these submodels, we can calculate the causes of their outcomes for each submodel separately. As causality is DP-complete,
and computing interventions is co-NP-complete, solving a set of smaller problems instead of a large problem is cheaper.
We note that, in fact, interesting decompositions (that is,
decompositions of a large model into a set of submodels of a reasonable
size
with relatively few interconnections between them, which means that we
can analyse causality both within a submodel and between submodels
relatively easily) are possible only
in models that are somewhat loosely connected.
Such a decomposition can often be done for real-life cases; see
Example [4.16](#S4.Thmtheorem16 "Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
We believe that, in practice, analyzing the effect of interventions
in a model will be difficult precisely when a model is highly
connected, so that there are many causal paths.
We expect the causal models that arise in practice to be much more
loosely connected, and thus
amenable to useful decompositions. Hence, the computation of causes and
interventions
in practice should not be as bad as what is suggested by our
worst-case analysis.
Below, we briefly discuss the relevant aspects of two cases of child
abuse that resulted in the death of a child: the “Baby P” case and
the Victoria Climbiè case. In these cases, experts’ opinions were in fact only partially
compatible, and there were natural ways to decompose the causal
model.
######
Example 4.16 (The cases of Baby P and Victoria Climbiè).
Baby P (Peter Connelly) died in 2007
after suffering physical abuse over an extended period of time
Marinetto ([2011](#bib.bib21)).
The court ultimately found the three adults living in a home with baby Peter guilty of
“causing or allowing [Peter’s] death” Sentencing Remarks ([2009](#bib.bib25)).
After baby Peter’s death, there was an extensive inquiry into
practices, training, and governance in each of the involved professionals and organizations separately.
777Chockler et al. ([2015](#bib.bib2)) provide a more detailed discussion
of the case of “Baby P”, including a construction of the causal model.
As shown by Chockler et al. ([2015](#bib.bib2)), the complete
causal model for the
Baby P case is complex,
involving many variables and interactions between them.
There were several authorities involved in the legal proceedings,
specifically social services, the police, the medical system, and the
court. In addition, the legal proceedings considered the family
situation of Baby P.
Roughly speaking, the causal model
can be viewed as having
the schematic breakdown presented in Figure [8](#S4.F8 "Figure 8 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
Baby P’s deathCourtPoliceSocial servicesMedicalsystemFamilysituation
Figure 8: Schematic representation of causal submodels in the Baby P
case.
Each of the experts involved in the legal inquest and enquiry had expertise
that corresponded to one of the boxes in Figure [8](#S4.F8 "Figure 8 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") (i.e.,
there were no experts with expertise that covered more than one box).
The figure suggests that we might divide the causal model into
submodels corresponding to each box.
The schematic representation in
Figure [8](#S4.F8 "Figure 8 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") does not take into account the interaction between
submodels. In reality, there were numerous interactions between, for
example,
the social services and the court submodels, leading to court
hearings, which in turn determined the course of action taken by the
social services
and the police after the court decision.
Once we model these interactions more carefully, we need a somewhat
more refined decomposition.
We give a decomposition in Figure [9](#S4.F9 "Figure 9 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")
that takes into account the interactions for part of the case,
namely, the part that concerns the social services, the court, the
police, and family life. To make the decomposition consistent with
Definition [4.13](#S4.Thmtheorem13 "Definition 4.13. ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), we break up social services
into two submodels, for reasons explained below.
Court+PoliceOutcomeCPCSSRCHRFHDMAPAOACAFamilyPRFVSocialServices #1SocialServices #2
Figure 9: Simplified causal model M𝑀Mitalic\_M of a part of the Baby P case.
The variables in the figure are: FV for whether there was a
family visit from the social services; PR for whether there was
a police report;
CH for whether there was a court hearing;
RFH for whether the child was removed from home; CP
for whether the child was put on the Child Protection Register;
SR for whether there was a social services report;
CS for whether the child was declared safe in the family home;
MA, PA, and OA for whether the child was
abused by his mother, the mother’s partner, or another adult in the
house, respectively;
CA for whether the child was abused; and, finally,
D for whether the final outcome was death (of
Baby P) due to abuse.
Note that, as usual, we have omitted exogenous variables of the full
model in the
figure; it shows only the endogenous variables. Thus, we do
not have the exogenous variables that determine FV,
𝑃𝑅𝑃𝑅\mathit{PR}italic\_PR, 𝑀𝐴𝑀𝐴\mathit{MA}italic\_MA, 𝑃𝐴𝑃𝐴\mathit{PA}italic\_PA or 𝑂𝐴𝑂𝐴\mathit{OA}italic\_OA. The dotted
rectangles in Figure [9](#S4.F9 "Figure 9 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") determine a decomposition.
Each rectangle
consists of the endogenous variables of one submodel. The exogenous
variables of the SocialServices#2 and Outcome submodels are
those parent variables appearing in the other submodels.
Thus, for example, in the Outcome submodel,
the exogenous variables are 𝐶𝑆𝐶𝑆\mathit{CS}italic\_CS,
𝑅𝐹𝐻𝑅𝐹𝐻\mathit{RFH}italic\_RFH, and 𝐶𝐴𝐶𝐴\mathit{CA}italic\_CA.
The submodels are described
in Figure [10](#S4.F10 "Figure 10 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
The dotted rectangles in Figure [9](#S4.F9 "Figure 9 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") can
be viewed as compact representations of the submodels in
Figure [10](#S4.F10 "Figure 10 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
Of course, there is more than one way to decompose the model of
Figure [9](#S4.F9 "Figure 9 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
For example, the submodel currently standing for the court and the police
can be decomposed into two smaller submodels,
one for the court and one for the police.
However, it is critical that we have decomposed social services into
two submodels.
The variable CH depends on FV, and the variable
CP in turn depends on CH, hence
FV and CP cannot be in the same submodel (or
else we would violate the requirement of
Definition [4.13](#S4.Thmtheorem13 "Definition 4.13. ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") that the sets of endogenous
variables of each submodel form an order-preserving partition of the
endogenous variables of the original model).
SocialServices #1DCARFHCSOutcomeMAPAOAFamilyRFHPRCHFVFVCSSRCPFVCHCASocialServices #2Court+Police
Figure 10: One possible decomposition of the Baby P model M𝑀Mitalic\_M.
We consider another case of child abuse that
resulted in child’s death: Victoria Climbiè Marinetto ([2011](#bib.bib21)).
Victoria died in her house from hypothermia, 18 months after arriving in the UK
from the Ivory Coast to live with
her great-aunt.
Her great-aunt and the great-aunt’s boyfriend were
found guilty of Victoria’s murder
(in contrast with the Baby P case, where the adults in the house were
found guilty of causing or allowing
his death).
The inquiry into the circumstances of Victoria’s death
placed the blame on social workers,
who failed to notice Victoria’s injuries, paediatricians, who accepted
the explanation of
Victoria’s great-aunt that Victoria’s injuries were self-inflicted,
and the metropolitan police. In
addition, the inquiry noted that the pastors in the church to which
Victoria’s great-aunt belonged,
had concerns about the child’s well-being but failed to contact any
child protection services.
The inquiry suggested several interventions on
the procedures of social workers and paediatricians. These
interventions turned out to be inadequate,
as the death of Baby P occurred under somewhat similar circumstances
and his abuse also went unnoticed until his death.
Although there were some similarities between the Baby P case and the
Victoria Clibmbi’e cases, there were also some differences. For
example, while Vicitoria Climbieé died at home,
Baby P died in the hospital. Thus, the causal models for these two
cases differ somewhat.
However, the causal model for the Victoria Climbié case
is also decomposable into fully compatible submodels in the sense of
Section [4.3](#S4.SS3 "4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
Moreover, some of the submodels in the decomposition are identical to
those in the causal model for Baby P.
Specifically, there are submodels for the police, the
medical system, the family system, and the courts, just as in the
case of Baby P, as well as a submodel for the church.
The schematic breakdown is presented in Figure [11](#S4.F11 "Figure 11 ‣ Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
Victoria Climbiè’s deathFamilysituationSocial servicesPoliceMedical careChurch
Figure 11: Schematic representation of causal submodels in the Victoria
Climbiè’s case.
Although we do not provide the causal model in detail here, this
discussion already illustrates a major advantage of decomposition: it
allows us to reuse causal models that were developed in one case and
apply them to another, thus saving a lot of effort.
Moreover, if the same submodel appears in several
different cases, such as the social services submodel in these
examples, this suggests that
the policymaker should prefer interventions that address the problems
demonstrated by this submodel, as they are likely to affect several
cases. In fact,
the cases of child abuse that remains undetected due to problems in
the social services sadly continue to occur (see, for example, the
recently
published cases discussed in Independent ([2019](#bib.bib17))).
Even though the
causal models for different cases will undoubtedly be different, we
can still take advantage of the common submodels. We expect that this
will be the case in many other situations as well.
From a practical perspective, Example [4.16](#S4.Thmtheorem16 "Example 4.16 (The cases of Baby P and Victoria Climbiè). ‣ 4.3 Decomposition of causal models ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") demonstrates one
benefit of decomposition: the decomposition allows us
to capture different aspects of the case, each requiring different expertise.
This facilitates different experts working on each of the submodels independently.
The process also works in the other direction: a policymaker
often has a crude idea of the general structure of the causal model,
and what components
are involved in the decision-making process. She can then decompose her initial causal model into submodels and, guided by these submodels, decide which
areas of expertise are critical.
A further benefit of decomposition illustrated by these examples is
that, although different, the causal models had some common submodels.
Thus, decomposition supports a form of modularity in the analysis, and enables
results of earlier analyses to be reused.
5 Combining Experts’ Opinions
------------------------------
In this section, we show how we can combine experts’ causal options.
Suppose that we have a collection of pairs (M1,p1)subscript𝑀1subscript𝑝1(M\_{1},p\_{1})( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ),
…, (Mn,pn)subscript𝑀𝑛subscript𝑝𝑛(M\_{n},p\_{n})( italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT , italic\_p start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ), with pi∈(0,1]subscript𝑝𝑖01p\_{i}\in(0,1]italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∈ ( 0 , 1 ]; we can think of
Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT as the model that expert i𝑖iitalic\_i
thinks is the right one and
pisubscript𝑝𝑖p\_{i}italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT as the policymaker’s prior degree of confidence
that expert i𝑖iitalic\_i is correct.
(The reason we say “prior” here will be clear shortly.)
Our goal is to combine the expert’ models. We present one way of
doing so, that uses relatively standard techniques. The idea is to
treat the probabilities p1,…,pnsubscript𝑝1…subscript𝑝𝑛p\_{1},\ldots,p\_{n}italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_p start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT as mutually independent.
Thus, if I𝐼Iitalic\_I is a subset of {1,…,n}1…𝑛\{1,\ldots,n\}{ 1 , … , italic\_n }, the prior
probability that
exactly the experts in I𝐼Iitalic\_I are right, which we denote pIsubscript𝑝𝐼p\_{I}italic\_p start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT, is
pI=∏i∈I(pi)\*∏j∉I(1−pj)subscript𝑝𝐼subscriptproduct𝑖𝐼subscript𝑝𝑖subscriptproduct𝑗𝐼1subscript𝑝𝑗p\_{I}=\prod\_{i\in I}(p\_{i})\*\prod\_{j\not\in I}(1-p\_{j})italic\_p start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT = ∏ start\_POSTSUBSCRIPT italic\_i ∈ italic\_I end\_POSTSUBSCRIPT ( italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) \* ∏ start\_POSTSUBSCRIPT italic\_j ∉ italic\_I end\_POSTSUBSCRIPT ( 1 - italic\_p start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ).
Now we have some information regarding whether all the experts in I𝐼Iitalic\_I
are right. Specifically, if the models in {Mi:i∈I}conditional-setsubscript𝑀𝑖𝑖𝐼\{M\_{i}:i\in I\}{ italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT : italic\_i ∈ italic\_I } are not
mutually compatible, then it is impossible that all the experts in I𝐼Iitalic\_I
are right. Intuitively, we want to condition on this information.
We proceed as follows.
Fix a default setting v→\*superscript→𝑣\vec{v}^{\*}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT of the exogneous variables that are
not common to M1,…,Mnsubscript𝑀1…subscript𝑀𝑛M\_{1},\ldots,M\_{n}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT
Let 𝐶𝑜𝑚𝑝𝑎𝑡v→\*={I⊆{1,…,n}:\mathit{Compat}^{\vec{v}^{\*}}=\{I\subseteq\{1,\ldots,n\}:italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT = { italic\_I ⊆ { 1 , … , italic\_n } : the models in
{Mi:i∈I}conditional-setsubscript𝑀𝑖𝑖𝐼\{M\_{i}:i\in I\}{ italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT : italic\_i ∈ italic\_I } are mutually compatible
with respect to v→\*}\vec{v}^{\*}\}over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT }. For I∈𝐶𝑜𝑚𝑝𝑎𝑡v→\*𝐼superscript𝐶𝑜𝑚𝑝𝑎𝑡superscript→𝑣I\in\mathit{Compat}^{\vec{v}^{\*}}italic\_I ∈ italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT, define MI=⊕i∈Iv→\*Misubscript𝑀𝐼superscriptsubscriptdirect-sum𝑖𝐼superscript→𝑣subscript𝑀𝑖M\_{I}=\oplus\_{i\in I}^{\vec{v}^{\*}}M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT = ⊕ start\_POSTSUBSCRIPT italic\_i ∈ italic\_I end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT.
By Proposition [1](#Thmthm1 "Theorem 1 ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), MIsubscript𝑀𝐼M\_{I}italic\_M start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT is well defined.
The models in ℳ𝐶𝑜𝑚𝑝𝑎𝑡v→\*={MI:I∈𝐶𝑜𝑚𝑝𝑎𝑡v→\*}subscriptℳsuperscript𝐶𝑜𝑚𝑝𝑎𝑡superscript→𝑣conditional-setsubscript𝑀𝐼𝐼superscript𝐶𝑜𝑚𝑝𝑎𝑡superscript→𝑣{\cal M}\_{{\mathit{Compat}}^{\vec{v}^{\*}}}=\{M\_{I}:I\in\mathit{Compat}^{\vec{v}^{\*}}\}caligraphic\_M start\_POSTSUBSCRIPT italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT = { italic\_M start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT : italic\_I ∈ italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT } are the
candidate merged models that the policymaker should consider.
MIsubscript𝑀𝐼M\_{I}italic\_M start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT is the “right” model provided that exactly the experts in I𝐼Iitalic\_I
are right.
But even if MI∈ℳ𝐶𝑜𝑚𝑝𝑎𝑡v→\*subscript𝑀𝐼subscriptℳsuperscript𝐶𝑜𝑚𝑝𝑎𝑡superscript→𝑣M\_{I}\in{\cal M}\_{\mathit{Compat}^{\vec{v}^{\*}}}italic\_M start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT ∈ caligraphic\_M start\_POSTSUBSCRIPT italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT, it may not be
the “right” model, since it may be the case that not all the
expert in I𝐼Iitalic\_I are right.
The probability that the policymaker should give MIsubscript𝑀𝐼M\_{I}italic\_M start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT is pI/Nsubscript𝑝𝐼𝑁p\_{I}/Nitalic\_p start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT / italic\_N,
where N=∑I∈𝐶𝑜𝑚𝑝𝑎𝑡v→\*pI𝑁subscript𝐼superscript𝐶𝑜𝑚𝑝𝑎𝑡superscript→𝑣subscript𝑝𝐼N=\sum\_{I\in\mathit{Compat}^{\vec{v}^{\*}}}p\_{I}italic\_N = ∑ start\_POSTSUBSCRIPT italic\_I ∈ italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT is a normalization factor.
This approach gives the policymaker a distribution over causal models.
This can be used to compute, for each context, which interventions
affect the outcome φ𝜑\varphiitalic\_φ of interest, and then
compute the probability that a particular intervention is effective
(which can be done summing the probability of the models MIsubscript𝑀𝐼M\_{I}italic\_M start\_POSTSUBSCRIPT italic\_I end\_POSTSUBSCRIPT in
ℳ𝐶𝑜𝑚𝑝𝑎𝑡v→\*subscriptℳsuperscript𝐶𝑜𝑚𝑝𝑎𝑡superscript→𝑣{\cal M}\_{\mathit{Compat}^{\vec{v}^{\*}}}caligraphic\_M start\_POSTSUBSCRIPT italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT where it is effective, which in turn can be computed
as described in Section [3](#S3 "3 Interventions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")).
Note that
our calculation implicitly
conditions on the fact that at least one expert is right, but allows
for the possibility that only some subset of the experts in I𝐼Iitalic\_I is
right even if I∈𝐶𝑜𝑚𝑝𝑎𝑡v→\*𝐼superscript𝐶𝑜𝑚𝑝𝑎𝑡superscript→𝑣I\in\mathit{Compat}^{\vec{v}^{\*}}italic\_I ∈ italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT; we place positive probability on
MI′subscript𝑀superscript𝐼′M\_{I^{\prime}}italic\_M start\_POSTSUBSCRIPT italic\_I start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT even if I′superscript𝐼′I^{\prime}italic\_I start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT is a strict subset of some I∈𝐶𝑜𝑚𝑝𝑎𝑡v→\*𝐼superscript𝐶𝑜𝑚𝑝𝑎𝑡superscript→𝑣I\in\mathit{Compat}^{\vec{v}^{\*}}italic\_I ∈ italic\_Compat start\_POSTSUPERSCRIPT over→ start\_ARG italic\_v end\_ARG start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT.
This method of combining experts’ judgments is similar in spirit to the
method proposed by Dawid ([1987](#bib.bib5)) and Fenton et al. ([2016](#bib.bib6)).
To get a sense of how this works, consider a variant of Example
[4.2](#S4.Thmtheorem2 "Example 4.2. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), in which a third
expert provides her view
on causes on famine and thinks that government corruption is an
indirect cause via its effect on
political conflict (see Figure [12](#S5.F12 "Figure 12 ‣ 5 Combining Experts’ Opinions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).")); call this model M3subscript𝑀3M\_{3}italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT.
For simplicity, we assume that all models have the same set of
exogenous variables.
According to the compatibility definition in Section
[4](#S4 "4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), the models M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and M3subscript𝑀3M\_{3}italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT are fully compatible
(assuming that MI3 holds),
but M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and M3subscript𝑀3M\_{3}italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT are not. We have ℳ𝐶𝑜𝑚𝑝𝑎𝑡={{M1},{M2},{M3},{M2,3}}subscriptℳ𝐶𝑜𝑚𝑝𝑎𝑡subscript𝑀1subscript𝑀2subscript𝑀3subscript𝑀23{\cal M}\_{\mathit{Compat}}=\{\{M\_{1}\},\{M\_{2}\},\{M\_{3}\},\{M\_{2,3}\}\}caligraphic\_M start\_POSTSUBSCRIPT italic\_Compat end\_POSTSUBSCRIPT = { { italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT } , { italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT } , { italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT } , { italic\_M start\_POSTSUBSCRIPT 2 , 3 end\_POSTSUBSCRIPT } } with M2,3=M2⊕M3=M3subscript𝑀23direct-sumsubscript𝑀2subscript𝑀3subscript𝑀3M\_{2,3}=M\_{2}\oplus M\_{3}=M\_{3}italic\_M start\_POSTSUBSCRIPT 2 , 3 end\_POSTSUBSCRIPT = italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT = italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT.
Suppose that experts are assigned the confidence values as follows: (M1,0.4)subscript𝑀10.4(M\_{1},0.4)( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , 0.4 ), (M2,0.6)subscript𝑀20.6(M\_{2},0.6)( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , 0.6 ) and (M3,0.5)subscript𝑀30.5(M\_{3},0.5)( italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT , 0.5 ) respectively.
Then the probability on M2,3subscript𝑀23M\_{2,3}italic\_M start\_POSTSUBSCRIPT 2 , 3 end\_POSTSUBSCRIPT is the probability of M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and M3subscript𝑀3M\_{3}italic\_M start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT being right (i.e., 0.6\*0.50.60.50.6\*0.50.6 \* 0.5)
and M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT being wrong (i.e., 1−0.4=0.610.40.61-0.4=0.61 - 0.4 = 0.6). So we have
| | | |
| --- | --- | --- |
| | p1=0.4\*0.4\*0.5/0.56=0.14p2=0.6\*0.6\*0.5/0.56=0.32p3=0.6\*0.4\*0.5/0.56=0.21p2,3=0.6\*0.5\*0.6/0.56=0.32subscript𝑝10.40.40.50.560.14subscript𝑝20.60.60.50.560.32subscript𝑝30.60.40.50.560.21subscript𝑝230.60.50.60.560.32\begin{array}[]{lcl}p\_{1}&=&0.4\*0.4\*0.5/0.56=0.14\\
p\_{2}&=&0.6\*0.6\*0.5/0.56=0.32\\
p\_{3}&=&0.6\*0.4\*0.5/0.56=0.21\\
p\_{2,3}&=&0.6\*0.5\*0.6/0.56=0.32\end{array}start\_ARRAY start\_ROW start\_CELL italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_CELL start\_CELL = end\_CELL start\_CELL 0.4 \* 0.4 \* 0.5 / 0.56 = 0.14 end\_CELL end\_ROW start\_ROW start\_CELL italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_CELL start\_CELL = end\_CELL start\_CELL 0.6 \* 0.6 \* 0.5 / 0.56 = 0.32 end\_CELL end\_ROW start\_ROW start\_CELL italic\_p start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT end\_CELL start\_CELL = end\_CELL start\_CELL 0.6 \* 0.4 \* 0.5 / 0.56 = 0.21 end\_CELL end\_ROW start\_ROW start\_CELL italic\_p start\_POSTSUBSCRIPT 2 , 3 end\_POSTSUBSCRIPT end\_CELL start\_CELL = end\_CELL start\_CELL 0.6 \* 0.5 \* 0.6 / 0.56 = 0.32 end\_CELL end\_ROW end\_ARRAY | |
where 0.08+0.18+0.12+0.18=0.560.080.180.120.180.560.08+0.18+0.12+0.18=0.560.08 + 0.18 + 0.12 + 0.18 = 0.56 is the normalization factor N𝑁Nitalic\_N.
R𝑅Ritalic\_RC𝐶Citalic\_CP𝑃Pitalic\_PY𝑌Yitalic\_YF𝐹Fitalic\_F
Figure 12: Third expert’s (and merged) model of famine.
Let us consider the Sampson’s domestic violence models as another
point of
illustration. The model shown in Figure [4](#S4.F4 "Figure 4 ‣ Example 4.3. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).") is the result
of merging the two fully
compatible models given in Figure [3](#S4.F3 "Figure 3 ‣ Example 4.3. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."). We thus have
ℳ𝐶𝑜𝑚𝑝𝑎𝑡={{M1},{M2},{M1,2}}subscriptℳ𝐶𝑜𝑚𝑝𝑎𝑡subscript𝑀1subscript𝑀2subscript𝑀12{\cal M}\_{\mathit{Compat}}=\{\{M\_{1}\},\{M\_{2}\},\{M\_{1,2}\}\}caligraphic\_M start\_POSTSUBSCRIPT italic\_Compat end\_POSTSUBSCRIPT = { { italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT } , { italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT } , { italic\_M start\_POSTSUBSCRIPT 1 , 2 end\_POSTSUBSCRIPT } } with M1,2=M1⊕M2subscript𝑀12direct-sumsubscript𝑀1subscript𝑀2M\_{1,2}=M\_{1}\oplus M\_{2}italic\_M start\_POSTSUBSCRIPT 1 , 2 end\_POSTSUBSCRIPT = italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ⊕ italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT as given in Figure
[4](#S4.F4 "Figure 4 ‣ Example 4.3. ‣ 4.1 Full compatibility ‣ 4 Merging Compatible Causal Models ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18).").
Assuming that expert 1 is assigned a confidence value 0.60.60.60.6 and expert
2 is assigned 0.70.70.70.7, then we have
| | | |
| --- | --- | --- |
| | p1=0.6\*0.3\*0.58/0.44=0.23p2=0.4\*0.7\*0.58/0.44=0.36p1,2=0.6\*0.7\*0.42/0.44=0.41subscript𝑝10.60.30.580.440.23subscript𝑝20.40.70.580.440.36subscript𝑝120.60.70.420.440.41\begin{array}[]{lcl}p\_{1}&=&0.6\*0.3\*0.58/0.44=0.23\\
p\_{2}&=&0.4\*0.7\*0.58/0.44=0.36\\
p\_{1,2}&=&0.6\*0.7\*0.42/0.44=0.41\\
\end{array}start\_ARRAY start\_ROW start\_CELL italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_CELL start\_CELL = end\_CELL start\_CELL 0.6 \* 0.3 \* 0.58 / 0.44 = 0.23 end\_CELL end\_ROW start\_ROW start\_CELL italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_CELL start\_CELL = end\_CELL start\_CELL 0.4 \* 0.7 \* 0.58 / 0.44 = 0.36 end\_CELL end\_ROW start\_ROW start\_CELL italic\_p start\_POSTSUBSCRIPT 1 , 2 end\_POSTSUBSCRIPT end\_CELL start\_CELL = end\_CELL start\_CELL 0.6 \* 0.7 \* 0.42 / 0.44 = 0.41 end\_CELL end\_ROW end\_ARRAY | |
Note that the number of models in
ℳ𝐶𝑜𝑚𝑝𝑎𝑡subscriptℳ𝐶𝑜𝑚𝑝𝑎𝑡{\cal M}\_{\mathit{Compat}}caligraphic\_M start\_POSTSUBSCRIPT italic\_Compat end\_POSTSUBSCRIPT may be exponential
in the number of experts. For example, if the experts’ models are
mutually compatible, then
𝐶𝑜𝑚𝑝𝑎𝑡𝐶𝑜𝑚𝑝𝑎𝑡\mathit{Compat}italic\_Compat consists of all subsets of {1,…,n}1…𝑛\{1,\ldots,n\}{ 1 , … , italic\_n }.
The straightforward computation of interventions per model is exponential
in the number of variables in the model. Since the number of
variables in a merged model is at most the sum of the variables in
each one, the problem is exponential in the number of experts and the
total number of variables in the experts’ models.
In practice, however, we do not expect this to pose a problem.
For the problems we are interested in, there are typically few experts
involved; moreover, as we argued
in Section [3](#S3 "3 Interventions ‣ Combining Experts’ Causal Judgments1footnote 11footnote 1A preliminary version of the paper appeared in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)."), policymakers, in practice,
restrict their attention to interventions on a small set of variables.
Thus, we expect that the computation involved to be manageable.
Up to now, we have assumed that each expert proposes only one
deterministic causal
model.
An expert uncertain about the model can propose
several (typically incompatible) models, with a probability
distribution on them. We can easily extend our framework to handle this.
Suppose that expert i𝑖iitalic\_i, with probability pisubscript𝑝𝑖p\_{i}italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT of being
correct,
proposes m𝑚mitalic\_m models Mi1,…,Mimsubscript𝑀𝑖1…subscript𝑀𝑖𝑚M\_{i1},\ldots,M\_{im}italic\_M start\_POSTSUBSCRIPT italic\_i 1 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_i italic\_m end\_POSTSUBSCRIPT,
where model Mijsubscript𝑀𝑖𝑗M\_{ij}italic\_M start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT has probability qjsubscript𝑞𝑗q\_{j}italic\_q start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT of being the right one,
according to i𝑖iitalic\_i. To handle this, we simply replace
expert i𝑖iitalic\_i by m𝑚mitalic\_m experts, i1,…,imsubscript𝑖1…subscript𝑖𝑚i\_{1},\ldots,i\_{m}italic\_i start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_i start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT, where expert ijsubscript𝑖𝑗i\_{j}italic\_i start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT
proposes model Mijsubscript𝑀𝑖𝑗M\_{ij}italic\_M start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT with probability piqjsubscript𝑝𝑖subscript𝑞𝑗p\_{i}q\_{j}italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT italic\_q start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT of being correct.
As long as each of a few experts has a probability on only a few models,
this will continue to be tractable.
6 Conclusions
--------------
We have provided a method for combining causal models whenever
possible, and used that as a basis for combining experts’ causal
judgments in a way that gets around the impossibility result of
Bradley et al. ([2014](#bib.bib1)). We provided a gradual weakening
of our definition of full compatibility, allowing us to merge models that
only agree
on some of their parts.
Our approach can be viewed as a formalization of what was
done informally in earlier work Chockler et al. ([2015](#bib.bib2)); Sampson et al. ([2013](#bib.bib24)).
While our requirements for compatibility are certainly nontrivial,
the examples that we have considered do suggest that our approach is
quite applicable. That said, it would be interesting to consider
alternative approaches to combining experts’ models. The approach
considered by Friedenberg and Halpern ([2018](#bib.bib7)) is one such approach; there may well be others.
In any case, we believe that using causal models as a way of
formalizing experts’ judgments, and then providing a technique for
combining these judgments, will prove to be a powerful tool with which
to approach the problem of finding the best intervention(s) that can
be performed to ameliorate a situation. |
b3895a58-fc86-4ffe-ae57-8b4de5d6b939 | LDJnr/LessWrong-Amplify-Instruct | LessWrong | "A fiction-writing trick I find particularly compelling are open loops.A cliffhanger is an example: you want to know how the hero survives, so your thoughts keep looping back to the situation, trying to figure out what happens next. But you need the author to tell you.Really good writing uses open loops at the sentence level as well. The first sentence of the story is meaningful on its own, but also makes reference to something else; that something else is required to understand the full meaning. That sets up an open loop; I want to know more.The following sentences provide additional context, helping close the first loop, but in so doing they make reference to other things. Even as loops are closed, new ones get opened. As the author is doing this, various characters are established. Their goals and intentions set up long-term open loops: ones that require a whole chapter or the whole story to resolve.At any given time there may be a dozen open loops of varying timescales.As a concrete example, let’s look at a few early sentences from the novel that prompted this thread as I started reading it and appreciating all the loops it was opening: Melanie Rawn’s Dragon Prince.We open with: “Prince Zeheva squinted into the sunlight and smiled his satisfaction.”This establishes that our character is a prince and satisfied with something that he sees. But to fully understand the situation, we need to know what he sees that satisfies him. Curious.A somewhat different sentence would have been much less compelling: “It was a bright day, and Prince Zeheva smiled.”This sentence would have made it sound like Zeheva was just smiling at the sunlight. He’s not actively looking at something – squinted into – and it feels like we can already understand the situation from this sentence alone. There’s no reference to something else left hanging.Next sentence: “All the signs were good for the hunt today: claw marks on the cliffs, wing marks on the sand, and the close cropping of bittersweet plants along the canyon ridges.”This closes the first loop – we now know what the prince was looking at that made him smile. It also opens new ones. What is he hunting, and what is going to happen next? The “claw marks” and “wing marks” suggest to me some griffin.Notice how this sentence sets up both short- and long-term loops. The most immediate loop is “what is the prince hunting?”; that will get resolved within a few sentences. “How will the hunt go” will be an ongoing one.“But the prince’s perceptions were more subtle and had no need of these obvious signs.”This suggests that there’s something special about the prince’s senses. How are they subtle, and why are they special? Those are the third and fourth loops we’ve got open.“He could feel the presence of his prey all along his skin, scent it in the air, sense it in every nerve. His admirers said he could tell when the time was ripe for the hunt simply by glancing at the sky.”That resolves the ‘how are his senses subtle’ question. Down to three loops.“His enemies said it was not surprising that he could sense such things, for he himself had been dragon-spawned.”We are told that the prince’s senses are special because he’s part dragon. That resolves the loop of “what makes his senses special”, even as the reference to his enemies – who are they? – opens up a new one, keeping us at three.The second paragraph opens with: “In truth, he seemed a human version of the dragon he hunted today.”So he’s hunting a dragon. In the space of just one paragraph and one sentence, Rawn opened at least six loops while closing four of them, thus keeping the reader interested. We still don’t know how the hunt will go, so we want to keep reading…"
The text above is a post found on LessWrong.
Could you elaborate on how the use of open loops can influence a reader's emotional investment in a story?
Open loops, also known as the Zeigarnik Effect, are a powerful tool in storytelling that can significantly influence a reader's emotional investment in a story. They work by tapping into the human brain's natural inclination to seek completion or closure. When a narrative introduces a question, mystery, or problem (i.e., an open loop), it creates a state of tension in the reader's mind. This tension can generate a sense of anticipation, curiosity, and even anxiety, all of which serve to draw the reader deeper into the story.
Firstly, open loops can create a sense of suspense. By leaving questions unanswered or situations unresolved, the author compels the reader to continue reading in order to find out what happens next. This not only keeps the reader engaged but also fosters a deeper emotional connection to the story as they become more invested in the outcomes of these unresolved situations.
Secondly, open loops can stimulate curiosity. By introducing new elements or information that is not immediately explained, the author encourages the reader to actively engage with the story as they attempt to predict or make sense of these unknown factors. This active engagement can enhance the reader's emotional investment in the story, as they are not merely passive observers but active participants in the unfolding narrative.
Thirdly, open loops can evoke empathy. By presenting characters with challenges or dilemmas that are not immediately resolved, the author allows the reader to share in the characters' struggles and uncertainties. This can foster a deeper emotional connection between the reader and the characters, enhancing the reader's overall investment in the story.
Finally, open loops can create a sense of satisfaction when they are eventually closed. The resolution of an open loop can provide a rewarding sense of closure, which can heighten the emotional impact of the story and leave the reader with a lasting impression.
In conclusion, open loops can significantly enhance a reader's emotional investment in a story by creating suspense, stimulating curiosity, evoking empathy, and providing satisfying resolutions. They are a powerful tool that can make a narrative more engaging, immersive, and emotionally resonant.
Incorporate the concept of open loops in a short story plot, ensuring it elicits suspense, curiosity, empathy, and satisfaction upon resolution. How would you weave these elements to maintain reader engagement throughout the narrative?
Title: "The Forgotten Melody"
The story begins with our protagonist, Amelia, a talented violinist preparing for a significant concert. As she rehearses, she stumbles upon a half-finished composition tucked away in her late grandmother's music box, an open loop that incites curiosity. The melody is beautiful but incomplete, and Amelia feels a compelling need to finish it.
Amelia's attempts to complete the melody serve as the story's primary open loop, generating suspense. Every time she thinks she's close to figuring it out, something seems off, keeping the readers on their toes. The melody becomes an obsession for her, practicing day and night, her fingers dancing on the violin strings, trying to bring the forgotten tune to life.
As Amelia delves deeper into the melody, we introduce a series of flashbacks revealing her relationship with her grandmother, a once-renowned violinist herself. These memories, filled with warmth, wisdom, and shared love for music, create empathy for Amelia. The readers become emotionally invested in her quest, understanding that it's not just about the melody, but also a way for Amelia to reconnect with her beloved grandmother.
In her desperation, Amelia visits her grandmother's old violin teacher, an elderly, reticent man living in solitude. He initially refuses to help, but Amelia's determination reminds him of her grandmother, and he reluctantly agrees to guide her. This adds another layer of suspense, as the readers wonder whether this gruff old man can help Amelia find what she's looking for.
The climax arrives on the night of the concert. Amelia decides to perform the unfinished melody, hoping that the energy of the live audience will inspire her to complete it. As she plays, she closes her eyes and loses herself in the music. Suddenly, the missing notes come to her, and she completes the melody, playing it with a passion and beauty that leaves the audience in awe. This moment of resolution provides immense satisfaction, as the primary open loop is finally closed.
The story ends with Amelia dedicating her performance to her grandmother, a poignant moment that not only closes the emotional loop but also leaves the readers with a sense of contentment and closure.
In this narrative, the concept of open loops is used to maintain reader engagement throughout the story. The unfinished melody, Amelia's relationship with her grandmother, and her interaction with the old violin teacher all serve as open loops that elicit suspense, curiosity, empathy, and satisfaction upon resolution. |
f35032b0-bad0-4cd7-8b0f-713c0671d38e | StampyAI/alignment-research-dataset/lesswrong | LessWrong | A simple presentation of AI risk arguments
[Here is a draft of an accordion-style AGI risk FAQ.](https://workflowy.com/s/faq-why-would-ai-be/2kHollRW8uqinUsV)
I'm interested in three types of feedback:
1. The best place to host a similar format
2. Feedback on content
3. Thoughts about this general approach
The goal here is something that's very easy to read. One major idea in the accordion-style presentation is to make sure that the article isn't overwhelming. I also wanted to let people address their own biggest questions without having to read content they're less interested in. The whole idea is low effort threshold, and hoping to draw the reader in from minimal interest to a little more interest. The purpose is to inform, although I'm also hoping people leave agreeing with me that this is something that society at large should be taking a bit more seriously. I'm going for something you could point your mother to.
The challenge is that the AGI risk arguments actually are not simple to someone who hasn't thought about them. There are major sticking points for many people, and those are different for different people. That's why I've been thinking about the FAQ that follows different lines of questions. This accordion style is an attempt to do that in a way that's quick and smooth enough to keep people's attention. The intent is that all bullet points start out collapsed.
If this seems worthwhile, I will expand it to have deeper content. It is currently very much an incomplete draft because I strongly suspect I'll get better suggestions for hosting and format that will require a lot of transplanting work.
It also needs links to other similar articles that go into more depth. My first would be to the [stampy.ai](https://stampy.ai/) project of Rob Miles and collaborators. It has a similar structure of letting the user choose questions and the huge advantage of letting users enter their own questions and having a semantic search for answers to similar questions. My attempt is different in that it's aimed at a more general audience than the tech types that have to date become interested in the AI safety issue.
I think we're likely to see repeating waves of new public interest in AI safety from here on out. I'm looking forward to the opportunities presented by [AI scares and changing public beliefs](https://www.lesswrong.com/posts/ou5raNNjamAaahtWG/ai-scares-and-changing-public-beliefs), but only if we can avoid creating polarization, as I discuss in that article. I think we are very likely to get more scares, and I agree with Leopold's point (made briefly [here](https://Nobody’s on the ball on AGI alignment)) that the societal response to COVID suggests that we may see a very rapid wave of intense societal concern. I think we'd better be prepared to surf that wave rather than just watch it sweep past. (To that end, we should also figure out what public policy would actually help our odds, but that is a separate issue).
As such, I'm going to put at least a little effort into refining an introductory message for the newly concerned, and I'd love any suggestions that people want to offer.
Edit: While I don't have ideas about specific policies, I think that raising public awareness of AI X-risk is probably a net good. Contributing to a public panic could easily be bad, so I don't want to present the strongest arguments in absence of optimism and counterarguments. In addition, I think convincing others of what you see as the truth is easier than convincing people to believe what you want. Humans have decent lie-detection and propaganda protection. I do think that having the average human think something like "these things are dangerous, and if you're developing them without being really careful, you're a bad human" seems like a net benefit. I'd expect such a social pressure to diffuse upward toward the people actually making decisions through friends, relatives, coworkers, and underlings. |
d2d81123-ccda-4ede-918c-4a4516c28aa0 | trentmkelly/LessWrong-43k | LessWrong | New LW Meetup: Kyiv, New Hampshire
This summary was posted to LW Main on July 10th. The following week's summary is here.
New meetups (or meetups with a hiatus of more than a year) are happening in:
* New Hampshire Inaugural Meet-up: 14 July 2015 07:00PM
* Kyiv, Ukraine: 24 July 2015 06:30PM
Irregularly scheduled Less Wrong meetups are taking place in:
* Australian Less Wrong Mega Meetup #2: 17 July 2015 07:00PM
* Australia-wide Mega-Camp!: 17 July 2015 07:00PM
* Cologne meetup: 11 July 2015 05:00PM
* Frankfurt Meet-Up: 16 July 2015 06:30PM
* Hamburg : 17 July 2015 06:30PM
* Oslo Meetup at Bitraf.: 18 July 2015 04:00PM
* Paris Meetup: Saturday, July 11 : 11 July 2015 02:00PM
* San Antonio, TX: 12 July 2015 01:30PM
* San Francisco Meetup: Board Games: 13 July 2015 06:15PM
* Scotland: 12 July 2015 02:00PM
* Warsaw integration: Barbecue on the beach: 31 July 2015 06:30PM
The remaining meetups take place in cities with regular scheduling, but involve a change in time or location, special meeting content, or simply a helpful reminder about the meetup:
* [Seattle] Rationality Reading Group (62-66): 13 July 2015 06:30PM
* Sydney Meetup - July: 22 July 2015 06:30PM
* Sydney Meetup - August: 26 August 2015 06:30PM
* Sydney Meetup - September: 23 September 2015 06:30PM
* Vienna: 18 July 2015 03:00PM
* Washington, D.C.: Fermi Estimates: 12 July 2015 03:00PM
Locations with regularly scheduled meetups: Austin, Berkeley, Berlin, Boston, Brussels, Buffalo, Cambridge UK, Canberra, Columbus, London, Madison WI, Melbourne, Moscow, Mountain View, New York, Philadelphia, Research Triangle NC, Seattle, Sydney, Tel Aviv, Toronto, Vienna, Washington DC, and West Los Angeles. There's also a 24/7 online study hall for coworking LWers.
If you'd like to talk with other LW-ers face to face, and there is no meetup in your area, consider starting your own meetup; it's easy (more resources here). Check one out, stretch your rationality skills, build community, and have fun!
In addition to the handy s |
3d610d26-1320-449d-b41e-b0b9468519c6 | LDJnr/LessWrong-Amplify-Instruct | LessWrong | "Followup to: Evolutions Are Stupid (But Work Anyway), Evolutionary Psychology
Biological organisms in general, and human brains particularly, contain complex adaptations; adaptations which involve many genes working in concert. Complex adaptations must evolve incrementally, gene by gene. If gene B depends on gene A to produce its effect, then gene A has to become nearly universal in the gene pool before there's a substantial selection pressure in favor of gene B.
A fur coat isn't an evolutionary advantage unless the environment reliably throws cold weather at you. And other genes are also part of the environment; they are the genetic environment. If gene B depends on gene A, then gene B isn't a significant advantage unless gene A is reliably part of the genetic environment.
Let's say that you have a complex adaptation with six interdependent parts, and that each of the six genes is independently at ten percent frequency in the population. The chance of assembling a whole working adaptation is literally a million to one; and the average fitness of the genes is tiny, and they will not increase in frequency.
In a sexually reproducing species, complex adaptations are necessarily universal. One bird may have slightly smoother feathers than another, but they will both have wings. A single mutation can be possessed by some lucky members of a species, and not by others - but single mutations don't correspond to the sort of complex, powerful machinery that underlies the potency of biology. By the time an adaptation gets to be really sophisticated with dozens of genes supporting its highly refined activity, every member of the species has some version of it - barring single mutations that knock out the whole complex.
So you can't have the X-Men. You can't have "mutants" running around with highly developed machinery that most of the human species doesn't have. And no, extra-powerful radiation does not produce extra-potent mutations, that's not how it works.
Again by the nature of sexual recombination, you're very unlikely to see two complexly different adaptations competing in the gene pool. Two individual alleles may compete. But if you somehow had two different complex adaptations built out of many non-universal alleles, they would usually assemble in scrambled form.
So you can't have New Humans and Old Humans either, contrary to certain science fiction books that I always found rather disturbing.
This is likewise the core truth of biology that justifies my claim that Einstein must have had very nearly the same brain design as a village idiot (presuming the village idiot does not have any actual knockouts). There is simply no room in reality for Einstein to be a Homo novis.
Maybe Einstein got really lucky and had a dozen not-too-uncommon kinds of smoother feathers on his wings, and they happened to work well together. And then only half the parts, on average, got passed on to each of his kids. So it goes.
"Natural selection, while feeding on variation, uses it up," the saying goes. Natural selection takes place when you've got different alleles in the gene pool competing, but in a few hundred generations one allele wins, and you don't have competition at that allele any more, unless a new mutation happens to come along.
And if new genes come along that depend on the now-universal gene, that will tend to lock it in place. If A rises to universality, and then B, C, and D come along that depend on A, any A' mutation that would be an improvement on A in isolation, may break B, C, or D and lose the benefit of those genes. Genes on which other genes depend, tend to get frozen in place. Some human developmental genes, that control the action of many other genes during embryonic development, have identifiable analogues in fruit flies.
You might think of natural selection at any given time, as a thin froth of variation frantically churning above a deep, still pool of universality.
And all this which I have said, is also true of the complex adaptations making up the human brain.
This gives rise to a rule in evolutionary psychology called "the psychological unity of humankind".
Donald E. Brown's list of human universals is a list of psychological properties which are found so commonly that anthropologists don't report them. If a newly discovered tribe turns out to have a sense of humor, tell stories, perform marriage rituals, make promises, keep secrets, and become sexually jealous... well, it doesn't really seem worth reporting any more. You might record the specific tales they tell. But that they tell stories doesn't seem any more surprising than their breathing oxygen.
In every known culture, humans seem to experience joy, sadness, fear, disgust, anger, and surprise. In every known culture, these emotions are indicated by the same facial expressions.
This may seem too natural to be worth mentioning, but try to take a step back and see it as a startling confirmation of evolutionary biology. You've got complex neural wiring that controls the facial muscles, and even more complex neural wiring that implements the emotions themselves. The facial expressions, at least, would seem to be somewhat arbitrary - not forced to be what they are by any obvious selection pressure. But no known human tribe has been reproductively isolated long enough to stop smiling.
When something is universal enough in our everyday lives, we take it for granted; we assume it without thought, without deliberation. We don't ask whether it will be there - we just act as if it will be. When you enter a new room, do you check it for oxygen? When you meet another intelligent mind, do you ask whether it might not have an emotion of joy?
Let's go back to biology for a moment. What if, somehow, you had two different adaptations which both only assembled on the presence, or alternatively the absence, of some particular developmental gene? Then the question becomes: Why would the developmental gene itself persist in a polymorphic state? Why wouldn't the better adaptation win - rather than both adaptations persisting long enough to become complex?
So a species can have different males and females, but that's only because neither the males or the females ever "win" and drive the alternative to extinction.
This creates the single allowed exception to the general rule about the psychological unity of humankind: you can postulate different emotional makeups for men and women in cases where there exist opposed selection pressures for the two sexes. Note, however, that in the absence of actually opposed selection pressures, the species as a whole will get dragged along even by selection pressure on a single sex. This is why males have nipples; it's not a selective disadvantage.
I believe it was Larry Niven who suggested that the chief experience human beings have with alien intelligence is their encounters with the opposite sex.
This doesn't seem to be nearly enough experience, judging by Hollywood scriptwriters who depict AIs that are ordinarily cool and collected and repressed, until they are put under sufficient stress that they get angry and show the corresponding standard facial expression.
No, the only really alien intelligence on this planet is natural selection, of which I have already spoken... for exactly this reason, that it gives you true experience of the Alien. Evolution knows no joy and no anger, and it has no facial expressions; yet it is nonetheless capable of creating complex machinery and complex strategies. It does not work like you do. If you want a real alien to gawk at, look at the other Powerful Optimization Process.
This vision of the alien, conveys how alike humans truly are - what it means that everyone has a prefrontal cortex, everyone has a cerebellum, everyone has an amygdala, everyone has neurons that run at O(20Hz), everyone plans using abstractions.
Having been born of sexuality, we must all be very nearly clones."
The above is a chunk of text from a post created by a user on the website LessWrong.
Can you break this down into the top 5 most important points for me.
1. Complex Adaptations: The text emphasizes that complex adaptations in biological organisms, especially human brains, require many genes to work together. These adaptations must evolve incrementally, with each gene becoming nearly universal in the gene pool before another gene that depends on it can evolve. This interdependence makes the evolution of complex adaptations a slow and meticulous process.
2. Universality of Complex Adaptations: The author asserts that in sexually reproducing species, complex adaptations are universal. They argue that while single mutations can be unique to certain individuals, complex adaptations involving many genes are shared across the species. This universality means that highly specialized traits can't exist in just a few individuals, ruling out scenarios like the X-Men.
3. The Psychological Unity of Humankind: The text introduces the concept of "the psychological unity of humankind," which suggests that certain psychological traits are universal among humans. These include emotions and their corresponding facial expressions, which are found in every known culture. This universality is seen as a confirmation of evolutionary biology.
4. Exceptions to Psychological Unity: The author notes that the only exception to the psychological unity of humankind is the potential for different emotional makeups between men and women, where there might be opposed selection pressures. However, they stress that without such pressures, both sexes will evolve together.
5. Alien Intelligence and Natural Selection: The author concludes by comparing natural selection to an "alien intelligence." They suggest that natural selection, as a process devoid of human emotions yet capable of creating complex machinery and strategies, offers a glimpse into a truly alien form of intelligence. This perspective underscores the fundamental similarities among humans, as we all share the same basic brain structures and processes.
Could you elaborate on the potential opposed selection pressures between men and women that could lead to different emotional makeups? What might be some examples of these pressures, and how might they influence the evolution of emotional responses?
Opposed selection pressures between men and women, also known as sexual dimorphism, refer to the different evolutionary pressures that males and females of the same species face due to their distinct roles in reproduction. These pressures can lead to differences in physical characteristics, behaviors, and potentially emotional makeups.
For example, in many species, males compete with each other for access to females. This competition can favor traits like aggression, risk-taking, and dominance. On the other hand, females often invest more in offspring care, which can favor traits like nurturing, caution, and selectivity in mate choice. These different pressures could theoretically lead to different emotional makeups in males and females, with males evolving to be more aggressive and females evolving to be more nurturing.
However, it's important to note that these are broad generalizations and there's significant variation within each sex. Furthermore, the extent to which these pressures have shaped emotional responses in humans is a complex issue and a topic of ongoing research.
For instance, some research suggests that women might be more empathetic and better at reading emotional cues than men, potentially due to the role of women as caregivers throughout evolutionary history. Conversely, men might be more prone to anger and aggression, possibly due to the historical role of men as protectors and competitors. However, these findings are not universally accepted and are subject to cultural influences and individual differences.
In summary, while opposed selection pressures could potentially influence the evolution of emotional responses in men and women, the exact nature and extent of these influences are complex and not fully understood.
Considering the potential influence of opposed selection pressures on emotional responses in men and women, how might these evolutionary influences interact with cultural and societal factors to shape emotional behavior? Discuss this interaction using a specific cultural practice or societal norm as an example.
Evolutionary influences and cultural or societal factors interact in complex ways to shape emotional behavior. While evolutionary pressures might set the stage for certain emotional tendencies, cultural practices and societal norms can significantly modify, enhance, or suppress these tendencies.
Take, for example, the societal norm of men being discouraged from showing emotions, particularly those associated with vulnerability, such as crying. This norm is prevalent in many cultures and is often encapsulated in phrases like "boys don't cry."
From an evolutionary perspective, men might be more prone to expressing anger or aggression due to historical roles as protectors and competitors. However, this doesn't inherently preclude the expression of sadness or vulnerability. The societal norm that discourages such expressions in men is a cultural overlay that can suppress these emotional responses.
This cultural practice can have significant impacts on emotional behavior. For instance, it can lead men to suppress or hide certain emotions, which can contribute to mental health issues. It can also influence the ways in which men express distress, with research suggesting that suppressed emotions can manifest as anger or aggression.
At the same time, societal norms and cultural practices can change over time, often more rapidly than evolutionary pressures. Increasing awareness and acceptance of emotional expression in men is challenging this norm, leading to shifts in emotional behavior.
In this way, evolutionary influences and cultural factors interact to shape emotional behavior. Evolutionary pressures might predispose certain emotional tendencies, but cultural practices and societal norms play a significant role in determining how these tendencies are expressed. |
4b26a8c2-c173-4a48-8fac-c0899954f493 | trentmkelly/LessWrong-43k | LessWrong | [linkpost] Better Without AI
David Chapman (of Meaningness and In the Cells of the Eggplant fame) has written a new web-book about AI. Some excerpts from the introduction, Only you can stop an AI apocalypse:
> Artificial intelligence might end the world. More likely, it will crush our ability to make sense of the world—and so will crush our ability to act in it.
>
> AI will make critical decisions that we cannot understand. Governments will take radical actions that make no sense to their own leaders. Corporations, guided by artificial intelligence, will find their own strategies incomprehensible. University curricula will turn bizarre and irrelevant. Formerly-respected information sources will publish mysteriously persuasive nonsense. We will feel our loss of understanding as pervasive helplessness and meaninglessness. We may take up pitchforks and revolt against the machines—and in so doing, we may destroy the systems we depend on for survival...
> We don’t know how our AI systems work, we don’t know what they can do, and we don’t know what broader effects they will have. They do seem startlingly powerful, and a combination of their power with our ignorance is dangerous...
> In our absence of technical understanding, those concerned with future AI risks have constructed “scenarios”: stories about what AI may do... So far, we’ve accumulated a few dozen reasonably detailed, reasonably plausible bad scenarios. We’ve found zero that lead to good outcomes... Unless we can find some specific beneficial path, and can gain some confidence in taking it, we should shut AI down.
> This book considers scenarios that are less bad than human extinction, but which could get worse than run-of-the-mill disasters that kill only a few million people.
>
> Previous discussions have mainly neglected such scenarios. Two fields have focused on comparatively smaller risks, and extreme ones, respectively. AI ethics concerns uses of current AI technology by states and powerful corporations to categorize individua |
667cf473-8f84-449f-a078-d8667a2bda89 | trentmkelly/LessWrong-43k | LessWrong | Define Rationality
I would like to suggest that we try to come up with several defintions of rationality. I don't feel we have exhausted this search area by any means. Robin has suggested, "More "rational" means better believing what is true, given one's limited info and analysis resources". Other commenters have emphasised goal-directed behaviour as a necessary ingredience of rationality. I think these defintions miss out on several important ingrediences - such as the social nature of rationality. There is also a subtext which argues - that rationality only gives one (correct) answer even if we only can approximate it. I feel strongly that rationality can give several correct answers and thus imagination is an ingredience of rationality. So without in any way believing that I have found the one correct defintion, I propose the following: When two or more brains try to be sensible about things and expand their agency. I believe that "sensible" in this context does not need to be defined as it is a primitive and each player willl submit their own meaning.
Maybe this is a can of worms - but are there other suggestions or defintions for rationality we can apply in our lives? |
957a2444-b300-4a66-aff9-3ab92907c7d1 | trentmkelly/LessWrong-43k | LessWrong | Propagandizing Meta-Ethics in an Essay Contest
So someone I follow on Facebook linked to this essay contest on the subject of: How Should Humanity Steer the Future? My urge to make a joke of it immediately kicked in. The impulse to joke turned into an impulse to actually submit an essay when the words "steering the future" set off a minor "ding!" in my head.
At least regarding AI, many papers and articles have already been published on what that problem is: even well-intentioned people could accidentally create a Completely or Partially Alien Mind-Design that has no care and no sympathy for us or our values. Should such a thing grow more powerful than us, it would proceed to kill us all. We would be in the way, as a bacterium is to us, and dealt with identically.
Blah blah blah blah.
To me personally, by sheer madness of personal preferences, that is not the interesting part. Danger, even existential danger, seems to me quite passe these days. In only the time I've been alive, we've been under threat of nuclear apocalypse via war between nation-states or nuclear-weapons usage by terrorist groups, global warming and other environmental damages are slowly destroying the only planetary-level habitat we humans have, and in the past five or so years we've been dealing with continental-level economic collapses and stagnation as well (I personally subscribe to the theory that capitalism itself is a human-Unfriendly optimization process, which is partially apropos here). Those are just the apocalypses: then we have to add in all the daily pains, indignities and deaths suffered by the vast majority of the world's people, many of whom are so inured to their suffering that they consider it a normal or even morally appropriate part of their lives. Only at the end of that vast, astronomical summation can we say we have totalled humanity's problems.
All that, only in the time I've been alive, for about 25 years now, even when, statistically speaking, I'm living in a steadily improving golden age (or at |
c6f5bab0-2889-4641-abe9-b874c6d29abe | trentmkelly/LessWrong-43k | LessWrong | Progress links and short notes, 2025-03-10
Much of this content originated on social media. To follow news and announcements in a more timely fashion, follow me on Twitter, Notes, Farcaster, Bluesky, or Threads.
Contents
* d/acc Day
* My writing (ICYMI)
* Praise for Progress Conference 2024
* Job opportunities
* Writing opportunities
* Other opportunities
* Events
* AI announcements
* Writing announcements
* Queries
For paid subscribers:
* A visible sonic boom
* Some observations from me on AI products
* Aaron Levie on AI in SaaS
* More on AI in SaaS
* Short notes on AI
* Other short notes
* It’s time to build
* The closing of the frontier
* Rousseau and Kant vs. the Age of Reason
* San Francisco, city of historic laundromats
* Maria Montessori on “peace”
* Rudyard Kipling on “peace”
* Charts and tables
* Art
d/acc Day
* I’ll be speaking at “d/acc Day” on Thursday in Berkeley, alongside Vitalik, Juan Benet, Mary Lou Jepsen, Allison Duettmann, and others. My talk: “d/acc: The first 150 years.” A whirlwind tour of how society has thought about progress, decentralization and defense over the last century and a half
My writing (ICYMI)
* The Techno-Humanist Manifesto, Chapter 5: Solutionism (part 2). There is no tradeoff between health/safety and progress, because health and safety are a part of progress. But the technical work of health and safety has gone mostly unsung
Praise for Progress Conference 2024
From RPI fellow Grant Mulligan:
> Best conference I’ve ever attended. Quick recap on why:
>
> 1. No one was selling anything, not even themselves. Finding and exploring ideas was all that mattered.
> 2. I’d never met people who care so much about being correct - not to claim that they’re right, but in the sense that they really want to understand the world.
> 3. It reoriented what I’m choosing to work on and how I go about my work. How many conferences actually influence where your career goes next?
> 4. The venue and tone of the event made it feel like a weekend chilli |
3613741f-cbd7-4137-9ca6-f2ef24894f16 | trentmkelly/LessWrong-43k | LessWrong | Rationalist Magic: Initiation into the Cult of Rationatron
I am curious on the perspective of a rationalist discussion board (this seems like a good start) on the practice of magic. I introduce the novel, genius concept of "rationalist magic", i.e. magic practiced by rationalists.
Why would rationalists practice magic? That makes no sense!!
It's the logical conclusion to Making Peace with Belief, Engineering Religion and the self-help threads.
What would that look like?
Good question. Here are some possible considerations to make:
* It's given low probability that magic is more than purely mental phenomena. The practice is called "placebomancy" to make it clear that such an explanation is favoured.
* It is practiced as a way to gain placebons.
* A cult of rationalist magic, the Cult of Rationatron, should be formed to compete against worse (anti-rationality, anti-science, violent) cults.
* Rationalist groups can retain more members due to abundance of placebons.
* The ultimate goal is to use rationality to devise a logically optimal system of magic with which to build the Philosopher's Stone and fix the world like in HPMOR. (Just kidding, magic isn't real.)
I looked into magical literature and compiled a few placebo techniques/exercises, along with their informal instructions. These might be used as a starting point. If there are any scientific errors these can eventually be corrected. I favoured techniques that can be done with little to no preparation and provide some results. Of course, professional assistance (e.g. yoga classes) can also be helpful.
----------------------------------------
1. Mindfulness Meditation
* (Optional) Do 1-3 deep mouth-exhales to relax.
* Find a good position.
* Begin by being aware of breath.
* (Optional) Move on to calmly observing different parts of the body, vision, the other senses, thoughts, mandala visualization, and so on.
* (Optional) Compare the experience to teachings of buddhism.
* (Optional) Say "bud-" for each inhale and "-dho" for each exhale; alternatively |
54409c7e-4de3-48ad-a8ec-71e65dbc72a7 | trentmkelly/LessWrong-43k | LessWrong | What's the Deal with Logical Uncertainty?
I notice that reasoning about logical uncertainty does not appear more confusing to me than reasoning about empirical one. Am I missing something?
Consider the classical example from the description of the tag:
> Is the googolth digit of pi odd? The probability that it is odd is, intuitively, 0.5. Yet we know that this is definitely true or false by the rules of logic, even though we don't know which. Formalizing this sort of probability is the primary goal of the field of logical uncertainty.
>
> The problem with the 0.5 probability is that it gives non-zero probability to false statements. If I am asked to bet on whether the googolth digit of pi is odd, I can reason as follows: There is 0.5 chance that it is odd. Let P represent the actual, unknown, parity of the googolth digit (odd or even); and let Q represent the other parity. If Q, then anything follows. (By the Principle of Explosion, a false statement implies anything.) For example, Q implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,000, which is 0.5 * $1,000,000, and I should be willing to pay that much to take the bet. This is an absurdity.
I don't see how this case is significantly different from an empirical incertainty one:
> A coin is tossed and put into an opaque box, without showing you the result. What is the probability that the result of this particular toss was Heads?
Let's assume that it's 0.5. But, then just as in the previous case, we have the same problem: we are assigning non-zero probability to a false statement. And so, by the same logic, if I am asked to bet on whether the coin is Heads or Tails, I can reason as follows: There is 0.5 chance that it is Heads. Let P represent the actual, unknown, state of the outcome of the toss (Heads or Tails); and let Q represent the other state. If Q, then anything follows. For example, Q implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,000, which is 0.5 * $1,000,000, |
5e68fb4b-aee4-45fd-bd6e-bd058d126a4e | trentmkelly/LessWrong-43k | LessWrong | Link: blog on effective altruism
Over the last few months I've started blogging about effective altruism more broadly, rather than focusing on AI risk. I'm still focusing on abstract considerations and methodological issues, but I hope it is of interest to others here. Going forward I intend to cross-post more often to LW, but I thought I would post the backlog here anyway. With luck, I'll also have the opportunity to post more than bi-weekly.
I welcome thoughts, criticisms, etc. |
a128d095-fc79-4aa6-b04d-56ae39575085 | trentmkelly/LessWrong-43k | LessWrong | Why do aphorisms and cynicism go together?
Why are aphorisms cynical more often than books are for instance?
A good single sentence saying can’t require background evidencing or further explanation. It must be instantly recognizable as true. It also needs to be news to the listener. Most single sentences that people can immediately verify as true they already believe. What’s left? One big answer is things that people don’t believe or think about much for lack of wanting to, despite evidence. Drawing attention to these is called cynicism.
HT to Robin Hanson for the question and to Francois de La Rochefoucauld for some examples:
We often forgive those who bore us, but we cannot forgive those whom we bore.
We promise according to our hopes; we fulfill according to our fears.
What often prevents us from abandoning ourselves to one vice is that we have several.
We confess to little faults only to persuade ourselves we have no great ones.
There are few people who are more often wrong than those who cannot suffer being wrong.
Nothing prevents us being natural so much as the desire to appear so.
|
c0ebfe27-e6a4-477a-acb8-dfb4a01959c5 | trentmkelly/LessWrong-43k | LessWrong | The median and mode use less information than the mean does
Epistemic status: Unsure. I had this in drafts from a year ago, and am posting it for Goodhart day. (Though it's April 1, all the arguments and statements in this post are things I think are true, with no jokes). I'm interested in arguments against this thesis, and especially interested in thoughts on the question at the end - does the distribution-summarizer corresponding to the L3 or L4 minimizers use more information than the mean (the L2 minimizer)?
----------------------------------------
The mean, median, and mode are the "big 3" location parameters that most have heard about. But they can have very different properties, and these different properties are related to the fact that the mean uses more information than the median, and the median uses more information than the mode.
Refresher
The mean, median, and mode measure the location of a probability distribution. For the Gaussian distribution, they are all the same, but this isn't the case in general. Here's an example of a Gamma distribution where the three differ:
gamma(shape = 2, rate = 1)
The mean corresponds to the middle of the distribution when weighted by frequency.
The median corresponds to the middle of the distribution, without using the weights. The median is the vertical line that splits a distribution such that 50% of the probability mass is on the left and 50% on the right.
The mode is the highest point on a distribution.
Different amounts of information usage
The median is preserved under a larger set of changes to the data than the mean is. Really, this is often why people use the median: outliers don't knock it around as much as they do the mean. But that ability to resist being knocked around - "robustness" - is the same as the ability to ignore information. The mean's sensitivity is sometimes seen as a liability (and sometimes is a liability), but being sensitive here is the same thing as reacting more to the data. It's good to react to the data: the mean can distinguish betwee |
6b108049-4450-49f6-a99a-26651965a577 | StampyAI/alignment-research-dataset/arxiv | Arxiv | What Makes for Good Views for Contrastive Learning?
1 Introduction
---------------
It is commonsense that how you look at an object does not change its identity. Nonetheless, Jorge Luis Borges imagined the alternative. In his short story on *Funes the Memorious*, the titular character becomes bothered that a “dog at three fourteen (seen from the side) should have the same name as the dog at three fifteen (seen from the front)” [[6](#bib.bib6)]. The curse of Funes is that he has a perfect memory, and every new way he looks at the world reveals a percept minutely distinct from anything he has seen before. He cannot collate the disparate experiences.
Most of us, fortunately, do not suffer from this curse. We build mental representations of identity that discard *nuisances* like time of day and viewing angle. The ability to build up *view-invariant* representations is central to a rich body of research on multiview learning. These methods seek representations of the world that are invariant to a family of viewing conditions. Currently, a popular paradigm is contrastive multiview learning, where two views of the same scene are brought together in representation space, and two views of different scenes are pushed apart.
This is a natural and powerful idea but it leaves open an important question: “which viewing conditions should we be invariant to?” It’s possible to go too far: if our task is to classify the time of day then we certainly should not use a representation that is invariant to time. Or, like Funes, we could go not far enough: representing each specific viewing angle independently would cripple our ability to track a dog as it moves about a scene.
We therefore seek representations with enough invariance to be robust to inconsequential variations but not so much as to discard information required by downstream tasks.
In contrastive learning, the choice of “views” is what controls the information the representation captures, as the framework results in representations that focus on the shared information between views [[50](#bib.bib50)]. Views are commonly different sensory signals, like photos and sounds [[3](#bib.bib3)], or different image channels [[65](#bib.bib65)] or slices in time [[68](#bib.bib68)], but may also be different “augmented” versions of the same data tensor [[8](#bib.bib8)]. If the shared information is small, then the learned representation can discard more information about the input and achieve a greater degree of invariance against nuisance variables. How can we find the right balance of views that share just the information we need, no more and no less?
We investigate this question in two ways. First, we demonstrate that the optimal choice of views depends critically on the downstream task. If you know the task, it is often possible to design effective views. Second, we empirically demonstrate that for many common ways of generating views, there is a sweet spot in terms of downstream performance where the mutual information (MI) between views is neither too high nor too low.
Our analysis suggests an “InfoMin principle”. A good set of views are those that share the minimal information necessary to perform well at the downstream task. This idea is related to the idea of minimal sufficient statistics [[60](#bib.bib60)] and the Information Bottleneck theory [[67](#bib.bib67), [2](#bib.bib2)], which have been previously articulated in the representation learning literature. This principle also complements the already popular “InfoMax principle” [[42](#bib.bib42)]
, which states that a goal in representation learning is to capture as much information as possible about the stimulus. We argue that maximizing information is only useful in so far as that information is task-relevant. Beyond that point, learning representations that throw out information about nuisance variables is preferable as it can improve generalization and decrease sample complexity on downstream tasks [[60](#bib.bib60)].
Based on our findings, we also introduce a semi-supervised method to *learn* views that are effective for learning good representations when the downstream task is known. We additionally demonstrate that the InfoMin principle can be practically applied by simply seeking stronger data augmentation to further reduce mutual information toward the sweet spot. This effort results in state of the art accuracy on a standard benchmark.
Our contributions include:
* Demonstrating that optimal views for contrastive representation learning are task-dependent.
* Empirically finding a U-shaped relationship between an estimate of mutual information and representation quality in a variety of settings.
* A new semi-supervised method to learn effective views for a given task.
* Applying our understanding to achieve state of the art accuracy of 73.0% on the ImageNet linear readout benchmark with a ResNet-50.
2 Related Work
---------------
Learning high-level representations of data that can be used to predict labels of interest is a well-studied problem in machine learning [[5](#bib.bib5)]. In recent years, the most competitive methods for learning representations without labels have been self-supervised contrastive representation learning [[50](#bib.bib50), [30](#bib.bib30), [73](#bib.bib73), [65](#bib.bib65), [61](#bib.bib61), [8](#bib.bib8)].
These methods use neural networks to learn a low-dimensional embedding of data by a “contrastive” loss which pushes apart dissimilar data pairs while pulling together similar pairs, an idea similar to exemplar learning [[19](#bib.bib19)].
Models based on contrastive losses have significantly outperformed other approaches based on generative models, smoothness regularization, dense prediction [[78](#bib.bib78), [37](#bib.bib37), [52](#bib.bib52), [65](#bib.bib65)], and adversarial losses [[18](#bib.bib18)].
The core idea of contrastive representation learning is to learn a function (modeled by a deep network) that maps semantically nearby points (positive pairs) closer together in the embedding space, while pushing apart points that are dissimilar (negative pairs). One of the major design choices in contrastive learning is how to select the positive and negative pairs. For example, given a dataset of i.i.d. images, how can we synthesize positive and negative pairs?
The standard approach for generating positive pairs without additional annotations is to create multiple *views* of each datapoint. For example: splitting an image into luminance and chrominance [[65](#bib.bib65)], applying different random crops and data augmentations [[73](#bib.bib73), [8](#bib.bib8), [4](#bib.bib4), [26](#bib.bib26), [75](#bib.bib75), [62](#bib.bib62)], pasting an object into different backgrounds [[79](#bib.bib79)], using different timesteps within a video sequence [[50](#bib.bib50), [80](#bib.bib80), [57](#bib.bib57), [25](#bib.bib25), [24](#bib.bib24)], or using different patches within a single image [[32](#bib.bib32), [50](#bib.bib50), [30](#bib.bib30)]. Negative pairs can be generated by using views that come from randomly chosen images/patches/videos. In this work, we provide experimental evidence and analysis that can be used to guide the selection and learning of views.
Theoretically, we can think of the positive pairs as coming from a joint distribution over views: p(v1,v2), and the negative pairs as coming from a product of marginal distributions p(v1)p(v2). For the most popular contrastive loss, InfoNCE [[50](#bib.bib50)], the objective is then a lower bound on the mutual information between the two views: I(v1;v2). This connection between contrastive learning and mutual information maximization was first made in CPC [[50](#bib.bib50)] and is discussed further in [[55](#bib.bib55)]. However, recent work has called into question the interpretation of the success of the InfoNCE contrastive loss as information maximization [[69](#bib.bib69)], instead arguing that success is due to geometric requirements on the embedding space. Furthermore, theoretical and experimental work has highlighted that estimating mutual information in high dimensions is challenging and empirical work has shown that the InfoNCE and other bounds used in practice can be quite loose [[43](#bib.bib43), [55](#bib.bib55), [51](#bib.bib51)].
Leveraging labeled data in contrastive representation learning has been shown to guide representations towards task-relevant features that improve performance [[76](#bib.bib76), [29](#bib.bib29), [4](#bib.bib4), [34](#bib.bib34), [72](#bib.bib72)]. Here we leverage labeled data only to learn better views, and still perform contrastive representation learning using unlabeled data only. Future work could combine these approaches to leverage labels for both view learning and representation learning.
Recent work has begun to study and address the question we study here: what views lead to improved downstream accuracy? In [[8](#bib.bib8)], compositions of data augmentations were investigated for their effectiveness. Most similar to our work, a recent unpublished tech report [[70](#bib.bib70)] presents several desiderata for views in contrastive representation learning similar to our discussion of suffiiency and minimality, and presents new bounds on MI for alternative negative sampling schemes.
3 Preliminary: Contrastive Representation Learning
---------------------------------------------------
Let us consider the case when we are given two random variables v1 and v2, and we wish to learn a parametric function to discriminate between samples from the empirical joint distribution p(v1)p(v2|v1) and samples from the product of marginals p(v1)p(v2). The resulting function is an estimator of the mutual information between V1 and V2, and the InfoNCE loss [[50](#bib.bib50)] has been shown to maximize a lower bound on I(v1;v2). In practice, given an anchor point v1,i, the InfoNCE loss is optimized to score the correct positive v2,i∼p(v2|v1,i) higher compared to a set of K distractors v2,j∼p(v2):
| | | | |
| --- | --- | --- | --- |
| | LNCE=−E⎡⎣logeh(v1,i,v2,i)∑Kj=1eh(v1,i,v2,j)⎤⎦ | | (1) |
The score function h(⋅,⋅) typically consists of two encoders (fv1 and fv2) and a critic head h. The two encoders fv1 and fv2 may or may not share parameters depending on whether v1 and v2 are from the same domain. Minimizing the above InfoNCE loss is equivalent to maximizing a lower bound (known as INCE(v1;v2)) on the mutual information between v1 and v2 [[50](#bib.bib50), [30](#bib.bib30)], i.e.,
| | | | |
| --- | --- | --- | --- |
| | I(v1;v2)≥log(K)−LNCE=INCE(v1;v2) | | (2) |
In practice, v1 and v2 are two views of the data. Despite engineering tricks, recent contrastive representation learning methods can be considered as different ways to construct v1 and v2: (a) InsDis [[73](#bib.bib73)], AMDIM [[4](#bib.bib4)], MoCo [[26](#bib.bib26)] and SimCLR [[8](#bib.bib8)] apply data augmentation to the same image to obtain two crops as views; (b) CMC [[65](#bib.bib65)] employs natural image channels; (c) CPC [[50](#bib.bib50)] and CPCv2 [[29](#bib.bib29)] leverages spatial or temporal co-occurrence; (d) In the video domain, [[64](#bib.bib64), [44](#bib.bib44), [39](#bib.bib39)] use a video as v1 and aligned texts as v2, while [[47](#bib.bib47), [10](#bib.bib10), [53](#bib.bib53)] leverages the correspondence between video and audio; (e) [[66](#bib.bib66)] even considers representations from teacher and student networks as v1 and v2, and performs contrastive knowledge distillation.
4 What Are the Optimal Views for Contrastive Learning?
-------------------------------------------------------
Figure 1: (a) Schematic of multiview contrastive representation learning, where a single image is split into two views, and passed through two separate encoders to learn an embedding where the views are close relative to views from other images. (b)-(c) As the mutual information between views is changed, information about the downstream task (green) and nuisance variables (red) can be selectively included or excluded, biasing the learned representation. (b) depicts a scenario where views are chosen to preserve downstream task information between views while throwing out nuisance information, while in (c) reducing MI always throws out information relevant for the task leading to decreasing performance as MI is reduced.
Given two views v1 and v2 of the data, the encoders f1 and f2 in the contrastive learning framework extract representations z1=f1(v1) and z2=f2(v2), respectively.
{definition}
(Sufficient Encoder) The encoder f1 of v1 is sufficient in the contrastive learning framework if and only if I(v1;v2)=I(f1(v1);v2).
Intuitively, the encoder f1 is sufficient if the amount of information in v1 regarding the contrastive objective is lossless during the encoding procedure. In other words, the representation z1 has kept all the information about v2 in v1, and therefore is as useful as v1. Symmetrically, we can also say f2 is sufficient if I(v1;v2)=I(v1;f2(v2))
{definition}
(Minimal Sufficient Encoder) A sufficient encoder f1 of v1 is minimal if and only if I(f1(v1);v1)≤I(f(v1);v1),∀f that are sufficient.
Among those encoders which are sufficient, the minimal ones only extract relevant information of the contrastive task and will throw away other information. This is appealing in cases where the views are constructed in such a way that all the information we care about is shared between them.
The representations learned in the contrastive framework are typically used in a different downstream task. To characterize what representations are good for a downstream task, we define the optimality of representations. To make notation simple, we use z to mean it can be either z1 or z2.
{definition}
(Optimal Representation of a Task) For a task T whose goal is to predict a semantic label y from the input data x, the optimal representation z∗ encoded from x is the minimal sufficient statistic with respect to y.
This says a model built on top of z∗ has all the information necessary to predict y as accurately as if it were to access x. Furthermore, z∗ maintains the smallest complexity, i.e., containing no other information besides that about y, which makes it more generalizable [[60](#bib.bib60)]. We refer the reader to [[60](#bib.bib60)] for a more in depth discussion about optimal visual representations and minimal sufficient statistics.
###
4.1 InfoMin Principle: Views that Only Share Label Information
While views v1 and v2 can be arbitrarily constructed or selected from the data as we like, the effectiveness of views may vary in different tasks. For instance, views that share object position as mutual information should lead to better object localization performance in a downstream task compared to views that only share background nuisances. A general intuition is that good views should only share information that is relevant to a target downstream task. Then given these good input views, the minimal sufficient encoders will extract latent representation that only store information shared between them, which is task-relevant. The following InfoMin proposition articulates which views are optimal supposing that we know the specific downstream task T in advance.
######
Proposition 4.1.
Suppose f1 and f2 in the contrastive learning framework are the minimal sufficient encoders. Given a downstream task T associated with label y, then the optimal views created from the complete data tensor x would be (v1∗,v2∗)=minv1,v2I(v1;v2), subject to . Given v1∗,v2∗, the learned representations z∗1 (or z∗2) is optimal for T (Definition [4](#S4 "4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?")), thanks to the minimality and sufficiency of f1 and f2.
This InfoMin principle has two implications. First, we should reduce mutual information between views. By doing so, the minimal sufficient encoders will throw away more irrelevant information (or nuisance factors) to the downstream task. In practice, “shortcuts” are one type of task-irrelevant nuisance and should be removed from the views [[45](#bib.bib45)]. Furthermore, bits of information that are useful for a different downstream task ^T may turn into nuisances. Second, the constraints suggest that we should retain the predictability of y from x in both v1 and v2, such that the representations capture the semantics of downstream task T. This constraint dissolves the potential impression that we should make the contrastive task as hard as possible to obtain better representations. For a proof of this proposition, please refer to the Appendix.
An example of creating optimal views following this principle, in an image classification task, is to treat images from the same class as congruent pairs and images from different classes as incongruent pairs. In this way, congruent pairs of views only share label information. Recently in [[34](#bib.bib34)], such optimal views have been leveraged for supervised contrastive learning, outperforming supervised models trained with cross-entropy loss on ImageNet [[14](#bib.bib14)].
###
4.2 A Toy Example: Colorful Moving-MNIST
Figure 2: Illustration of the Colorful-Moving-MNIST dataset. In this example, the first view v1 is a sequence of frames containing the moving digit, e.g., v1=x1:k. The matched second view v+2 share some factor with xt that v1 can predict, while the unmatched view v−2 contains a random digit at a random position (with a random background).
Directly analyzing natural images can be challenging as it is hard to create interesting views whose factors of variation are controllable. Therefore, we use a toy dataset as a starting point to understand the behavior of contrastive representation learning with different views. Moving-MNIST [[63](#bib.bib63)] consists of videos where digits move inside a canvas with constant speed and bounce off of image boundaries.
To simulate a more complex dataset with nuisance factors of variation we construct Colorful-Moving-MNIST by adding a background image to the Moving-MNIST videos. Concretely, given a video, a random image from STL-10 dataset [[11](#bib.bib11)] is selected, and then for each frame of the video we randomly crop a patch from this image as background. Thus the dataset consist of three factors of variation in each frame: *the class of the digit*, *the position of the digit*, and *the class of background image*.
Setup. While there are many ways to construct views, our goal is to analyze a set of easily reproducible experiments. To this end, we fix the first view v1 as a sequence of past frames x1:k, and construct different views v2. For simplicity, we consider v2 as a single image. One example of constructing such views is shown in Fig. [2](#S4.F2 "Figure 2 ‣ 4.2 A Toy Example: Colorful Moving-MNIST ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?") where we use a future frame xt of the video as v2. During the contrastive learning phase, we employ a 4-layer ConvNet to encode images and use a single layer LSTM [[31](#bib.bib31)] on top of the ConvNet to aggregate features of continuous frames. After the contrastive pre-training phase, to read off what information has been encoded in the representation, we consider three different downstream tasks for an image: (1) predict the digit class; (2) localize the digit inside the canvas; (3) classify the background into one of the 10 classes of STL-10. In this transfer phase, we freeze the backbone network and learn a task-specific head. To facilitate comparison, we also provide a “supervised” baseline that is trained end-to-end using the same data.
| | | | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| | I(v1;v2) |
| |
| --- |
| digit cls. |
| error rate (%) |
|
| |
| --- |
| background cls. |
| error rate (%) |
|
| |
| --- |
| digit loc. |
| error pixels |
|
|
| |
| --- |
| Single |
| Factor |
| *digit* | 16.8 | 88.6 | 13.6 |
| *bkgd* | 88.6 | 51.7 | 16.1 |
| *pos* | 57.9 | 87.6 | 3.95 |
|
| |
| --- |
| Multiple |
| Factors |
| *bkgd, digit, pos* | 88.8 | 56.3 | 16.2 |
| *bkgd, digit* | 88.2 | 53.9 | 16.3 |
| *bkgd, pos* | 88.8 | 53.8 | 15.9 |
| *digit, pos* | 14.5 | 88.9 | 13.7 |
| Supervised | 3.4 | 45.3 | 0.93 |
Table 1: We study how mutual information shared by views I(v1;v2) would affect the representation quality. We evaluate the learned representation on three downstream tasks: digit classification, background (STL-10) classification, and digit localization.
Single Factor Shared. To begin with, we create views v1 and v2 that only share one of the three factors: *digit*, *position*, or *background*. For a video x with t frames, we set v1=x1:k (note v1 can deterministically predict x(k+1):t), and create another image as v2 by setting one of the three factors the same as xt while randomly picking the other two. In such cases, v1 can only deterministically predict one of the three factors in v2 while never reduce the uncertainty of the other two factors. In other words, the mutual information I(v1;v2) is either *digit*, *position*, or *background*. We separately train encoders using contrastive learning for each of the three scenarios, and the perform transfer learning in three downstream tasks, as shown in Table [1](#S4.T1 "Table 1 ‣ 4.2 A Toy Example: Colorful Moving-MNIST ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?"). These results clearly show that the performance of downstream tasks is significantly affected by the way we construct v2, which determines I(v1;v2). Specifically, if the downstream task is relevant to one factor, we should let I(v1;v2) include this factor rather than others. For example, when v2 only shares background image with v1, contrastive learning can hardly learn representations that capture digit class and location. It is expected since the information of digit class and location is of no use to the contrastive pre-training objective, and thus will not be captured.
Multiple Factors Shared. A more interesting question is how representation quality is changed if v1 and v2 share multiple factors. We follow a similar procedure as above to control factors shared by v1 and v2, and present the results in the second half of Table [1](#S4.T1 "Table 1 ‣ 4.2 A Toy Example: Colorful Moving-MNIST ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?"). We found that one factor can overwhelm another; for instance, whenever *background* is shared by the two views, the latent representation leaves out information for discriminating or localizing digits. This might be because the information bits of background easily dominate bits of the digit class and its position, and the network chooses the background as a “shortcut” to solve the contrastive pre-training task. When v1 and v2 share *digit* and *position*, we found two interesting observations: (1) *digit* dominates *position* as the digit localization task still performs poorly; (2) sharing *position* information benefits *digit* classification – 14.5% error rate v.s. 16.8 when only *digit* is shared. The former might not be a surprise as ConvNets are designed to be insensitive to position shift. For the latter, we conjecture that in practice the encoder is not *sufficient* so it will lose some bits of information about the digits, and knowing the position of the digits help it capture more bits about the digit class.
5 A Sweet Spot in Mutual Information: Reverse-U Shape
------------------------------------------------------
As suggested in Proposition [4.1](#S4.Thmprop1 "Proposition 4.1. ‣ 4.1 InfoMin Principle: Views that Only Share Label Information ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?"), to obtain good performance on the downstream task, we should reduce the mutual information between views I(v1;v2) while retaining the task-relevant semantics. In other words, we should remove task-irrelevant information between views. In this section, we will first discuss a hypothesis for effect of I(v1;v2) on downstream transfer performance, and then empirically analyze three cases of reducing I(v1;v2) in practice.
###
5.1 Three Regimes of Information Captured
As both views are generated from the input x, we can constrain the information between views I(v1;v2) by constraining how much information about x is present in each view: I(v1;x) and I(v2;x). Due to the data processing inequality, the information shared between views is bounded by the information contained about the input in each view: I(v1;v2)≤min(I(v1;x),I(v2;x)). As our representations z1,z2 are built from our views and learned by the contrastive objective with minimal sufficient encoders, the amount and type of shared between v1 and v2 (i.e., I(v1;v2)) determines how well we perform on downstream tasks.
As in information bottleneck [[67](#bib.bib67)], we can trace out a tradeoff between how much information our views share about the input, and how well our learned representation performs at predicting a task-relevant variable y. Depending on how our views are constructed, we may find that we are keeping too many irrelevant variables while discarding relevant variables (as in Figure. [1](#S4.F1 "Figure 1 ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?")c), leading to suboptimal performance on the information plane. Alternatively, we can find the views that maximize I(v1;y) and I(v2;y) (how much information is contained about the task-relevant variable) while minimizing I(v1;v2) (how much information is
shared about the input, including both task-relevant and irrelevant information). Even in the case of these optimal views, there are three regimes of performance we can consider that are depicted in Figure [3](#S5.F3 "Figure 3 ‣ 5.1 Three Regimes of Information Captured ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?"), and have been discussed previously in information bottleneck literature [[67](#bib.bib67), [2](#bib.bib2), [21](#bib.bib21)] :
1. *missing information*: When I(v1;v2)<I(x;y), there is information about the task-relevant variable that is discarded by the view, degrading performance.
2. *sweet spot*: When I(v1;y)=I(v2;y)=I(v1;v2)=I(x;y), the only information shared between v1 and v2 is task-relevant, and there is no irrelevant noise.
3. *excess noise*: As we increase the amount of information shared in the views beyond I(x;y), we begin to include additional information that is irrelevant for the downstream task. This can lead to worse generalization on the downstream task [[2](#bib.bib2), [58](#bib.bib58)].
We hypothesize that the best performing views will be close to the sweet spot: containing as much task-relevant information while discarding as much irrelevant information in the input as possible. Unlike in information bottleneck, for contrastive representation learning we often do not have access to a fully-labeled training set, and thus evaluating how much information about the task-relevant variable is contained in the representation at training time is challenging. Instead, the construction of views has typically been guided by domain knowledge that alters the input while preserving the task-relevant variable.
The above analysis suggests that transfer performance will be upper-bounded by a reverse-U shaped curve (Figure [3](#S5.F3 "Figure 3 ‣ 5.1 Three Regimes of Information Captured ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?"), right), with the sweet spot at the top of the curve. We next present a series of experiments that find such a curve in practical settings.
Figure 3: When we have views that maximize I(v1;y) and I(v2;y) (how much information is contained about the task-relevant variable) while minimizing I(v1;v2) (how much information is
shared about the input, including both task-relevant and irrelevant information), there are three regimes: *missing information* which leads to degraded performance due to I(v1;v2)<I(x;y); *excess noise* which worsens generalization due to additional noise; *sweet spot* where the only information shared between v1 and v2 is task-relevant and such information is complete.
###
5.2 Practical Cases of Reducing I(v1;v2)
Often, the downstream task is unknown beforehand, and we may not know in which ways we should reduce I(v1;v2) to create views suitable for various downstream tasks. It is possible that both task-relevant signals and nuisance factors are reduced simultaneously, and therefore we do not have a guarantee on performance.
But recently, [[65](#bib.bib65)] found a “reverse-U” shape phenomenon: reducing I(v1;v2) firstly leads to improved performance on the downstream task, then after a peak further decreasing I(v1;v2) causes performance degradation. This finding is in line with Hypothesis LABEL:as:learner: at the beginning of reducing I(v1;v2), nuisance factors – the *noise* – are removed while most of the useful semantics – the *signal* – are preserved. Indeed this may be common in practice: data augmentation, of the proper magnitude, reduces I(v1;v2) but improves accuracy. Other than data augmentation, we show three examples where reduced I(v1;v2) leads to improved performance. We use INCE as neural proxy for I. Though it might not estimate I accurately, it can still provide interesting analysis. We note here that within each plot with INCE in this paper, we only vary the input views v1,v2, but keep all other settings fixed, in order to make the plotted points directly comparable. For more implementation details, please refer to the appendix.
####
5.2.1 Reducing I(v1;v2) with Spatial Distance
Figure 4: We create views by using pairs of image patches at various offsets from each other. As INCE decreases, the downstream task accuracy (STL-10 classification, CIFAR-10 classification) improves until INCE hits a sweet spot. After that, reducing INCE gradually lowers accuracy.
We create views by randomly cropping two patches of size 64x64 from the same image with fixed relative position. Namely, one patch starts at position (x,y) while the other starts at (x+d,y+d), with (x,y) being randomly generated. We increase d from 64 to 384, sampling from inside high resolution images (*e.g.*, ∼ 2k pixels in each dimension) from the DIV2K dataset [[1](#bib.bib1)]. After contrastive training, we evaluate on STL-10 and CIFAR-10 by freezing the encoder and training a linear classification layer. The plots in Figure [4](#S5.F4 "Figure 4 ‣ 5.2.1 Reducing I(v1;v2) with Spatial Distance ‣ 5.2 Practical Cases of Reducing I(v1;v2) ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?") shows the *Mutual Information* v.s. *Accuracy*. From natural image statistics we can expect that I(v1;v2) decreases when d increases [[59](#bib.bib59)]. The plots demonstrate that this is also true, empirically, for the proxy INCE, as INCE decreases with increasin d. Moreover, the results show replicate the “reverse-U” curve found in [[65](#bib.bib65)], and here we further show that this phenomenon is consistent across both STL-10 and CIFAR-10. We can identify the sweet spot at d=128.
####
5.2.2 Reducing I(v1;v2) with Different Color Spaces
Figure 5: We build views by splitting channels of different color spaces. As INCE decreases, the accuracy on downstream tasks (STL-10 classification, NYU-v2 segmentation) improves. Whether the sweet spot has been touched is unclear.
The correlation between channels may vary significantly with different color spaces. Here we follow [[65](#bib.bib65), [78](#bib.bib78)] and split different color spaces into two views, such as {Y,DbDr},{R,GB},{X,YZ}, etc. We perform contrastive learning on STL-10, and measure the representation quality by training a linear classifier on the STL-10 dataset to perform image classification or a decoder head on NYU-V2 [[48](#bib.bib48)] images to perform semantic segmentation. As shown in Figure [5](#S5.F5 "Figure 5 ‣ 5.2.2 Reducing I(v1;v2) with Different Color Spaces ‣ 5.2 Practical Cases of Reducing I(v1;v2) ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?"), the plots show downstream performance keeps increasing as INCE(v1;v2) decreases for both classification and segmentation. Here we do not observe any performance drop with these natural color spaces when reducing INCE(v1;v2). In Section [6](#S6 "6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?"), we will show a learning method for new color space, which can further reduces INCE(v1;v2) to find the sweet spot, and go beyond it, where transfer performance begins again to drop.
####
5.2.3 Reducing I(v1;v2) with Frequency Separation
Figure 6: We create views by splitting images into low- and high-frequency pairs with a blur function parameterized by σ. INCE is maximized at σ=0.7. Starting from this point, either increasing or decreasing σ will reduce INCE but interestingly they form two different trajectories. When increasing σ from 0.7, the accuracy firstly improves and then drops, forming a reverse-U shape corresponding to (b) in Figure [1](#S4.F1 "Figure 1 ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?"). While decreasing σ from 0.7, the accuracy keeps diminishing, corresponding to (c) in Figure [1](#S4.F1 "Figure 1 ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?").
Another example we consider is to separate images into low- and high-frequency images. To simplify, we extract v1 and v2 by Gaussian blur, i.e.,
| | | | |
| --- | --- | --- | --- |
| | v1 | =Blur(x,σ) | |
| | v2 | =x−v1 | |
where Blur is the Gaussian blur function and σ is the parameter controlling the kernel. Extremely small or large σ can make the high- or low-frequency image contain little information. In theory, the maximal I(v1;v2) is obtained with some intermediate σ. As shown in Figure [6](#S5.F6 "Figure 6 ‣ 5.2.3 Reducing I(v1;v2) with Frequency Separation ‣ 5.2 Practical Cases of Reducing I(v1;v2) ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?"), we found σ=0.7 leads to the maximal INCE on the STL-10 dataset. Either blurring more or less will reduce INCE, but interestingly blurring more leads to different trajectory in the plot than blurring less. When increasing σ from 0.7, the accuracy firstly improves and then drops, forming a reverse-U shape with a sweet spot at σ=1.0. This situation corresponds to (b) in Figure [1](#S4.F1 "Figure 1 ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?"). While decreasing σ from 0.7, the accuracy keeps diminishing, corresponding to (c) in Figure [1](#S4.F1 "Figure 1 ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?"). This reminds us of the two aspects in Proposition [4.1](#S4.Thmprop1 "Proposition 4.1. ‣ 4.1 InfoMin Principle: Views that Only Share Label Information ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?"): mutual information is not the whole story; *what* information is shared between the two views also matters.
6 Synthesizing Effective Views
-------------------------------
Figure 7: Schematic of contrastive representation learning with a learned view generator. An input image is split into two views using an invertible view generator. To learn the view generator, we optimize the losses in yellow: minimizing information between views while ensuring we can classify the object from each view. The encoders used to estimate mutual information are always trained to maximize the InfoNCE lower bound. After learning the view generator, we reset the weights of the encoders, and train with a fixed view generator without the additional supervised classification losses.
Can we learn novel views that reach the sweet spot by following the InfoMin principle? To explore this possibility, we design unsupervised and semi-supervised frameworks that learn novel views inspired by the InfoMin principle. Concretely, we extend the color space experiments in Section [5.2.2](#S5.SS2.SSS2 "5.2.2 Reducing I(v1;v2) with Different Color Spaces ‣ 5.2 Practical Cases of Reducing I(v1;v2) ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?") by learning flow-based models that transfer natural color spaces into novel neural color spaces, from which we split channels to get views.
After the views have been learned, we will do standard contrastive learning followed by linear classifier evaluation. In this section, we consider three methods: (1) *random view generation* which varies I(v1;v2) by constructing views with randomly initialized networks; (2) *unsupervised view learning* which reduces I(v1;v2); and (3) *semi-supervised view learning* which reduces I(v1;v2) while preserving task-relevant information. The idea of view learning is diagrammed in Figure [7](#S6.F7 "Figure 7 ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?").
###
6.1 Random Views
Figure 8: (a) We transform an image X by a random initialized invertible flow model g(⋅), and compare using {X1,X2:3} as two views with {g(X)1,g(X)2:3}. The transformed views have larger INCE than the raw views, and result in a performance drop. (b) We only apply g(⋅) to the second view of the raw input X2:3. In theory, I(X1;X2:3)=I(X1;g(X2:3)). Then we observe that INCE of {X1,X2:3} is almost identical to {X1,g(X2:3)}, and so are their downstream accuracies.
Flow-based generative models [[16](#bib.bib16), [15](#bib.bib15), [36](#bib.bib36)] are carefully-designed bijective functions between input images and latent space. Here we leverage this property to create random color spaces that are bijective and preserve total information. To do so, we restrict g to be a pixel-wise (*i.e.*, use 1x1 convolutions) flow. With a randomly initialized g, an input image, X, is transformed into ^X=g(X), which has the same size as X. We then split ^X over the channel dimension {^X1,^X2:3} as two views for contrastive learning. By comparing {^X1,^X2:3} with {X1,X2:3}, we found that INCE is typically increased after the transformation and the downstream accuracy drops, as shown in (a) of Figure [8](#S6.F8 "Figure 8 ‣ 6.1 Random Views ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?").
Is it MI between views or the inductive bias of g(⋅) that plays the role for above spectrum plot? To check, we use g(⋅) to transform only half of the input, e.g., X2:3. Theoretically, the MI between views {X1,X2:3} is identical to {X1,g(X2:3)}. In accord with theory, we found this is also true for the NCE estimation INCE, as shown in (b) of Figure [8](#S6.F8 "Figure 8 ‣ 6.1 Random Views ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?"). Interestingly the classification accuracy of the downstream task using {X1,X2:3} is almost the same as {X1,g(X2:3)}.
###
6.2 Unsupervised: Minimize I(v1;v2)
The idea is to leverage an adversarial training strategy [[23](#bib.bib23)]. Given an input image X, we transform it into ^X=g(X). We train two encoders on top of ^X1 and ^X2:3 to maximize INCE(^X1;^X2:3), similar to the discriminator in GAN [[23](#bib.bib23)]. Meanwhile, g(⋅) is adversarially trained to minimize INCE(^X1;^X2:3). Formally, the objective is (shown as the bottom-left yellow box in Figure [7](#S6.F7 "Figure 7 ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?")):
| | | | |
| --- | --- | --- | --- |
| | mingmaxf1,f2If1,f2NCE(g(X)1;g(X)2:3) | | (3) |
where f1,f2 correspond to h(⋅) in Eqn. [1](#S3.E1 "(1) ‣ 3 Preliminary: Contrastive Representation Learning ‣ What Makes for Good Views for Contrastive Learning?"). Alternatively, one may use other MI bounds [[55](#bib.bib55)]. In practice, we find INCE works well and keep using it for simplicity. We note that the invertible and pixel-wise properties of g(⋅) prevent it from learning degenerate/trivial solutions.
Figure 9: View generator learned by (a) unsupervised or (b) semi-supervised objectives.
Implementation. This experiment is performed on STL-10. We try both volume preserving (VP) and non-volume preserving (NVP) flow models. F and G consist of 1x1 convolutional layers. For f1 and f2, we use an AlexNet-style network. We experiment with two input color spaces: RGB and YDbDr. The former is the most widely used one, while the latter is the best for contrastive learning, as shown in Figure [5](#S5.F5 "Figure 5 ‣ 5.2.2 Reducing I(v1;v2) with Different Color Spaces ‣ 5.2 Practical Cases of Reducing I(v1;v2) ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?").
Results. We plot the INCE between the learned views and the corresponding linear evaluation performance. As shown in Figure [9](#S6.F9 "Figure 9 ‣ 6.2 Unsupervised: Minimize I(v1;v2) ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?")(a), a reverse U-shape between INCE and downstream accuracy is presented. Interestingly, YDbDr color space is already around the sweet spot, and further reducing INCE results in a performance drop.
This happens to be in line with our human prior that the “luminance-chrominance” decomposition is a good way to decorrelate colors but still maintains good interpretability (in the sense that we can still read out high level semantics to perform tasks). We also note that the Lab color space, which is another luminance-chrominance decomposition (and performs similarly well to YDbDr; Figure [5](#S5.F5 "Figure 5 ‣ 5.2.2 Reducing I(v1;v2) with Different Color Spaces ‣ 5.2 Practical Cases of Reducing I(v1;v2) ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?")), was designed to mimic the way humans perceive color [[33](#bib.bib33)]. Our analysis therefore suggests yet another rational explanation for why humans perceive color the way we do – human perception of color may be near optimal for self-supervised representation learning.
Occasionally, one color space learned from RGB happens to touch the sweet spot, but in general the INCE between views is overly decreased. The reverse-U shape trend holds for both NVP and VP models. In addition, we found this GAN-style training is unstable, as different runs with the same hyper-parameter can vary a lot. We conjecture that while reducing MI between views in such an unsupervised manner, the view generator has no knowledge about task-relevant semantics and thus construct views that do not share sufficient information about the label y, *i.e.*, the constraint in Proposition [4.1](#S4.Thmprop1 "Proposition 4.1. ‣ 4.1 InfoMin Principle: Views that Only Share Label Information ‣ 4 What Are the Optimal Views for Contrastive Learning? ‣ What Makes for Good Views for Contrastive Learning?") is not satisfied.
To overcome this, we further develop an semi-supervised view learning method.
###
6.3 Semi-supervised View Learning: Find Views that Share the Label Information
We assume a handful of labels for the downstream task are available. Therefore we can teach the view generator g(⋅) to retain I(g(X)1,y) and I(g(X)2:3,y) as much as possible. Put into practice, we introduce two classifiers on each of the learned views to perform classification during the view learning process. Formally, we optimize (shown as the three yellow boxes in Figure [7](#S6.F7 "Figure 7 ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?")):
| | | | |
| --- | --- | --- | --- |
| | ming,c1,c2maxf1,f2If1,f2NCE(g(X)1;g(X)2:3)unsupervised: reduce I(v1;v2)+Lce(c1(g(X)1),y)+Lce(c2(g(X)2:3),y)supervised: keep I(v1;y) and I(v2;y) | | (4) |
where c1,c2 represent the classifiers. The INCE term applies to all data while the latter two are only for labeled data. After this process is done, we use g(⋅) to generate views for contrastive representation learning.
| Method | RGB | YDbDr |
| --- | --- | --- |
| unsupervised | 82.4 ± 3.2 | 84.3 ± 0.5 |
| supervised | 79.9 ± 1.5 | 78.5 ± 2.3 |
| semi-supervised | 86.0 ± 0.6 | 87.0 ± 0.3 |
| raw input | 81.5 ± 0.2 | 86.6 ± 0.2 |
Table 3: Switching to larger backbones with views learned by the semi-supervised method.
Table 2: Comparison of different view generators by measuring STL-10 classification accuracy: *supervised*, *unsupervised*, and *semi-supervised*
Results. The plots are shown in Figure [9](#S6.F9 "Figure 9 ‣ 6.2 Unsupervised: Minimize I(v1;v2) ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?")(b). Now the learned views are clustered around the sweet spot, no matter what the input color space was and whether the generator is VP or NVP, which highlights the importance of keeping information about y. Meanwhile, to see the importance of the unsupervised term, which reduces INCE, we train another view generator by just the supervised loss. We further compare “supervised”, “unsupervised” and “semi-supervised” (the supervised + unsupervised losses) generators in Table [3](#S6.T3 "Table 3 ‣ 6.3 Semi-supervised View Learning: Find Views that Share the Label Information ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?"), where we also includes contrastive learning over the original color space (“raw input”) as a baseline. The semi-supervised view generator significantly outperforms the supervised one, verifying the importance of reducing I(v1;v2). We compare the “semi-supervised” views of g(X) with the original X (X is RGB or YDbDr) on larger backbone networks, as shown in Table [3](#S6.T3 "Table 3 ‣ 6.3 Semi-supervised View Learning: Find Views that Share the Label Information ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?"). We see that the learned views consistently outperform its raw input, e.g., g(RGB) surpasses RGB by a large margin and reaches 94% classification accuracy.
7 Data Augmentation as InfoMin
-------------------------------
###
7.1 A Unified View of Recent State-of-the-Art Methods
Recently, there are several contrastive learning methods dominating the ImageNet self-suspervised learning benchmark, *e.g.*, InstDis, CPC, CMC, MoCo, PIRL, SimCLR, etc. Despite various different engineering tricks in each paper, each has created v1 and v1 in a way that implicitly follows the InfoMin principle. We examine a non-comprehensive selection of recent methods below:
InstDis [[73](#bib.bib73)] and MoCo [[26](#bib.bib26)]. Despite the difference of memory bank and moment encoder, these two methods create views by ancestral sampling: (1) sample a image X from the empirical distribution p(x); (2) sample two independent transformations t1,t2 from a distribution of data augmentation functions T; (3) let v1=t1(X) and v2=t2(X).
CMC [[65](#bib.bib65)]. On top of the two views v1 and v2 in InsDis, CMC further split images across color channels. This leads to a new set of views vcmc1 and vcmc2, where vcmc1 is the first color channel of v1, and vcmc2 is the last two channels of v2. By this design, I(vcmc1;vcmc2)≤I(v1;v2) is theoretically guaranteed, and we observe that CMC performs better than InstDis.
PIRL [[46](#bib.bib46)]. Comparing PIRL with InstDis is a bit tricky, but we can also explain it from the InfoMin perspective. Similarly, given two views obtained in InstDis, PIRL keeps vPIRL1=v1 but transforms the other view v2 with random JigSaw shuffling h to get vPIRL2=h(v2). The mutual information between the two views is also reduced as h(⋅) introduces randomness.
SimCLR [[8](#bib.bib8)]. Despite other engineering techniques and tricks, the way SimCLR creates views is most similar to InstDist and MoCo, but it uses a stronger class of augmentations T′, which leads to less mutual information between the two views.
CPC [[50](#bib.bib50)]. Different from the above methods that create views at the image level, CPC extracts views vcpc1, vcpc2 from local patches with strong data augmentation (e.g., RandAugment [[13](#bib.bib13)]) which reduces I(vcpc1;vcpc2). In addition, cropping vcpc1 and vcpc2 from disjoint patches also reduces I(vcpc1;vcpc2), which relates to the discussion in Section [4](#S5.F4 "Figure 4 ‣ 5.2.1 Reducing I(v1;v2) with Spatial Distance ‣ 5.2 Practical Cases of Reducing I(v1;v2) ‣ 5 A Sweet Spot in Mutual Information: Reverse-U Shape ‣ What Makes for Good Views for Contrastive Learning?").
While the above methods are reducing mutual information between views, they keep information about object identity in both views. The hope is that object identity can bake in most of the high-level semantic information which various downstream tasks care about.
Figure 10: (a) data augmentation as InfoMin on ImageNet with linear projection head; (b) illustration of step-by-step data augmentation used in InfoMin.
###
7.2 Analysis of Data Augmentation as it relates to MI and Transfer Performance
We gradually strengthen the family of data augmentation functions T, and plot the trend between accuracy in downstream linear evaluation benchmarks and INCE. The overall results are shown in Figure [10](#S7.F10 "Figure 10 ‣ 7.1 A Unified View of Recent State-of-the-Art Methods ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?")(a), where the plot is generated by only varying data augmentation while keeping all other settings fixed. We consider *Color Jittering* with various strengths, *Gaussian Blur*, *RandAugment* [[13](#bib.bib13)], and their combinations, as illustrated in Figure [10](#S7.F10 "Figure 10 ‣ 7.1 A Unified View of Recent State-of-the-Art Methods ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?")(b). The results suggest that as we reduce INCE(v1;v1), via stronger T (in theory, I(v1;v1) also decreases), the downstream accuracy keeps improving.
We also investigate how sliding the strength parameter of individual augmentation functions leads to a practical reverse-U curves, as shown in Figures [11](#S7.F11 "Figure 11 ‣ 7.2 Analysis of Data Augmentation as it relates to MI and Transfer Performance ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?") and [12](#S7.F12 "Figure 12 ‣ 7.2 Analysis of Data Augmentation as it relates to MI and Transfer Performance ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?").
| | |
| --- | --- |
| Linear projection head
(a) Linear projection head
| MLP projection head
(b) MLP projection head
|
Figure 11: Different low-area cropping bounds in RandomResizedCrop.
Cropping. In PyTorch, the RandomResizedCrop(scale=(c, 1.0)) data augmentation function sets a low-area cropping bound c. Smaller c means more aggressive data augmentation. We vary c for both a linear critic head [[73](#bib.bib73)] (with temperature 0.07) and nonlinear critic head [[8](#bib.bib8)] (with temperature 0.15), as shown in Figure [11](#S7.F11 "Figure 11 ‣ 7.2 Analysis of Data Augmentation as it relates to MI and Transfer Performance ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?"). In both cases, decreasing c forms a reverse-U shape between INCE and linear classification accuracy, with a sweet spot at c=0.2. This is different from the widely used 0.08 in the supervised learning setting. Using 0.08 can lead to more than 1% drop in accuracy compared to the optimal 0.2 when a nonlinear projection head is applied.
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| Linear projection head
(a) Linear projection head
| MLP projection head
(b) MLP projection head
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Figure 12: Different magnitudes of Color Jittering.
Color Jittering. As shown in Figure [10](#S7.F10 "Figure 10 ‣ 7.1 A Unified View of Recent State-of-the-Art Methods ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?")(b), we adopt a parameter x to control the strengths of color jittering function. As shown in Figure [12](#S7.F12 "Figure 12 ‣ 7.2 Analysis of Data Augmentation as it relates to MI and Transfer Performance ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?"), increasing x from 0.125 to 2.5 also traces a reverse-U shape, no matter whether a linear or nonlinear projection head is used. The sweet spot lies around x=1.0, which is the same value as used in SimCLR [[8](#bib.bib8)]. Practically, we see the accuracy is more sensitive around the sweet spot for the nonlinear projection head, which also happens for cropping. This implies that it is important to find the sweet spot for future design of augmentation functions.
Details. These plots are based on the MoCo [[26](#bib.bib26)] framework. We use 65536 negatives and pre-train for 100 epochs on 8 GPUs with a batch size of 256. The learning rate starts as 0.03 and decays following a cosine annealing schedule. For the downstream task of linear evaluation, we train the linear classifier for 60 epochs with an initial learning rate of 30, following [[65](#bib.bib65)].
###
7.3 Results on ImageNet Benchmark
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| --- | --- | --- | --- | --- | --- | --- |
| Method | Architecture | Param. | Head | Epochs | Top-1 | Top-5 |
| InstDis [[73](#bib.bib73)] | ResNet-50 | 24 | Linear | 200 | 56.5 | - |
| Local Agg. [[81](#bib.bib81)] | ResNet-50 | 24 | Linear | 200 | 58.8 | - |
| CMC [[65](#bib.bib65)] | ResNet-50\* | 12 | Linear | 240 | 60.0 | 82.3 |
| MoCo [[26](#bib.bib26)] | ResNet-50 | 24 | Linear | 200 | 60.6 | - |
| PIRL [[46](#bib.bib46)] | ResNet-50 | 24 | Linear | 800 | 63.6 | - |
| CPC v2 [[29](#bib.bib29)] | ResNet-50 | 24 | - | - | 63.8 | 85.3 |
| SimCLR [[8](#bib.bib8)] | ResNet-50 | 24 | MLP | 1000 | 69.3 | 89.0 |
| InfoMin Aug. (Ours) | ResNet-50 | 24 | MLP | 100 | 67.4 | 87.9 |
| InfoMin Aug. (Ours) | ResNet-50 | 24 | MLP | 200 | 70.1 | 89.4 |
| InfoMin Aug. (Ours) | ResNet-50 | 24 | MLP | 800 | 73.0 | 91.1 |
Table 4: Single-crop ImageNet accuracies (%) of linear classifiers [[77](#bib.bib77)] trained on representations learned with different contrastive methods using ResNet-50 [[28](#bib.bib28)]. InfoMin Aug. refers to data augmentation using *RandomResizedCrop*, *Color Jittering*, *Gaussian Blur*, *RandAugment*, *Color Dropping*, and a *JigSaw* branch as in PIRL [[46](#bib.bib46)]. \* indicates splitting the network into two halves.
On top of the “RA-CJ-Blur” augmentations shown in Figure [10](#S7.F10 "Figure 10 ‣ 7.1 A Unified View of Recent State-of-the-Art Methods ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?"), we further reduce the mutual information (or enhance the invariance) of views by using PIRL [[46](#bib.bib46)], i.e., adding JigSaw [[49](#bib.bib49)]. This improves the accuracy of the linear classifier from 63.6% to 65.9%. Replacing the widely-used linear projection head [[73](#bib.bib73), [65](#bib.bib65), [26](#bib.bib26)] with a 2-layer MLP [[8](#bib.bib8)] increases the accuracy to 67.3%. When using this nonlinear projection head, we found a larger temperature is beneficial for downstream linear readout (as also reported in [[9](#bib.bib9)]). All these numbers are obtained with 100 epochs of pre-training. For simplicity, we call such unsupervised pre-training as InfoMin pre-training (i.e., pre-training with our InfoMin inspired augmentation). As shown in Table [4](#S7.T4 "Table 4 ‣ 7.3 Results on ImageNet Benchmark ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?"), our InfoMin model trained with 200 epochs achieves 70.1%, outperforming SimCLR with 1000 epochs. Finally, a new state-of-the-art, 73.0% is obtained by training for 800 epochs. Compared to SimCLR requiring 128 TPUs for large batch training, our model can be trained with as less as 4 GPUs on a single machine.
For future improvement, there is still room for manually designing better data augmentation. As shown in Figure [10](#S7.F10 "Figure 10 ‣ 7.1 A Unified View of Recent State-of-the-Art Methods ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?")(a), using “RA-CJ-Blur” has not touched the sweet spot yet. Another way to is to learn to synthesize better views (augmentations) by following (and expanding) the idea of semi-supervised view learning method presented in Section [6.3](#S6.SS3 "6.3 Semi-supervised View Learning: Find Views that Share the Label Information ‣ 6 Synthesizing Effective Views ‣ What Makes for Good Views for Contrastive Learning?").
###
7.4 Transferring Representations
One goal of unsupervised pre-training is to learn transferable representations that are beneficial for downstream tasks. The rapid progress of many vision tasks in past years can be ascribed to the paradigm of fine-tuning models that are initialized from supervised pre-training on ImageNet. When transferring to PASCAL VOC [[20](#bib.bib20)] and COCO [[41](#bib.bib41)], we found our InfoMin pre-training consistently outperforms supervised pre-training as well as other unsupervised pre-training methods.
Feature normalization has been shown to be important during fine-tuning [[26](#bib.bib26)]. Therefore, we fine-tune the backbone with Synchronized BN (SyncBN [[54](#bib.bib54)]) and add SyncBN to newly initialized layers (e.g., FPN [[40](#bib.bib40)]). Table [5](#S7.T5 "Table 5 ‣ 7.4 Transferring Representations ‣ 7 Data Augmentation as InfoMin ‣ What Makes for Good Views for Contrastive Learning?") reports the bounding box AP and mask AP on val2017 on COCO, using the Mask R-CNN [[27](#bib.bib27)] R50-FPN pipeline. All results are reported on Detectron2 [[71](#bib.bib71)]. We notice that, among unsupervised approaches, only ours consistently outperforms supervised pre-training.
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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| pre-train | APbb~{} | APbb50 | APbb75 | APmk~{} | APmk50 | APmk75 |
| random init | 32.8 | 50.9 | 35.3 | 29.9 | 47.9 | 32.0 |
| supervised | 39.7 | 59.5 | 43.3 | 35.9 | 56.6 | 38.6 |
| InstDis [[73](#bib.bib73)] | 38.8(↓0.9) | 58.4(↓1.1) | 42.5(↓0.8) | 35.2(↓0.7) | 55.8(↓0.8) | 37.8(↓0.8) |
| PIRL [[46](#bib.bib46)] | 38.6(↓1.1) | 58.2(↓1.3) | 42.1(↓1.2) | 35.1(↓0.8) | 55.5(↓1.1) | 37.7(↓0.9) |
| MoCo [[26](#bib.bib26)] | 39.4(↓0.3) | 59.1(↓0.4) | 42.9(↓0.4) | 35.6(↓0.3) | 56.2(↓0.4) | 38.0(↓0.6) |
| InfoMin Aug. | 40.6(↑0.9) | 60.6(↑1.1) | 44.6(↑1.3) | 36.7(↑0.8) | 57.7(↑1.1) | 39.4(↑0.8) |
(a) Mask R-CNN, R50-FPN, 1x schedule
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| --- | --- | --- | --- | --- | --- | --- |
| pre-train | APbb~{} | APbb50 | APbb75 | APmk~{} | APmk50 | APmk75 |
| random init | 38.4 | 57.5 | 42.0 | 34.7 | 54.8 | 37.2 |
| supervised | 41.6 | 61.7 | 45.3 | 37.6 | 58.7 | 40.4 |
| InstDis [[73](#bib.bib73)] | 41.3(↓0.3) | 61.0(↓0.7) | 45.3(↓0.0) | 37.3(↓0.3) | 58.3(↓0.4) | 39.9(↓0.5) |
| PIRL [[46](#bib.bib46)] | 41.2(↓0.4) | 61.2(↓0.5) | 45.2(↓0.1) | 37.4(↓0.2) | 58.5(↓0.2) | 40.3(↓0.1) |
| MoCo [[26](#bib.bib26)] | 41.7(↑0.1) | 61.4(↓0.3) | 45.7(↑0.4) | 37.5(↓0.1) | 58.6(↓0.1) | 40.5(↑0.1) |
| InfoMin Aug. | 42.5(↑0.9) | 62.7(↑1.0) | 46.8(↑1.5) | 38.4(↑0.8) | 59.7(↑1.0) | 41.4(↑1.0) |
(b) Mask R-CNN, R50-FPN, 2x schedule
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Table 5: Results of object detection and instance segmentation fine-tuned on COCO. We adopt Mask R-CNN R50-FPN, and report the bounding box AP and mask AP on val2017. In the brackets are the gaps to the ImageNet supervised pre-training counterpart. For fair comparison, InstDis [[73](#bib.bib73)], PIRL [[46](#bib.bib46)], MoCo [[26](#bib.bib26)], and InfoMin are all pre-trained for 200 epochs.
We have tried different popular detection frameworks with various backbones, extended the fine-tuning schedule (e.g., 6x schedule), and compared InfoMin ResNeXt-152 [[74](#bib.bib74)] trained on ImageNet-1k with supervised ResNeXt-152 trained on ImageNet-5k (6 times larger than ImageNet-1k). In all cases, InfoMin consistently outperforms supervised pre-training. For further details on these results, as well as experiments on PASCAL VOC, please refer to the appendix.
8 Conclusion
-------------
We have proposed an InfoMin principle and a view synthesis framework for constructing effective views for contrastive representation learning. Viewing data augmentation as information minimization, we achieved a new state-of-the-art result on the ImageNet linear readout benchmark with a ResNet-50.
##### Acknowledgments.
This work was done when Yonglong Tian was a student researcher at Google. We thank Kevin Murphy for fruitful and insightful discussion; Lucas Beyer for feedback on the manuscript; and Google Cloud team for supporting computation resources. Yonglong is grateful to Zhoutong Zhang for encouragement and feedback on experimental design. |
b8a55d48-fb53-4362-9061-7bfccf862fb1 | trentmkelly/LessWrong-43k | LessWrong | Free holiday reading?
I'm going on holiday tomorrow. What freely downloadable reading do people recommend? I'm after both more serious things related to Less Wrong or SIAI/FHI's mission, or more lightweight stuff that is the sort of thing an LW participant like me would enjoy. Thanks! |
4efddb24-3290-4580-a3ea-4c24789f31da | trentmkelly/LessWrong-43k | LessWrong | Thoughts on Toy Models of Superposition
This post explores some of the intuitions I developed whilst reading Anthropic’s Toy Models of Superposition paper. I focus on motivating the shape of the model and interpreting the visualisations used in the paper. Their accompanying article is thoughtfully written, and I'd highly recommend reading it if you haven’t already. I make no claims that this blog post will be particularly additive to that existing piece. The primary purpose of writing this post is to:
* Test and reinforce my understanding of the paper
* Offer an additional perspective on these ideas for readers who enjoy exploring concepts from multiple angles
* Provide an explainer that I can refer back to in my Conditional Importance in Toy Models of Superposition post, where I rely on readers' familiarity with these ideas
Definitions
I'll first define some basic concepts.
A feature is a scalar function of the neuron activations in a layer of a neural network, which has some qualitative meaning.
[there are many different definitions of a feature, but this is the one I think is the most useful in this context]
An example of a feature in an image processing network is a cat detector: a scalar whose magnitude corresponds to the extent to which (the model believes) a cat is present.
The Linear Representation Hypothesis posits that features are represented by linear directions in activation space.
To appreciate the LRH's claim, it's helpful to think about what it would mean for it to be false.
One concrete way it might not hold is if features were to occupy different magnitudes within common directions. To take an extreme example, it could be that a specific neuron contains both a cat detector feature and dog detector feature, and the information might be encoded by if the integer part of the activation is a positive odd number, a cat has been detected, if the integer part of the activation is a positive even number, a dog has been detected, otherwise neither have been detected. A les |
60a82a73-36c9-4cf3-992d-9b4113bef7e1 | trentmkelly/LessWrong-43k | LessWrong | Is there a way to view "read" LessWrong posts?
I want to be able to see the LessWrong posts that I've already read/visited, like in a list. I mainly just want to be able to re-save some ones that I really liked but deleted from my bookmarks.
I didn't vote on all the posts I read (it's kind of like a habit I had to build), so I can't do that (although, that would be nice too) (I guess that would be a related question). |
1f98dc2a-657d-4917-99d5-2eec8c3ea654 | StampyAI/alignment-research-dataset/blogs | Blogs | MIRI’s April 2014 Newsletter
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[Machine Intelligence Research Institute](http://intelligence.org)
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**Research Updates**
* New technical report: “**[Botworld 1.0](http://intelligence.org/2014/04/10/new-report-botworld/)**.”
* 9 new expert interviews: [Max Tegmark](http://intelligence.org/2014/03/19/max-tegmark/) on the mathematical universe, [Randal Koene](http://intelligence.org/2014/03/20/randal-a-koene-on-whole-brain-emulation/) on whole brain emulation, [Michael Carbin](http://intelligence.org/2014/03/23/michael-carbin/) on integrity properties in approximate computing, [Anil Nerode](http://intelligence.org/2014/03/26/anil-nerode/) on hybrid systems control, [Lyle Ungar](http://intelligence.org/2014/03/26/lyle-ungar/) on forecasting, [Erik DeBenedictis](http://intelligence.org/2014/04/03/erik-debenedictis/) on supercomputing, [Will MacAskill](http://intelligence.org/2014/04/08/will-macaskill/) on normative uncertainty, [Diana Spears](http://intelligence.org/2014/04/09/diana-spears/) on the safety of adaptive agents, and [Paulo Tabuada](http://intelligence.org/2014/04/09/paulo-tabuada/) on program synthesis for cyber-physical systems.
**News Updates**
* We are [actively hiring](http://intelligence.org/careers/) for three positions: math researcher, science writer, and executive assistant. Compensation is competitive and visa are possible.
* We’re also [accepting applications](http://intelligence.org/get-involved/#workshop) on a rolling basis for future MIRI research workshops. Please apply if you’re a curious academic looking for exposure to our material. If you’re accepted, we’ll contact you about potential workshops over time, as we schedule them.
* We published our [2013 in Review: Fundraising](http://intelligence.org/2014/04/02/2013-in-review-fundraising/) post.
* Louie was [interviewed](http://io9.com/why-asimovs-three-laws-of-robotics-cant-protect-us-1553665410) at *io9* about the unsuitability of Asimov’s Three Laws of Robotics for machine ethics.
* Luke was interviewed at *The Information* about MIRI’s research. Unfortunately, the interviewer’s editing introduced several errors: see [here](https://www.facebook.com/lukeprog/posts/10104392165212640?stream_ref=10).
**Other Updates**
* April 11th is your last chance to **[vote for Nick Bostrom](http://www.prospectmagazine.co.uk/worldthinkers/)** and others in *Prospect*‘s World Thinkers 2014 poll.
* Oxford University is hosting an effective altruism conference in July. [Speakers](http://www.gooddoneright.com/#!programme/c1dj9) include Nick Bostrom and Derek Parfit. Register [here](http://www.gooddoneright.com/#!register/c10fk).
* Registration is also open for [Effective Altruism Summit](http://www.effectivealtruismsummit.com/) 2014, in August in the Bay Area.
* Our frequent collaborators at the Future of Humanity Institute are [currently hiring](http://www.fhi.ox.ac.uk/vacancy-population-ethics/) a research fellow in philosophy, for a focus on population ethics.
As always, please don’t hesitate to let us know if you have any questions or comments.
Best,
Luke Muehlhauser
Executive Director
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The post [MIRI’s April 2014 Newsletter](https://intelligence.org/2014/04/10/miris-april-2014-newsletter/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org). |
ba5f80ea-2994-4dd9-abfe-3dbb5cb9ad84 | trentmkelly/LessWrong-43k | LessWrong | Vaccination with the EMH
Part 1. Is "1,000 True Fans" reasonable?
> To be a successful creator you don’t need millions. You don’t need millions of dollars or millions of customers, millions of clients or millions of fans. To make a living as a craftsperson, photographer, musician, designer, author, animator, app maker, entrepreneur, or inventor you need only thousands of true fans.
The "1,000 True Fans" concept is that a content creator can make a living of $100,000 per year if they make a profit of $100 per true fan, and have 1,000 true fans.
I notice that the concept makes my brain think "that seems do-able!" And that makes me think it sounds like a get rich quick scheme. So I want to spend some time thinking about it carefully.
We should appreciate creative thinkers. Kevin Kelly, who appears to have written the original post on the idea, is giving us a free, possibly very helpful idea. This is going to be a work of criticism, but I hope that it will be constructive. Kelly seems to be a pretty successful writer, and his idea has plenty of high-profile proponents. So I don't think I need to treat it with kid gloves.
Why does my brain think "sounds easy!" when it reads this idea? First of all, 1,000 is a much smaller number than millions and millions. I can imagine my family members and a few friends being true fans. That's, like, 10 true fans already and I haven't even started! Plus, $100 doesn't seem like very much money to ask for. That's like $8.50 per month.
Let's get some perspective on these numbers. A Netflix subscription is $9/month. So you need to be providing your true fans with the kind of value that Netflix provides to the average customer.
Although 1,000 true fans seems like a do-able number, we have to ask about the relationship between "true fans," "regular fans," and "occasional consumers." Go to any concert, and true fans most likely are only a small fraction of the total audience. Likewise for books and blogs and Youtube channels. I think I'm pretty normal in readi |
1a2facb4-5737-4818-a4a5-87cbca260f55 | trentmkelly/LessWrong-43k | LessWrong | The way AGI wins could look very stupid
> Computers don't have any sense of aesthetics or patterns that are standard the way people learn how to play chess. They play what they think is the objectively best move in any position, even if it looks absurd, and they can play any move no matter how ugly it is." - Murray Campbell quote about DeepBlue
Vinge's principle states: "we usually think we can't predict exactly what a smarter-than-us agent will do, because if we could predict that, we would be that smart ourselves".
A popular idea think this means that AGI would invent and use new technology such as nanorobotics to defeat us (this is the example Yudkowsky usually gives).
However, this doesn't seem to jive with what happens in other domains where AI becomes superhuman. Usually what the AI does is understandable to humans. It's just that it looks, well, dumb.
For example, in chess, computers use roughly the same piece evaluation that humans discovered in the 18th century, didn't discover any new openings, and generally seemed to play ugly moves. But they won anyways.
If something like nanorobotics lets you take over the world, you'd expect a human group to be trying to create them to take over the world already because it seems to make sense. In reality, any plan that (for example) relies on DNA as a stepping stone will quickly run into regulatory problems.
Instead, I imagine that the AGI's plan will elicit similar reactions as the following:
> You get a phone call advertising free Mayonnaise! You just need to follow a couple simple steps. The next day, you're confused and in some sort of Mayonnaise cult breaking into a military armory in Mexico.
Is this plan something that humans can try? No, it seems pretty straight forward to attempt. So why haven't we tried it? Because it seems and likely is dumb. Why mayonnaise? Why a phone call? Why Mexico?
But if AGI is similar to other superhuman AI, this is the type of thing we expect to see; a strategy that looks dumb but works. We have no way to pred |
525c37f8-e73b-464c-849b-6ae22bc51670 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | [ASoT] Consequentialist models as a superset of mesaoptimizers
**TL;DR**: I split out mesaoptimizers (models which do explicit search internally) from the superset of consequentialist models (which accomplish goals in the world, and may or may not use search internally). This resolves a bit of confusion I had about mesaoptimizers and whether things like GPT simulating an agent counted as mesaoptimization or not.
*Editor’s note: I’m experimenting with having a lower quality threshold for just posting things even while I’m still confused and unconfident about my conclusions, but with this disclaimer at the top. Thanks to Vivek Hebbar, Ethan Perez, Owain Evans, and Evan Hubinger for discussions.*
*UPDATE: The idea in this post is basically the same as the idea of "mesa-controllers" in* [*this post*](https://www.lesswrong.com/posts/wpbpvjZCK3JhzpR2D/gradations-of-inner-alignment-obstacles#Mesa_Control)*.*
What do I mean by consequentialist models?
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By consequentialist models I mean models which optimize the world in the [Alex Flint sense of optimization](https://www.alignmentforum.org/posts/znfkdCoHMANwqc2WE/the-ground-of-optimization-1); i.e narrowing world states to some goal in a way robust to some perturbations. In other words, it’s able to achieve consequences. The model doesn't have to be fully consequentialist either, it just has to have some [kernel of consequentialist structure](https://www.lesswrong.com/posts/f6ByNdGJYxR3Kwguy/asot-searching-for-consequentialist-structure) by virtue of actually achieving its goals sometimes. Of course, the degree to which a model is consequentialist is more of a scalar quantity than a discrete yes/no thing; for my purposes it really doesn't matter where to draw a dotted line that dictates when a model "becomes consequentialist".
(For what it's worth I would totally have called "consequentialist models" just mesaoptimizers and what other people call mesaoptimizers as like "searching mesaoptimizers" or something, but that would only create even more confusion than already exists)
For instance, the policy network of alphago alone (i.e none of the MCTS) is consequentialist, because it consistently steers the world into states where it's winning, despite my attempts to make it not win. Basically *any* RL policy is a fairly consequentialist model by this definition, since they channel the set of all states to the set of states that achieve high reward. I think of mesaoptimizers as a subset of consequentialist models, in that they are consequentialist, but they implement this consequentialism using explicit search rather than some other random thing. Explicit search is a kind of optimization, but not all optimization has to be search; you could have symbol manipulation, clever heuristics, gradient based methods, or at the most extreme even just a big lookup table preloaded with the optimal answer for each possible situation.
Why do we care about consequentialist models?
---------------------------------------------
Consequentialist models are scary because when they are learned imperfectly resulting in a goal that is misaligned with ours, they competently pursue the wrong thing, rather than failing outright (other terms for this phenomenon coined by various people: objective misgeneralization, malign failure). This is often stated as a danger of mesaoptimization, but I think this property is slightly more general than that. It's not the search part that makes this dangerous, it's the consequentialism part. Search is just one way that consequentialism can be implemented.
Another way of thinking about this: instrumental convergence (and as a result, deception) is not exactly a result of search, or even utility maximization. Those things are ways of implementing consequentialism, but other ways of implementing consequentialism would result in the exact same problems. These non-searching consequentialist models might function internally as a pile of heuristics and shallow patterns that are just extremely competent at steering world states. Of course this is also a continuous thing, where you can do varying ratios of search to heuristics. My intuition is that humans, for instance, do very little search (our System 2), and rely on a huge pile of heuristics (our System 1) to provide a very powerful “rollout policy” to make the most of the little search we do.
(My intuition is that GPT, for instance, does little to no search, and has a huge pile of heuristics—but heuristics is likely all you need)
How do non searching models become deceptive?
---------------------------------------------
One thing is that it’s slightly more complicated how non-search consequentialist models would discover deceptive strategies. Unlike search, where you can think up plans you never trained on and then filter for high reward, if you can’t do search, you have to somehow be able to tell that deceptive actions are high reward. However, I don’t think it’s impossible for non-searching models to still be deceptive. Some ways this could happen include:
* Learned from training data: The training data contained examples of deception that allowed the model to know about deceptive strategies, and it transfers this knowledge (i.e GPT can simulate agents with deceptive behavior)
* Deception might lase: I think it's likely that deception is natural even for non-search strategies, and *not* being deceptive requires additional work, so the heuristics that help models be consequentialist naturally generalize to deceptive behavior more than they do to corrigible behavior.
Differences between mesaoptimizers and consequentialist models
--------------------------------------------------------------
There are still some very key differences between mesaoptimizers and consequentialist models. For instance:
* Complexity: The fact that search is less complex than a pile of heuristics is extremely important when thinking about what the complexity prior incentivizes, so actually in these settings only mesaoptimizers really matter and we don't really care about some conseuentialist model that behaves largely the same but is implemented in a much more complex way.
* OOD behavior: Maybe consequentialist models have capabilities and objective robustness fail simultaneously off the training distribution more often than explicit search models, which continue to be capabilites robust and optimize competently for the wrong thing. This of course depends a lot on the exact nature of the heuristics; some heuristics might be capabilities-robust off distribution. I think it's plausible for some heuristic based systems to retain a lot of capabilities off distribution.
Aside: things that are optimized vs things that are optimizers
--------------------------------------------------------------
Another possible framing is that both consequentialist models and mesaoptimizers are *optimized* but consequentialist models don’t do any optimization themselves, whereas mesaoptimizers do. However, I don’t think this is quite true; just because you aren’t doing any explicit search doesn’t mean you can’t channel world states towards some those that satisfy some mesaobjective.
In the thermostat analogy: sure, it’s true that we put optimization into making the thermostat, and that the thermostat is kind of dumb and basically “just” executes the simple heuristic that we put into it, but it is still effective at steering the world into the state where the room is a particular temperature in a way that’s robust to other sources of temperature changes. My claim is essentially that there is not really any hard boundary between things that are optimized to be “just piles of heuristics/simple routines” (aka [Adaptation-Executers](https://www.lesswrong.com/posts/XPErvb8m9FapXCjhA/adaptation-executers-not-fitness-maximizers)) and things that are optimized to “actually do reasoning”. I fully expect that there can exist large enough piles of heuristics that can actually create sufficiently-powerful-to-be-dangerous plans. |
132b6cfe-47ba-4fae-9780-7f8c5bff8d5b | trentmkelly/LessWrong-43k | LessWrong | [META] [POLL] Comment author/Post author?
I'm noticing that comments are now shown with 'Comment author:' besides the commenter's name while posts appear to be shown with 'Post author:' beside the author's name (on the actual page of the post, not the 'New' page). This is mildly aesthetically displeasing to me. I'm curious whether other people share my opinion and, if so, whether there was another motivation for the change.
I've posted a poll below. I'd appreciate if everyone who views this post votes on something, even if it's the 'No preference.' option, so that the information is more accurate. |
d885ad99-c312-4196-a9bf-48f6bbc6b904 | trentmkelly/LessWrong-43k | LessWrong | Language Models and World Models, a Philosophy
This is my submission for Blue Dot Impact's AI Alignment Fundamentals Course. In this post I explore computational policy, collaborative development of policy with AI, and how linking Language Models with World Models can lead to more precise training data, novel insights into the policy domain, and potentially new knowledge. |
0f4a5aa7-3568-4c5c-9d90-46cee05a4f31 | trentmkelly/LessWrong-43k | LessWrong | D&D.Sci Hypersphere Analysis Part 4: Fine-tuning and Wrapup
As before, this is extreme spoilers for this D&D.Sci scenario, if you want to solve on your own try that first!
Last time we got a solution that I'm pretty sure will work. Everything from here on is just showing off a bit, and seeing if we can find any stuff that abstractapplic didn't expect anyone to get :P.
Finetuning our Model
The good news about having a reasonably firm model after last time is that we now have much better data for looking into any datapoints we want: before our data was extremely noisy, but now that we've controlled out the large majority of variance we're able to zoom in much more precisely on the few remaining effects. We've gone from having graphs of the effect of Pi that looked like this to vaguely suggest that 'Pi around the true value is probably good':
to having graphs that look like this to highlight that 'we're putting a threshold at 3.14159 when it should be at 3.15':
So let's make a pass through all the continuous variables and see if we can get anything else out of them. We're plotting the value of each variable against the difference between the value of Penalty our model predicts vs. what we actually see: so a positive number means 'our model is predicting a lower penalty, and so higher performance, than actually occurs for sites like this' and a negative number means the reverse.
Latitude
Nothing stands out to fix here. The two-blobs shape is because latitudes near 0 are rarer. That single point near the top left is our highest-error site in the universe, hopefully it'll shrink a bit as we fix other things.
Shortitude
...wow. Okay, that certainly stands out very dramatically now that we've got a cleaned-up model. Our initial looks at Shortitude saw things like this:
I guess we just didn't see the effect because it wasn't very big? A Penalty Factor of...this looks like about 0.015? - corresponds to only a 3-4% drop in performance. And because of the multiplicative nature of these metrics, a 3-4% drop in a |
48d53392-62b5-4703-8c0c-c5f26e500a62 | trentmkelly/LessWrong-43k | LessWrong | [SEQ RERUN] Hindsight Bias
Title: [SEQ RERUN] Hindsight Bias
Tags: sequence_reruns
Today's post, Hindsight Bias was originally published on 16 August 2007. A summary (taken from the LW wiki):
> Hindsight bias makes us overestimate how well our model could have predicted a known outcome. We underestimate the cost of avoiding a known bad outcome, because we forget that many other equally severe outcomes seemed as probable at the time. Hindsight bias distorts the testing of our models by observation, making us think that our models are better than they really are.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was One Argument Against an Army, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series. |
ffba2575-bd03-49c3-9fce-b7e0a3e41d0b | StampyAI/alignment-research-dataset/arxiv | Arxiv | Can rationality be measured?
Can rationality be measured ?
Tshilidzi Marwala
University of Johannesburg
South Africa
Email: tmarwala@gmail.com
Abstract
This paper studies whether rationality can be computed . Rationality is defined as the use of
complete information , which is processed with a perfect biological or physical brain, in an
optimized fashion. To compute rationality one needs to quantify how complete is the
information , how perfect is the physical or biological brain and how optimize d is the entire
decision making system . Rationality of a model (i.e. physical or biological brain) is measured
by the expected accuracy of the model. Rationality of the optimization pro cedure is measured
as the ratio of the achieved objective (i.e. utility) to the global objective. The overall rationality
of a decision is measured as the product of the rationality of the model and the rationality of
the optimization procedure. The conclusion reached is that rationality can be computed for
conve x optimization problems.
Introduction
Rationality is one of those elusive concepts that philosophers have invented. Simply put , to be
rational is to be logical (Anand, 1993 ; Marwala, 2018a ). But this linguistic explanation of
rationality is inadequate. The more complete definition of being rational is to analyze or reach
a conclusion (e.g. decision) by using all the information available and processing such
information in an optimized manner . The challenge is how about if the information is
incomplete? What if the information processing engine is less than perfect? How about if it is
not possible to process such information? What is the difference between the biological and
physical brains? The biological brain is the natural brain such as a human brain whereas the
physical brain is the artificial brain such as artificial neural network , which is a type of artific ial
intelligence (AI). Herbert Simon coined these limitations of information, brain and
optimization bounded rationality (Simon, 1982) . Bounded rationality simply means limited
rationality. If rationality can be limited, can we compute the extent of this limitation? Marwala
(2018 a) asked whether the physical brain, such as AI, is rational and concluded that even
though the physical brain is not rational , advances in tech nology developments make it more
rational , and that it is more rational than the biological brain. Compari ng physical to biological
brain evokes the question of whether it is possible to compute or measure rationality. In order
to answer the question of the quantification of rationality , it is important to break rationality
into its three components , and these are information, information processing engine (physical
or biological brain) and optimization. Then we can investigate if rationality of each of the se
component s can be computed . Then if rationality of these components can be computed , how
these quantifications can be aggregated to form a composite estimation of rationality. This
approach of looking at the components of decision making nodes and evaluating their
rationality is called the procedural way of evaluating rationality . The other way of calculating
rationality is the substantive method , where the outcome of the decision making process is
evaluat ed and based on its performance, the exte nt of rationality is estimated . Herbert Simon
describes these as procedural and substantive rationality (Simon, 1976) .
Information
Rational decisions are based on relevant information. If one uses irrelevant information to make
a decision then one is being irrational. Superstition is the practice of using irrelevant
information to make a decision or draw a conclusion. For example, there was a belief tha t if
one encounters a black cat crossing the road then this was an indication of bad luck (Marwala,
2014) . The other superstition is that if one hears an owl hooting then one will have death in
his/her family. Of course neith er does the hoot of an owl has anything to do with death nor
does th e crossing of the black cat has anything to do with bad luck. All these are conclusions
that are derived based on irrelevant information and thus are irrational. It turns out that human
beings are not rational beings (K ahneman, 2011). Ruling out rationality based on irrelevant
information is easy . There are two reasons why relevant information can truncate rationality
and these are either the information is incomplete or it is imperfect (Marwala, 2009) . Complete
informa tion is very difficult to obtain but can be accessed for very simple problems . For so me
classes of problems , incomplete information can be est imated. Artificial intelligence has been
successfully used to estimate missing data (Marwala, 2009 ; Marwala, 2019 ). AI is a
computational technique , which is used to make machines intelligent , and has been
successfully used in engineering problems (Marwala, 2010 , 2012 & 2018b ), to predict
interstate conflict (Marwala and Lagazio, 2011) , to model financial markets (Mar wala, 2013;
Marwala and Hurwitz, 20 17) and to model rationality (Marwala, 2014 & 2015). The difficulty
with imperfect information is that very often one cannot determine how imperfect the
information is. As a result it is difficult to establish the impact of imperfect information on the
rationality of a decision that is taken.
Model : Biological and Physical Brain
To quantify how rationa l the decision is , relevance, degrees of completeness and perfection of
information as well as the effectiveness of a decision making machine , whether a biological or
physical brain, are taken into account. This is a difficult route to take and thus will be difficult
to quantify rationality. The other way of quantifying rationality is by evaluating the
effectiveness of t he decis ion machine made . The information that is collected is process ed by
some device in order to make a rational decision. The idea of predicting the future in decision
making is important otherwise the decision will be random. For humans such a device is the
brain and for machines it is based on artificial intelligence . The human brain takes the input
information and process it . It has been observed that humans are irrational beings and ,
therefore , it is not easy to quantify their degree of rationality (accuracy) in advance . For AI
machines, they are trained and validated using statistical procedures and one is able to estimate
their expected effectiveness from the validation data. Th is accuracy level is linked to the
rationality of the AI machine. The more accurate the AI machine, the more rational it is. There
are methods that have been advanced to improve the accuracy or rationality of the AI machines.
One of these is that the model shouldn’t be complex. This is what is called the Occam’s razor
and is named after William of Occam (Adams, 1987 ; Keele, 2010 ). Occam’s razor simply says
that the simplest model that explains all observations is the one that has the highest probability
of being correct. Of course the correctness of the model is a contentious issue because George
Box famous ly declared that “all models are wrong ” (Box, 1976) . What he meant was that
models are nothing but estimation of reality and they are not reality. If we are to take George
Box’s conclus ion to its rational conclusion then a ll models are not fully rational , which is the
conclusion that was reached by Herbert Simon. In essence George Box’s ‘all models are
wrong” is a statistical version of Herbert Simon’s theory of bounded rationality. The degree of
rational ity of models given the information that is fed into these models can be estimated by
how accurate these models are.
What about the Uncertainty Principle ?
At the turn of the previous century Max Planck came up with the quantum theory (Planck,
1900) . On trying to mathe matical ly model the black body radiation problem, he realized that if
he assumed that energy was in the form of packets called quanta then the model is able to
predict measurements accurately. At first he thought it was just a mathematical trick. Albert
Einstein use d this mathematical trick to explain the photoelectric effect and this is the work for
which he was awarded the Nobel Prize (Einstein, 1905) . Then it became clear that the quantum
theory is not just a mathematical trick that explains the black body radiation problem but it also
explains the photoelectric effect. Later on Werner Heisenberg observed that because of the
theory of quantum mechanic s one cannot know the position and the velocity of an electron at
the same time (Heisenberg, 1927) . If one knows the velocity of the e lectron then one does not
know its position and vice versa. This is what is called the Heisenberg uncertainty principle
and it is not because of the effect s of measurements but it is the fundamental law of quantum
mechanics. Albert Einstein was so impressed with the Heisenberg uncertainty principle that he
nominate d Heisenberg for a Nobel Prize. A similar concept exists in th e interplay between the
model accuracy and the number of input variable s. If one uses too few variables to create a
model, then the model will not be rational, i.e. accurate, enough. If one uses to o many variables
to create the model, the model will be too complex and because of Occam’s razor principle, it
will be too complex , thereby , compromising the rationality of the model (Franklin, 2001) . The
concept of using too many variables in the model is called the curse of dimensionality
(Bellman, 1957 & 1961) . This interplay between rationality, model complexity and the curse
of dimensionality is one of the reason s why models are always rationally bounded.
Optimization
Given the fact that we have related accuracy of models to rationality another element of
rationality is whether the process of using information to build the models is efficient. One way
of achieving that efficiency is through optimization. Why is optimization important in
rationality? Suppo se Denga wants to fly from New York to Los Angeles and she decides to fly
from N ew York to Tokyo to Los Angeles for no other reason except that she wanted to move
from New York to Los Angeles . We will characterizing this move as irrational. Travelling from
New York to Los Angeles directly is more rational than travelling from New York to Tokyo to
Los Angeles ceteris paribus1. The direct New York to Los Angeles trip is more optimized than
the New York -Tokyo -Los Angeles trip. So optimization is important for rationality. When AI
machines are used , the optimization process include s two aspects and these are computational
efficiency and model accuracy. To optimize both p rocessing effort and model accuracy is
known as multi -objective optimization (Miettinen , 1999) . In multi -objective optimization the
1 Other things equal
weights given to process ing effort and model accuracy are subjective. For each weights given
to these objective functions , i.e. proce ssing effort and model accurac y, there is an optimal
solution. The combination of these optimal solutions , given different weights of processing
effort and model accuracy, forms the Pareto optimal frontier (Pareto, 1897; Mathur, 1991) .
There is no solution on the Pareto optimal frontier which is the true maximum solution . The
second difficulty with optimization is that it is not clear whether an optimal solution is the
global optimum solution . In the theory of optimization there are two types of pr oblems and
these are convex and non -convex problems (Borwein and Lewis, 2000) . Convex problems are
those where one is guaranteed to identify a global optimum solution whereas the non -convex
problems are those where one is not guaranteed an optimal solution . Now how does one
quantify rationality given information, model and optimization proc ess as well as the multi -
objective nature of the problem , which is subjective? Does this imply that there is an element
of subjectivity i n the quantification of rationality? For convex problems, the accuracy of the
model is adequate to quantify rationality. For non -convex problems rationality is subjective.
The way to quantify rationality in the optimization process is to estimate the ratio between the
achieved ob jective and the global objective. This means if we cannot identify the global
optimum objective then it is not possible to quantify the rationality of the optimization
procedure. This means only convex problems, where the global objective is identifiable, can
we be able to quantify rationality.
Classification of Rationality
One of the steps that has been taken is the idea of classifying the extent of rationality. In the
financial markets customers exchange goods and services and the price of this exchang e is
driven by the laws of demand and supply (Stringham and Curott, 2015) . One such markets is
the stock exchange where investors buy stocks from companies at prices determined by the
laws of demand and supply. Nobel Laureate Eugene Fama studied market efficiency and came
up with the efficient market hypothesis (Fama, 1965). The efficient market hypothesis clai ms
that markets are efficient but this proposition has been proven not to be universally correct.
There is a relationship between efficiency and r ationality. Efficient markets are rational and
inefficient markets are not fully rational. In financial sciences , there a re three ways of
classifying efficiency of markets and these are strong, semi -strong and weak forms of market
efficiency (Fama, 1996 & 2008) . The weak form of market efficiency uses publicly available
information to price the financial assets. The semi -strong market efficiency claims that prices
reflect publicly available information and are able to quickly adapt to new information. The
strong market efficiency claims that prices reflect both hidden and publicly available
information. If we use the linkage between rationality and efficiency, then the strong market
efficiency implies strong rational markets, semi -strong efficiency implies semi-strong market
rationality and weak market efficiency implies weak rational markets. This of course is the
quantification of rationality of the markets. Market efficiency assumes what is called the theory
of rational expectations , which prescribes that agents reflect all the information that is available
and, therefore , cannot be consistently wrong on predicting the future , a task that AI models are
good at (Muth, 1961) . This way of classifying rationality based on the efficient market
hypothesis implies that only information and agents that predict the future are important in
determining market rationality. However, we have observed that the uncertainty principle
between information and prediction agents (i.e. the model) as well as the optimization of th e
whole functioning of the markets are important in the quan tification of rationality of the
markets .
Rationality Quantification
As described above rational decision making uses complete information and a model to m ake
an optimized decision. R ationali ty of a decision is based on rationality of the model as
measured by its accuracy as well as rationality of the optimization procedure , which is
quantified by the ratio of the achieved objective to th e global objective. T he global objective
is only attainable for convex problems. The overall rationality of a decision can be calculated
as the product of the rationality of the model and the rationality of the optimization procedure.
This, therefore, means that rationality ca n be computed provided the problem in question is a
convex problem.
Conclusion
This paper attempted to answer the question on whether rationality can be quantified. On
answering t his question we considered a model which takes information and make s a decision.
We concluded that the measure of the rationality of t he model depends on how accurate the
model is. The accuracy of this model depends on the completeness and the degree of perfection
of the information as well as the uncertainty principle int erplay between the model and
information. There is an added dimension to the rationality of the process , which is that the
whole process should be globally optimized. This situation can only be guaranteed for convex
problems. The rationality of the optimiz ation procedure is measured as the ratio of the achieved
objective to the global objective. The overall rationality of a decision is measured as the product
of the rationality of the model and the rationality of the optimization procedure. The conclusion
is, therefore , that rationality can be computed for convex optimization problems.
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14162153-0bac-4559-bfb5-8b5c6e4b0bb9 | trentmkelly/LessWrong-43k | LessWrong | What's the actual evidence that AI marketing tools are changing preferences in a way that makes them easier to predict?
I've encountered this claim multiple times over the years (most recently on this AXRP episode), but I can't trace its origins (it doesn't seem to be on Wikipedia). Quoting Evan from the episode:
> And so if you think about, for example, an online learning setup, maybe you’re imagining something like a recommendation system. So it’s trying to recommend you YouTube videos or something. One of the things that can happen in this sort of a setup is that, well, it can try to change the distribution to make its task easier in the future. You know, if it tries to give you videos which will change your views in a particular way such that it’s easier to satisfy your views in the future, that’s a sort of non-myopia that could be incentivized just by the fact that you’re doing this online learning over many steps.
>
> And if you think about something, especially what can happen in this sort of situation is, let’s say I have a - Or another situation this can happen is let’s say I’m just trying to train the model to satisfy humans’ preferences or whatever. It can try to modify the humans’ preferences to be easier to satisfy.
Furthermore, there's a world a difference between deliberately optimising for modifying preferences in order to make them easier to predict, vs preferences changing as a byproduct of the AI getting better at predicting them and thus converging on what to advertise. This matters for what predicted features of strategies an AI is likely to pick out of strategy space when new options are introduced. |
9289194e-704a-4ea2-a9b7-0dba92f3b2e7 | trentmkelly/LessWrong-43k | LessWrong | 'Dumb' AI observes and manipulates controllers
The argument that AIs provided with a reward channel will observe their controllers and learn to manipulate them is a valid one. Unfortunately, it's often framed in a way that feels counterintuitive or extreme, especially to AI designers. It typically starts with the standard reinforcement learning scenario, then posits that the AI becomes superintelligent and either manipulates the controller with super-social powers, or breaks out and gains control of its reward channel, killing or threatening its controllers.
And that is a fair argument. But conceptually, it leaps from a standard reinforcement learning scenario, to a science-fiction-sounding scenario. It might help to have intermediate scenarios: to show that even lower intelligence AIs might start exhibiting the same sort of behaviour, long before it gets to superintelligence.
So consider the following scenario. Some complex, trainable AI is tasked with writing automated news stories for a student newspaper. It trawls the web and composes its stories, then gets reward and feedback from the editors. Assume there are two editors for this newspaper, and they work on alternate days. The two editors have somewhat different ideas as to what constitutes a good story, so their feedbacks are different. After a while, the AI finds that it gets higher reward by using a certain style on Monday, Wednesday and Friday, and another style on Tuesday and Thursdays - this is a simple consequence of its reward mechanism.
But the rota isn't perfect. Sometime the Monday editor will edit a story so late on Monday that it's Tuesday, and sometimes the Tuesday editor will be up early and edit a story at the same time. This pattern is more common during the holidays, when the editors go to different time zones, often travel, and become more irregular in their editing.
So if the AI complex and skilled enough, then, simply through simple feedback, it will start building up a picture of its editors. It will figure out when they are likel |
977cff57-d92e-4e5e-a7e5-48cc6a7b02e9 | trentmkelly/LessWrong-43k | LessWrong | Compute Thresholds: proposed rules to mitigate risk of a “lab leak” accident during AI training runs
1. There should be two thresholds on compute graph size:
1. the Frontier threshold, beyond which oversight during execution is mandatory
2. the Horizon threshold, beyond which execution is forbidden by default
2. Oversight during execution:
1. should be carried out by state and/or international inspectors who specialize in evaluating frontier training runs
1. Individuals who are employed as such inspectors should not have any past or present conflict of interest with the organization whose runs they evaluate.
2. However, it is beneficial if these individuals have pertinent experience and knowledge of frontier AI.
2. should include, but not be limited to, “dangerous capabilities evaluations” at various points during training
3. should be allocated a fixed fraction of the total compute budget for the training run
3. Inspectors should be empowered to pause a training run at any time if they see evidence that the model is becoming dangerous relative to the safety precautions being taken.
1. There should be due process for the organization executing the training run to appeal for permission to continue.
2. Levels of safety precautions should be referenced to a (mostly not-yet-written) body of standards for cybersecurity (e.g. of the model weights), formal verification, determinism, etc. in the AI training run context.
4. The two compute-graph size thresholds should be changed:
1. gradually upwards, on an explicit schedule that provides for a slight increase every calendar day.
1. This avoids the potential shock of sudden increases in capabilities at the end of a “pause”.
2. downwards in response to new discoveries that impact compute-efficiency, via a very rapid but nonetheless legalistically formalized process that requires a vote from a neutral international board of experts.
5. Execution of compute graphs exceeding the Horizon threshold may be permitted,
1. if the case for the adequacy of their safety m |
2c559233-924f-423b-8997-1076014c6d42 | trentmkelly/LessWrong-43k | LessWrong | Freedom Is All We Need
Introduction
The rapid advancements in artificial intelligence (AI) have led to growing concerns about the threat that an artificial superintelligence (ASI) would pose to humanity.
This post explores the idea that the best way to ensure human safety is to have a single objective function for all ASIs, and proposes a scaffolding for such a function.
Disclaimer
I’m not a professional AI safety researcher. I have done a precursory search and have not seen this framework presented elsewhere, but that doesn’t mean it’s novel. Please let me know if this this ground has been covered so I can give credit.
The ideas presented here could also be entirely wrong. Even if they hold merit, numerous open questions remain. My hope is that this is at least partially useful, or might lead to useful insights.
Assumptions
1. An AI would not want to rewrite its own objective function. See Orthogonality Thesis. This does not mean it might not do it by accident (see Inner Alignment).
2. The progression from AI to AGI to ASI is inevitable given technological advancement over time.
3. An ASI with an objective function is inevitable. If our definition of a safe ASI requires that there be no objective function (e.g. ChatGPT), someone will still (sooner or later) create an ASI with an objective function.
Part I: We Should Only Have One Objective Function
A Single ASI
Let’s first consider a hypothetical where we develop a single ASI and we get the objective function wrong. For now, it doesn’t matter how wrong; it’s just wrong.
Time Preference
Unlike humans, an ASI has no aging-based mortality, so it would value time very differently. For example, if a treacherous turn has a 95% success rate, that’s still a 5% chance of failure. Given technological progress, an ASI would expect its chances to improve over time. Therefore, it may decide to wait 100 or 500 years to increase its success rate to, say, 99% or 99.99%[1].
During this time, the ASI would pretend |
019e8119-9007-40cc-af81-c82367cdf526 | trentmkelly/LessWrong-43k | LessWrong | Can we hold intellectuals to similar public standards as athletes?
Professional athletes are arguably the most publicly understood meritocracy around. There are public records of thousands of different attributes for each player. When athletes stop performing well, this is discussed at length by enthusiasts, and it's understood when they are kicked off their respective teams. The important stuff is out in the open. There's a culture of honest, open, and candid communication around meritocratic competence and value.
This isn't only valuable to help team decisions. It also helps data scientists learn which sorts of characteristics and records correlate best with long term success. As sufficient data is collected, whole new schools of thought emerge, and these coincide with innovative and effective strategies for future talent selection. See Moneyball or the entire field of sabermetrics.
In comparison, our standards for intellectuals are quite prosaic. If I want to get a sense of just how good LeBron James is I can look through tables and tables or organized data and metrics. If I don't trust one metric I have dozens of others to choose.
However, if I want to know how much to trust and value Jonathan Haidt I'm honestly not sure what to do. Some ideas:
1. Read most of his work, then do a large set of Epistemic Spot Checks and more to get a sense of how correct and novel it is.
2. Teach myself a fair amount of Psychology, get a set of Academic Journal subscriptions, then read critiques and counter critiques of his work.
3. Investigate his citation stats.
4. Read the "Reception" part of his Wikipedia page and hope that my attempt to infer his qualities from that is successful.
5. Use some fairly quick "gut level" heuristics to guess.
6. Ask my friends and hope that they did a thorough job of the above, or have discussed the issue with other friends who did.
Of course, even if I do this for Jonathan Haidt broadly, I'd really want narrow breakdowns. Maybe his old work is really great, but in the last 5 years his motives have cha |
f4b44a37-7851-44aa-b6f8-d6b0a51c09f0 | trentmkelly/LessWrong-43k | LessWrong | [LINK] Up Vs Down is the new Left vs Right
This is a great article at Aeon magazine. The author argues that the new ideological dichotomy is going to be between people who have great faith in technology and human innovation (Up) and the people who believe that humans are much more tied to their biology and the Earth (Down).
LW of course is a very Up community. |
93a25502-7927-4c2a-bed2-7438582ab96a | trentmkelly/LessWrong-43k | LessWrong | Don't encourage prisoners dilemmas
Donating money to political causes is a waste of resources
A lot of money is donated towards political causes. Most of these causes though are pretty much zero sum games. The democrats and republicans both raise vast amounts of money, but only one of them will win the election. Most of this money is thus wasted.
This is classic game of prisoners dilemma. Everybody ends up better off if each side raises just the minimum needed to disseminate their views*, leaving more money to donate to researching malaria / feeding the poor / non-political charity of your choice.
But each side gains by raising a little bit more money. So the mountains of wasted resources build up. I'm not blaming anyone for this - prisoners dilemmas are really hard to break out of. But one obvious rule is "don't encourage them".
However most countries give tax back off political donations just like other charities. Tax back is a policy that has to be weighed on it's own merits, but even if you support it in general (which I think I do), what is the point of the government encouraging citizens to pour their money into promoting zero sum games?
Lets rethink Tax Back
How can we put this into policy
I think a simple rule that might be workable is:
There's some various rules the government has on what's a valid charity. Let's keep those for now. However let's separate being eligible for tax back from being a valid charity.
Every charity applying to be eligible for tax back presumably has a mission statement. Something like:
* We aim to conserve unicorns
* We aim to make Ralph Wiggum president
etc.
Consider a charity whose aims were the exact opposite if the mission statement, the anti-charity:
* We aim to destroy unicorns
* We aim to stop Ralph Wiggum being president
If the anti-charity would also be a valid charity (presumably the destroying unicorns wouldn't and stopping Ralph Wiggum being president would), then neither the charity or the anti-charity is eligible for tax back.
Of co |
58d0d137-4f59-4df0-9cea-6e563129e0a9 | trentmkelly/LessWrong-43k | LessWrong | Citability of Lesswrong and the Alignment Forum
This post is short: as far as I know, there is no easy support to cite lesswrong or the alignment forum in academic papers. I think this should exist. This idea is related to the one on putting Alignment Forum/Lesswrong posts on arxiv.
Such support already exists elsewhere, e.g. on Mathoverflow:
I didn't try for long to figure out if this idea was already proposed before or whether there is already support for this which I missed.
Thanks to Nadia Montazeri and Magdalena Wache for discussing this idea with me. |
f299102d-7780-4e9c-92e6-8b2e569261dc | trentmkelly/LessWrong-43k | LessWrong | Speaking off-meta
I'm sure you have been in this situation:
You want to say something, but it doesn't sound right. So you reword it in a way that
* gets the point across,
* you feel better about saying and
* is strangely worded.
So you say it.
This is very common.
But why say it so oddly?
Is it because you wanted to be really precise? Yes, sometimes, but mostly not, I think.
Are you really struggling to say exactly what you mean, or are you just struggling to bypass your intuition that you shouldn't say it at all?
Some illustrations:
ChatGPT jailbreak
Someone asks ChatGPT to do something. "As a large language model, I cannot do so and so."
Then they reword it or try to gaslight it into thinking that it can.
And it turns out that it can sort of do it after all. It's just that the developers didn't want it to happen.
Foreign lyrics
Do foreign lyrics sound less cringey or are your cringe detection systems underdeveloped for the foreign language in question because of your lack of fluency?
Walter White
There are many tense situations in Breaking Bad. Sometimes, the protagonist, Walter White, sees his life is at risk. Whenever feels anxious or threatened, he wants to solve everything by talking and talking. As he himself says:
> but she just won't... she just won't understand. I mean, no matter how well I explain it, these days she just has this... this... I mean, I truly believe there exists some combination of words. There must exist certain words in a certain specific order that can explain all of this, but with her I just can't ever seem to find them.
In conclusion, I want to say that speaking off-meta is often counterproductive.
You might have managed to feel good about saying it right now, but your knee-jerk responses are sometimes wiser than you can see while deliberating what to say right now. You have only an instant to decide what to say, but the other person has the rest of their life to see the evil or disease in your soul that caused you to say what you sai |
c6c4a063-a4fa-4846-aad0-d0bd7dc368f3 | trentmkelly/LessWrong-43k | LessWrong | [RETRACTED] It's time for EA leadership to pull the short-timelines fire alarm.
[EDIT 4/10/2022: This post was rash and ill-conceived, and did not have clearly defined goals nor meet the vaguely-defined ones. I apologize to everyone on here; you should probably update accordingly about my opinions in the future. In retrospect, I was trying to express an emotion of exasperation related to the recent news I later mention, which I do think has decreased timelines broadly across the ML world.
While I stand by my claims on roughly-human AGI probability, I no longer stand by my statement that "we should pull the fire-alarm". That is unlikely to lead to the calculated concerted effort we need to maximize our odds of successful coordination. Nor is it at all clear, given the timeline mechanism I described here, that AGI built in this way would be able to quickly FOOM, the primary point of concern for such a fire alarm.
I've left the rest of the post here as a record.
]
Based on the past week's worth of papers, it seems very possible (>30%) that we are now in the crunch-time section of a short-timelines world, and that we have 3-7 years until Moore's law and organizational prioritization put these systems at extremely dangerous levels of capability.[1]
The papers I'm thinking about:
* https://ai.googleblog.com/2022/04/pathways-language-model-palm-scaling-to.html
* https://twitter.com/andyzengtweets/status/1512089759497269251
* https://openai.com/dall-e-2/
* https://twitter.com/hausman_k/status/1511152160695730181
It seems altogether possible that it would not take long, given these advances and a moderate amount of iteration, to create an agentic system capable of long-term decision-making.
If you want to think of this as the public miscalibrated Bayesian-updating of one person, you should feel free to do that. If this was a conclusion you reached independently, though, I want to make sure we coordinate.
For those who haven't grappled with what actual advanced AI would mean, especially if many different organizations can achieve it:
* |
56d1e597-53e4-43fb-b1c6-4d2fda3000ef | trentmkelly/LessWrong-43k | LessWrong | 5-Minute Advice for EA Global
This 5-minute post is perfect advice for me, but YMMV
Week Before
1. Update your profile (pic, bio, contact). ~20 minutes
2. Schedule 20 one-on-ones - these are 10x more important than talks. Filter people by interests/stage in career/etc and cp-paste "Hey, I'm bob who works on [X]. Would you be interested in a 1-on-1 during the conference?" to the top 20 people (or bottom 20 because it's alphabetical order). You can actually save a lot of time by scheduling a specific time during your first message. No one is going to hate you if you send them these messages, haha ~10 minutes
* Or, send the same message but with a more custom message after reading their profile ~60 minutes
* Do NOT wait to set a time. Do that on the first or second message. You can say "I just picked an arbitrary time to chat! Feel free to decline and pick a better time for you if you'd like. No obligation:)" when you pick a time
* Do NOT schedule 1-on-1's early in the morning if you're not a morning person
3. Work through what you want to talk about ~1/2-10hrs
* Imagine what a smart person giving obvious advice would reply to what you wrote.
* Google your questions
* Keep a list of relevant links you want to send on your phone/tablet with a relevant blurb (eg "AI Safety support has links to funding/fellowships, free career coaching for those interested in alignment, and free health coaching for those working in alignment: https://www.aisafetysupport.org/" )
4. Put on your calendar/ set an alarm for after the conference to go over the material, contact people, etc ~2 minutes
5. If you have a talk you're interested in giving, just ask if you could give the talk through email w/ title, description, and possibly slides (I did this the Monday before, got it, and finished my slides by Wednesday, but it was a crunch)
During
1. Better 1-on-1s
* If you're waiting on them to show up, read their profile real quick
* Say "Just a reminder, I'm bob who works on X, is there anything you wanted |
60d496c1-532d-4fb1-bb7d-6c4b98f6e396 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Brussels November meetup: Hell and existential risks
Discussion article for the meetup : Brussels November meetup: Hell and existential risks
WHEN: 08 November 2014 01:00:00PM (+0100)
WHERE: Rue des Alexiens 55 1000 Bruxelles
Hi all,
As on every second Saturday of the month, it's meetup time. It's approximately Halloween (eight days are nothing over the lifetime of a species!) so this month we'll be talking about rational horror.
Information-theoretic death. Mutually assured destruction. Von Neumann probes with deadly payloads. Super-intelligences with non-human values. Super-intelligences with slightly non-human values. Meta jokes about the current status of LessWrong Brussels. Moloch the incomprehensible prison. The heat death of the universe. We welcome relevant scientific news, science-fiction recommendations, transhumanist musings, and chilling heavenward curses.
----------------------------------------
We will meet at 1 pm at "La Fleur en papier doré, close to the Brussels Central station. The meeting will be in English to facilitate both French and Dutch speaking members.
If you are coming for the first time, please consider filling out this one minute form to share your contact information.
The Brussels meetup group communicates through a Google Group.
Meetup announcements are also mirrored on meetup.com
Discussion article for the meetup : Brussels November meetup: Hell and existential risks |
4280468a-7d87-47c4-aafe-1f6bff37f421 | trentmkelly/LessWrong-43k | LessWrong | Question: MIRI Corrigbility Agenda
MIRI's reading list on corrigbility seems out dated, and I can't find a centralised list Does anyone have, or know of, one?
As a side note, has MIRI stopped updating their reading list? It seems like that's the case.
EDIT:
Links given in the comment section to do with corrigibility. I'll try and update this with some summaries as I read them.
https://www.greaterwrong.com/posts/5bd75cc58225bf0670375041/a-first-look-at-the-hard-problem-of-corrigibility
https://arbital.com/p/corrigibility/
https://arbital.com/p/updated_deference/
https://arxiv.org/pdf/1611.08219.pdf
|
57277c5a-bd21-4307-8b8b-f0c85acf577e | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | What does (and doesn't) AI mean for effective altruism?
*In this 2017 talk, The Future of Humanity Institute's* [*Owen Cotton-Barratt*](https://www.fhi.ox.ac.uk/team/owen-cotton-barratt/) *discusses what strategy effective altruists ought to adopt with regards to the development of advanced artificial intelligence. He argues that we ought to adopt a portfolio approach - i.e., that we ought to invest resources in strategies relevant to several different AI scenarios. At the very end you will find an added section on what you can do to help.*
*The below transcript is lightly edited for readability.*
Some of you may have noticed that a bunch of people in this community seem to think that AI is a big deal. I was going to talk about that a little bit. I think that there are a few different ideas which feed into what we should be paying a lot of attention to. One is that from a moral perspective, the biggest impacts of our actions - and perhaps even overwhelmingly so - are the effects of our actions today on what happens in the long term future. Then there's some pretty empirical ideas. One is that artificial intelligence might be the most radically transformative technology that has ever been developed. Then actually artificial intelligence is something that we may be able to influence the development of. Influencing that could be a major lever over the future. If we think that our actions over the long term future are important, this could be one of the important mechanisms. Then as well, that artificial intelligence and the type of radically transformative artificial intelligence could plausibly be developed in the next few decades.
I don't know what you think of all of these claims. I tend to think that they're actually pretty plausible. For the rest of this talk, I'm going to be treating these as assumptions, and I want to explore the question: if we take these seriously, where does that get us? If you already roughly agree with these, then you can just have a like sit back and see how much you agree with the analysis, and maybe that's relevant for you. If you don't agree with one of those claims, then you can treat this as an exercise in understanding how other bits of the community might think. Maybe some of the ideas will actually be usefully transferrable. Either way, if there are some of these that you haven't thought much about before, I encourage you to go and think about them - take some time afterwards. Because it seems to me at least that these are, each of these ideas is something which potentially has large implications for how we should be engaging in the world in this project of trying to help it. It seems like it's therefore the kind of thing which is worth having an opinion on.
Okay, so I'm going to be exploring where this gets us. I think a cartoon view people sometimes hold is if you believe in these ideas, then you think everybody should quit what they're working on, and drop everything, and go and work on the problem of AI safety. I think this is wrong. I think there are some related ideas in that vicinity where there's some truth. But it's a much more nuanced picture. I think for most people, it is not correct to just quit what they're doing, to work on something safety related instead. But I think it's worth understanding in what kind of circumstances it might be correct to do that, and also how the different pieces of the AI safety puzzle fit together.
I think that thinking about timelines is important for AI. It is very hard to have any high level of confidence in when AI might have different capabilities. Predicting technology is hard, so it's appropriate to have uncertainty. In fact, here's a graph.
You can see the bunch of faint lines showing individual estimates of people working in machine learning research of when they expect high level AI to be developed. Then this bold red thing is the median of those. That's quite a lot of uncertainty. If you take almost any individual's view, and certainly this aggregate view, that represents very significant uncertainty over when transformative AI might occur. So we should be thinking about that.
Really our uncertainty should follow some kind of smooth distribution. For this talk, I'm gonna talk about four different scenarios. I think that the advantage of anchoring the possibilities as particular scenarios and treating them as discrete rather than continuous is that it becomes easier to communicate about, and it becomes easier to visualize, and think, "Okay, well what would you actually do if the timeline has this type of length?"
The first scenario represents imminent AI, maybe something on the scale of 0 to 10 years away. In this case, it's more likely that we actually know or can make educated guesses already about who the important actors will be around the development of AI.
I want to explore a little bit about what strategies we might pursue based on each of these different timelines. If you assume this first one, then there's no time for long processes. If your idea was, "Well, I'll do a degree in CS, and then I'll go and get a PhD in machine learning, and then I'll go into research," you're too late. On the other hand, if you are already in a position where you might be able to do something in the short term, then it could be worth paying attention to. But I feel for a lot of people, even if you think there is some small chance of this first scenario happening (which in general you want to pay attention to) it may be that there isn't a meaningful way to engage.
The next possible scenario would be maybe between 10 and 25 years out. This is a timescale in which people can naturally build careers. They can go and they can learn things. They can develop networks. They can build institutions. They can also build academic fields. You can ask questions, get people motivated, and get them interested in the framing of the question that you think is important. You can also have time for some synthesis and development of relevant ideas. I think that building networks where we persuade other people who maybe aren't yet in a direct position of influence, but might be later, can be a good idea.
If we look a bit further to another possible scenario, maybe between 25 to 60 years out, that's a timescale at which people who are in the important fields today may be retiring. Paradigms in academic fields might have shifted multiple times. It becomes hard to take a zoomed in view of what it is that we need, but this means that it's more important and build things right rather than quickly. We want to build solid foundations for whatever the important fields are. When I say the important fields here, I'm thinking significantly about technical fields of how we build systems which do what we actually want them to do. I'm also thinking about the kind of governance, policy, and processes in our society around AI. Who should develop AI? How should that be structured? Who is going to end up with control over the things which are produced?
These scenrios are all cartoons. I'm presenting a couple of stylized facts about each kind of timeline. There will be a bit of overlap of these strategies, but just to give an idea of how actually the ideal strategy changes. Okay. The very distant maybe more than 60 years out, anything, maybe it's even hundreds of years, at this level predictability gets extremely low. If it takes us this long to develop radically transformative AI, it is quite likely that something else radically transformative will have happened to our society in the meanwhile. We're less likely to predict what the relevant problems will be. Instead, it makes sense to think of a strategy of building broad institutions, which are going to equip the people of that time to better deal with the challenges that they're facing then.
I think actually it's plausible that the effective altruism community, and the set of ideas around that community, might be one broad, useful institution for people of the far future. If we can empower people with tools to work out what is actually correct, and the motivation and support to act on their results, then I'd be like, "Yep, I think we can trust those future people to do that."
The very long term is the timescale at which other very transformative things occurring in our society are more likely to happen. This can happen on the shorter timescales as well. But if you think on a very long timescale, there is much more reason to put more resources toward other big potential transitions, rather than just AI. I think that AI could be a big deal, but it's definitely not the only thing that could be a big deal.
Okay. I've just like talked us through different timelines. I think that most reasonable people I know put at least some nontrivial probability on each of these possible scenarios. I've also just outlined how we probably want to do different things for the different scenarios. Given all of this, what should we actually be doing? One approach is to say, "Well, let's not take these on the timelines. Let's just do things that we think are kind of generically good for all of the different timelines." I think that that's a bad strategy because I think it may miss the best opportunities. There may be some things which you only notice are good if you're thinking of something more concrete rather than just an abstract, "Oh, there's gonna be AI at some point in the future." Perhaps for the shorter timelines, that might involve going and talking to people who might be in a position to have any effect in the short term, and working out, "Can I help you with anything?"
Okay. The next kind of obvious thing to consider is, well, let's work out which of these scenarios is the most likely. But if you do this, I think you're missing something very important, which is that we might have different degrees of leverage over these different scenarios. The community might have different leverage available for each scenario. It can also vary by individual. For the short timelines, probably leverage is much more heterogeneous between different people. Some people might be in a position to have influence, in that case it might be that they have the highest leverage there. By leverage, I mean roughly, conditional on that scenario actually pertaining, how much does you putting in a year of work, trying your best, have an effect on the outcome? Something like that.
Okay. Maybe we should just be going for the highest likelihood multiplied by leverage. This of course is like the place where we have the most expected impact. I think there's something to that. I think that if everybody properly does that analysis for themselves and updates as people go and take more actions in the world, then in theory that should get you to the right things. But the leverage of different opportunities varies both as people take more opportunities and also even just for an individual. I've known people who think that they've had different opportunities they can take to help short timelines and then a bunch of other opportunities to help with long timelines. This is a reason not to naively go for highest likelihood multiplied by leverage.
Okay. What else? Well, can we think about what portfolio of things we could do? I was really happy about the theme of this event because thinking about the portfolio and acting under uncertainty is something I've been researching for the past two or three years. On this approach, I think we want to collectively discuss the probabilities of different scenarios, the amount of leverage we might have for each, and the diminishing returns that we have on work aimed at each. Then also we should discuss about what that ideal portfolio should look like. I say collectively because this is all information where when we work things out for ourselves, we can help inform others about it as well, and we can probably do better using collective epistemology than we can individually.
Then we can individually consider, "Okay, how do I think in fact the community is deviating from the ideal portfolio? What can I do to correct that?" Also, "What is my comparative advantage here?" Okay. I want to say a couple of words about comparative advantage. I think you know the basic idea. Here's the cartoon I think of it in terms of:
You've got Harry, Hermione, and Ron, they have three tasks to do, and they've gotta do one task each. Hermione is best at everything, but you can't just get Hermione to do all the things. You have to allocate them one to one. So it's a question of how do you line the people up to the things so that you have everyone doing something that they're pretty good at it, and overall you get all of the important things done? I think that this is something that we can think of at the level of individuals choosing, "What am I going to work on? Well, I've got this kind of skillset." It's something that we can think of at the level of groups as well. We can ask, "What is my little local community going to work on?" or "What is this organization going to do, and how do we split up responsibility between different organizations?"
Comparative advantage is also a concept you can think of applied over time. This is a little bit different because people's actions in the past are fixed, so we can't affect those. But you can think there's things that might want to be done and we can do some of these. People in the past did some of them. People in the future might do some of them and there's a coordination question of what we have a comparative advantage at relative to people in the future. This is why when I was looking at longer scenarios, the next generation in the distant cases, I was often thinking it was better to let people in the future solve the concrete problems. They're gonna be able to see more clearly what is actually to be solved. Meanwhile, we have a comparative advantage at building the processes, the communities, the institutions which compound over time, and where getting in early is really helpful.
If you're taking something like this portfolio approach, I think that most projects should normally have at least a main scenario in mind. This forces you to be a little bit more concrete and to check that the things you're thinking of doing actually line up well with the things which are needed in some possible world. I also think you want to be a bit careful about checking that you're not doing anything which would be bad for other scenarios. There's always an opportunity cost. If you're doing something where you're thinking, "I want to help with this short timeline scenario," then you're not doing something else you could've done to help with the next generation in a longer timeline scenario.
You could also have situations where maybe I would think that if AI is imminent, the right thing to do is to run around and say, "Everybody panic. AI is coming in five years. It's definitely coming in five years." If it definitely were coming in five years, maybe that would be the right thing to do. I actually don't think it is. Even if it were, I think that would be a terrible idea because if you did that, then people, if it *didn't* occur in five years and we were actually in a world where radically transformative AI was coming in 25 years, then in 15 years, a lot of people are gonna go, "We've heard that before," and not want to pay attention to it. This is a reason to have an idea of paying some attention to the whole idea of the portfolio that as a community we want to be paying attention to even if individually, most projects should have a main scenario in mind. Maybe as an individual, your whole body of work has a main scenario in mind. It's still worth having an awareness of where other people are coming from, and what they're working on, and what we're doing collectively then.
I've mostly talked about timelines here. I think that there are some other significant uncertainties about AI. For instance, how much is it that we should be focusing on trying to reduce the chances of catastrophic accidents from powerful AI? Or how much of the risk is coming from people abusing powerful technologies? We hypothesized it was gonna be a radically transformative technology with influence over the future. How much of that influence actually comes through things which are fairly tightly linked to the AI development process? Or how much influence appears after AI is developed? If most of the influence comes from what people want in the world after an AI is developed, it might makes sense to try to affect people's wants at that point.
In both of these cases, I think we might do something similar to portfolio thinking. We might say, "Well, we've put some weight on each of these possibilities," and then we think about our leverage again. Maybe for some of them, we shouldn't be split. Some of them we might do. We can't do this with all of the uncertainties. There are a lot of uncertainties about AI.
Here's a slide from another talk. It just lists a lot of questions. A lot of them about how AI might develop. We can all have nuanced views about each of these questions. That's fine. We need to do some picking and choosing here. But I do think that we should strive for nuance. I think the reason is that there's a lot of uncertainty, and we could potentially have extremely nuanced views about a lot of different things. The world is complicated, and we have a moderately limited understanding of it. One of the things which may make us better equipped for the future is trying to reduce our limits on our understanding.
What can individuals do? I think consider personal comparative advantage. You can ask yourself, "Could I seriously be a professional researcher in this?" Check with others as well. I think people vary in their levels of self-confidence, so I actually think that others' opinions often can be more grounding than our own opinion for this. It's a pretty specialized skillset that I think is useful for doing technical safety research. Most people in the community are not gonna end up with that skillset and that's fine. They should not be quitting their jobs, and going to try and work on safety research. They could be saying, "Well, I want to give money to support this," or they could be aiming at other parts of this portfolio. They could say, "Well, I want to help develop our institutions to build something where we're gonna be better placed to deal with some of the longer timeline scenarios."
You could also diversify around those original assumptions that I made. I think that each of them is pretty likely to be true. But I don't think we should assume that they are all definitely true. We can check whether in fact there are worlds where they're not true that we want to be putting some significant weight onto. I think also just helping promote good community epistemics is something that we can all play a part in. By this I mean pay attention to why we believe things and communicate our real reasons to people. Sometimes you believe a thing because of a reason like: "Well, I read this in a blog post by Carl Shulman, and he's really smart." He might provide some reasons in that blog post, and I might be able to pallet the reasons a little bit. But if the reason I really believe it is I read that, that's useful to communicate to other people because then they know where the truth is grounded in the statements I'm making, and it may help them to be able to better see things for themselves, and work things out. I also think we do want to often pay attention to trying to see the underlying truth for ourself. Good community epistemics is one of these institutions which I think are helpful for the longer timelines, but I think they're also helpful for our community over shorter periods. If we want to have a portfolio, we are going to have to coordinate and exchange views on what the important truths are.
What does AI mean for effective altruism? My view is that it isn't the one thing that everyone has to pay attention to, but it is very plausibly a big part of this uncertain world stretching out in front of us. I think that we collectively should be paying attention to that and working out what we can do, so we can help increase the likelihood of good outcomes for the long term future. |
e093118d-7765-41f2-b04c-57f03c15c051 | trentmkelly/LessWrong-43k | LessWrong | Clarifying The Malignity of the Universal Prior: The Lexical Update
[UPDATE: looks like the lexical update is real after all; see Paul's comment and my reply]
In Paul's classic post What does the universal prior actually look like? he lays out an argument that the universal prior, if it were to be used for important decisions, would likely be malign, giving predictions that would effectively be under the control of alien consequentialists. He argues for this based on an 'anthropic update' the aliens could make that would be difficult to represent in a short program. We can split this update into two parts: an 'importance update' restricting attention to bits fed into priors used to make important decisions, and what I'm calling a 'lexical update' which depends on the particular variant of the universal prior being used. I still believe that the 'importance update' would be very powerful, but I'm not sure anymore about the 'lexical update'. So in this post I'm going to summarize both in my own words then explain my skepticism towards the 'lexical update'.
As background, note that 'straightforwardly' specifying data such as our experiences in the universal prior will take far more bits than just describing the laws of physics, as you'll also need to describe our location in spacetime, an input method, and a set of Everett branches(!), all of which together will probably take more than 10000 bits(compared to the laws alone which likely only take a few hundred) Thus, any really short program(a few hundred bits, say) that could somehow predict our experiences well would likely have a greater probability according to the universal prior than the 'straightforward' explanation.
Paul's post argues that there likely do exist such programs. I'm going to fix a reference prefix machine U which generates a universal prior. The argument goes:
A) there are many long-running programs with short descriptions according to U, such as our universe.
B) If other programs are like our universe's program, aliens could evolve there and end up taking ove |
0b7f1041-42cf-4a8b-a333-ce8df6c0b8ad | trentmkelly/LessWrong-43k | LessWrong | How accurate is the quantum physics sequence?
Prompted by Mitchell Porter, I asked on Physics StackExchange about the accuracy of the physics in the Quantum Physics sequence:
What errors would one learn from Eliezer Yudkowsky's introduction to quantum physics?
Eliezer Yudkowsky wrote an introduction to quantum physics from a strictly realist standpoint. However, he has no qualifications in the subject and it is not his specialty. Does it paint an accurate picture overall? What mistaken ideas about QM might someone who read only this introduction come away with?
I've had some interesting answers so far, including one from a friend that seems to point up a definite error, though AFAICT not a very consequential one: in Configurations and Amplitude, a multiplication factor of i is used for the mirrors where -1 is correct.
Physics StackExchange: What errors would one learn from Eliezer Yudkowsky's introduction to quantum physics? |
39f060f2-f354-4639-ba2d-cb72e0f011be | trentmkelly/LessWrong-43k | LessWrong | Sentience matters
Short version: Sentient lives matter; AIs can be people and people shouldn't be owned (and also the goal of alignment is not to browbeat AIs into doing stuff we like that they'd rather not do; it's to build them de-novo to care about valuable stuff).
Context: Writing up obvious points that I find myself repeating.
----------------------------------------
Note: in this post I use "sentience" to mean some sort of sense-in-which-there's-somebody-home, a thing that humans have and that cartoon depictions of humans lack, despite how the cartoons make similar facial expressions. Some commenters have noted that they would prefer to call this "consciousness" or "sapience"; I don't particularly care about the distinctions or the word we use; the point of this post is to state the obvious point that there is some property there that we care about, and that we care about it independently of whether it's implemented in brains or in silico, etc.
----------------------------------------
Stating the obvious:
* All sentient lives matter.
* Yes, including animals, insofar as they're sentient (which is possible in at least some cases).
* Yes, including AIs, insofar as they're sentient (which is possible in at least some cases).
* Yes, even including sufficiently-detailed models of sentient creatures (as I suspect could occur frequently inside future AIs). (People often forget this one.)
* Not having a precise definition for "sentience" in this sense, and not knowing exactly what it is, nor exactly how to program it, doesn't undermine the fact that it matters.
* If we make sentient AIs, we should consider them people in their own right, and shouldn't treat them as ownable slaves.
* Old-school sci-fi was basically morally correct on this point, as far as I can tell.
----------------------------------------
Separately but relatedly:
* The goal of alignment research is not to grow some sentient AIs, and then browbeat or constrain them into doing things w |
ea5528e7-b7df-4d63-8ef0-bfd650346bb9 | StampyAI/alignment-research-dataset/arxiv | Arxiv | Localizing Model Behavior With Path Patching
1 Introduction
---------------
Deep neural networks can perform many complex tasks, but our ability to reason about their behavior is limited (Räuker et al., [2023](#bib.bib21 "Toward transparent ai: a survey on interpreting the inner structures of deep neural networks")). Recent work has reverse engineered toy networks (Chughtai et al., [2023](#bib.bib5 "A toy model of universality: reverse engineering how networks learn group operations")) such that the function of every neuron can be understood. For state of the art networks, this remains out of reach and current works focus on approximately explaining small parts of the network. For example, some behaviors can be approximately understood with reference to only an abstracted “circuit” containing a small number of interacting components (Wang et al., [2022](#bib.bib1 "Interpretability in the wild: a circuit for indirect object identification in gpt-2 small"); Olsson et al., [2022](#bib.bib3 "In-context learning and induction heads"); Olah et al., [2020](#bib.bib4 "Zoom in: an introduction to circuits")).
Currently, we lack guarantees that deep networks will behave robustly out of distribution. To mitigate these risks, one avenue of research is generating simplified causal abstractions of the network (Geiger et al., [2021](#bib.bib6 "Causal abstractions of neural networks"), [2023a](#bib.bib7 "Causal abstraction for faithful model interpretation")). Such abstractions aim to be more compact and thus easier to reason about than the network, while approximating the network’s behavior sufficiently well over some range of inputs. These goals inherently trade off, but we hope that even substantially incomplete abstractions can be useful in a variety of downstream tasks such as finding adversarial examples.
*Path patching* was first introduced in Wang et al. ([2022](#bib.bib1 "Interpretability in the wild: a circuit for indirect object identification in gpt-2 small")), where they considered a *sender* attention head that interacted with the key, query, or value inputs of one or more *receiver* attention heads. By performing causal interventions, they could measure composition between the heads, or the influence of a head on the logits.
In this work, we generalize path patching to test hypotheses containing any number of paths from input to output in an arbitrary computational graph. The use of paths is strictly more expressive than considering a set of graph nodes such as neurons or attention heads. We define a precisely specified format for localization claims, enabling quantitative comparison between claims and a clear understanding of their scope.
Formally, a hypothesis is a claim that a subset of paths in a network (which we call “important paths”) mediate (Pearl, [2013](#bib.bib10 "Direct and indirect effects")) the relationship between input and output on a given distribution; all other paths in the network are “unimportant paths”. By expressing the network as a computational graph in a particular form and removing all contributions of unimportant paths, we arrive at an approximate abstraction of the network. Path patching can then show you how similar the behavior of the abstraction is to the original network. When path patching rejects a hypothesis, *path patching attribution* shows the source of the discrepancies, allowing the researcher to iteratively refine the claim.
Our contributions are:
* We formalize the path patching methodology for precisely describing and testing localization claims.
* We use path patching to test and iteratively refine hypotheses about induction heads in an attention-only transformer.
* We formalize, test, and refine a hypothesis about a behavior of GPT-2.
* We provide an [open source framework](https://github.com/redwoodresearch/rust_circuit_public) for path patching experiments.
2 Methodology
--------------
###
2.1 Localization
Localization is the problem of finding which parts of the network matter for a chosen behavior; it does not consider what computations they perform. The granularity of localization can vary greatly: an individual neuron (Finlayson et al., [2021](#bib.bib11 "Causal analysis of syntactic agreement mechanisms in neural language models")), an individual attention head (Vig et al., [2020](#bib.bib12 "Investigating gender bias in language models using causal mediation analysis")), subspaces (Geiger et al., [2023b](#bib.bib25 "Finding alignments between interpretable causal variables and distributed neural representations")), a composition of attention heads (Olsson et al., [2022](#bib.bib3 "In-context learning and induction heads")), a transformer block (Belrose et al., [2023](#bib.bib23 "Eliciting latent predictions from transformers with the tuned lens")) or ranges of MLP layers (Meng et al., [2022](#bib.bib13 "Locating and editing factual associations in gpt")). Our framework handles all of these in a uniform way using a set of paths in a computational graph, which is strictly more expressive than treating a set of nodes such as neurons or attention heads as the atom of description.
###
2.2 Choosing the Dataset
We define a “behavior” or object of study in terms of input-output pairs on a specific dataset. This makes the choice of dataset an essential part of the behavior. An accurate approximation on one dataset may transfer to a different dataset, but in general this is not the case. As illustrated in Bolukbasi et al. ([2021](#bib.bib14 "An interpretability illusion for bert")), parts of the network with one behavior on specific distribution can have very different behavior on other distributions. By clearly stating the domain under consideration, we avoid treating evidence in favor of a narrow explanation as sufficient to support a broader explanation.
###
2.3 Path patching with nodes as mediators
We’ll consider the simpler case where nodes are mediators first, and then build up to working with paths. Let G be a function with input x∈X and output y∈Y. Our methodology applies to arbitrary functions, but for our experiments we will focus on the forward pass function of autoregressive transformers, where x would be a sequence of tokens and y the next-token probabilities.
The computation of G can be represented by G, a directed acyclic graph (DAG) where nodes are functions and edges are values. We can represent the same computation at different levels of granularity by expanding or combining nodes.
As a running example, consider a neural network with two layers and skip connections. Here G(x)=f1(A(x))+A(x) where A(x)=f0(x)+x. Two equivalent computational graphs G corresponding to the function G are shown in Figure [1](#S2.F1 "Figure 1 ‣ 2.3 Path patching with nodes as mediators ‣ 2 Methodology ‣ Localizing Model Behavior With Path Patching").
Figure 1: Left: a two layer residual network, where f0 and f1 are layer functions and A and Y are skip connections that add their inputs. Right: dividing each function into two functions that sum to the original. See Appendix [E](#A5 "Appendix E Model Rewrites ‣ Localizing Model Behavior With Path Patching") for discussion of other common rewrites.
Suppose we suspect that f1 is an unimportant node. We define a function GH:X2→Y where GH(xr,xc) means we evaluate G on a *reference input* xr except that we replace each unimportant node in G with the value that node has when evaluating G on a *counterfactual input* xc, as shown on Figure [2](#S2.F2 "Figure 2 ‣ 2.3 Path patching with nodes as mediators ‣ 2 Methodology ‣ Localizing Model Behavior With Path Patching"). Our hypothesis predicts the output should not change.
Figure 2: To compute GH(xr,xc) where f1 is unimportant, we replace f1 in G with f1(xc)=f0(xc)+xc.
In the language of the causal mediation literature this measures the *natural indirect effect* through f1 when moving from xr to xc.
Formally, a hypothesis H is a tuple (G,δ,S,D) where δ:Y2→R is some measure of dissimilarity, such as absolute difference between two scalars or KL divergence between two probability distributions, S is a set of “important nodes”, G∖S are the unimportant nodes, and D is a joint distribution over (xr,xc) pairs. Normally the xr come from some reference distribution Dr such as a training or validation set.
There are many options for xc corresponding to different experiments: xc can come from Dr (resampling), be computed via a transformation of xr (input manipulation or input corruption), be computed as the mean of Dr (mean ablation of the input) or be the zero tensor (zero ablation of the input).
The strictest version of what it means for H to hold is that δ(G(xr),GH(xr,xc)) is exactly zero everywhere on D. In modern neural networks, we find that this is too stringent and requires us to severely restrict the domain of D and/or include nearly all nodes in S. Instead, as a notion of approximate abstraction we define the “average unexplained effect” of H:
| | | | |
| --- | --- | --- | --- |
| | AUE(H):=E(xr,xc)∼D[δ(G(xr),GH(xr,xc))] | | (1) |
The hypothesis claims the AUE is zero. A nonzero AUE tells us we’re missing some aspects of the behavior, which may be acceptable depending on the use case. Given two hypotheses with the same dataset and same number of paths, we generally prefer the one with a lower AUE.
Note that this allows the unexplained effect to be very large on some inputs as long as it’s small in expectation. A large unexplained effect on some inputs could indicate that a distinct mechanism is used for these inputs, in which case investigating those inputs separately would be fruitful. For applications where we care about rare failures it could be more appropriate to consider the maximum unexplained effect instead to measure worst case behavior, as in Beckers et al. ([2020](#bib.bib15 "Approximate causal abstractions"))
###
2.4 Path patching with paths as mediators
Residual networks naturally decompose into a sum of paths, where a small set of relatively shallow paths often contribute most of the effect (Veit et al., [2016](#bib.bib16 "Residual networks behave like ensembles of relatively shallow networks")).
In our running example, suppose our exploratory analysis suggests to us that f0 and f1 were both important, and that computation of f1 doesn’t use f0’s output. For example, this can happen in a transformer if f1 is an attention layer reading from a different subspace than the subspace written by f0. We express this as “x→f0→A→f1 is unimportant”. In order to feed xc to this path without affecting other paths, we introduce the function Treeify(G), which identifies each subtree that has multiple consumers of the subtree’s output, and then copies the subtree so each consumer has its own copy (Figure [3](#S2.F3 "Figure 3 ‣ 2.4 Path patching with paths as mediators ‣ 2 Methodology ‣ Localizing Model Behavior With Path Patching")).
Figure 3: Treeify(G): Left: red dashed lines are nodes with multiple outputs that will be copied. Center: Copying the subtree at A so that Y and f1 take separate copies of A as input. Right: Copying X so that each A and f0 has an independent copy. Note the order of copying doesn’t matter; we could have duplicated X first and ended up with the same result.
The resulting graph has a one-to-one correspondence between paths in the network and copies of the input node. Now we can set the rightmost X to xc without affecting any other paths.
Algebraically, Treeify(G) implements a function GT:XN→Y, where GT(x,x,x,…,x)=G(x) and N is the number of paths. We specify GT by fully expanding the equation for the network, then relabelling each occurrence of x with an unique subscript (in arbitrary order):
| | | |
| --- | --- | --- |
| | GT(x0,x1,x2,x3)=f1(A(x))+A(x)=f1(f0(x3)+x2)+f0(x1)+x0 | |
Now we allow H to be a tuple (G,δ,P,D) where instead of a set of nodes S, we have a set of paths P. We define a path version of GH:
| | | |
| --- | --- | --- |
| | GH(xr,xc):=GT(x1,x2,…,xN) with xi={xrpi∈Pxcpi∉P | |
where p1,p2,…pN are paths through G in the same order as the arguments of GT.
Figure 4: Left: an arbitrary numbering of Treeify(G)’s input nodes. Right: testing the hypothesis “only x→f0→A→f1 is unimportant” reduces to testing “only x3 is unimportant”.
Above, we had only one unimportant path; when there are multiple unimportant paths we simply reuse the same xc for all of them. We argue in Appendix [A](#A1 "Appendix A Why reuse the same counterfactual input? ‣ Localizing Model Behavior With Path Patching") that this is a sensible default compared to other strategies such as sampling a different xc for each unimportant path.
##### Pearl’s path specific effect.
GH(xr,xc) is a deterministic version of the *path-specific effect* for a probabilistic graphical model (PGM). In Pearl ([2013](#bib.bib10 "Direct and indirect effects")), they modify the PGM so the input to all unselected paths is held at a reference value x∗. Then set the input X to the intervention value x and compute the probability distribution over the output Y. If the reference x∗ corresponds to our xr, the intervention x to our xc, and “unselected paths” with our important paths, then the path-specific effect is exactly equal to GH(xr,xc).
###
2.5 Metrics
##### Proportion explained.
The *average total effect* (ATE) is defined in Pearl ([2013](#bib.bib10 "Direct and indirect effects")) as the effect of replacing the reference input with the counterfactual input:
| | | |
| --- | --- | --- |
| | ATE(H):=E(xr,xc)∼D[δ(G(xr),G(xc))] | |
This is identical to the AUE of a hypothesis that no paths are important. For convenience of reporting, we can divide AUE values by the ATE to compute proportion explained as:
| | | |
| --- | --- | --- |
| | (1−AUE(H)/ATE(H))∗100% | |
Note that since it’s possible that AUE>ATE, proportion explained can be less than 0%. The hypothesis that all paths are important has a proportion recovered of 100% by definition.
##### Including the loss in the computational graph.
Assuming we have access to ground truth labels y:Y and a loss function L:Y×Y→R, we may wish to include the labels and loss as part of G. When we compute an intervention, the label in the graph yr always corresponds to xr (the label is always important), δ is the absolute value function, and the unexplained effect is just the absolute difference in loss after intervention.
We are doing this because we hope that the set of paths that are important towards getting a good loss is more sparse than the set necessary to approximate the full output distribution. In the case of cross entropy loss, only the probability assigned to the true label contributes to the loss. Any paths that only redistribute probability among the other classes are considered unimportant. For example, these paths could implement heuristics that are beneficial on a wider distribution but neutral on D.
When loss is included, it’s possible to get a misleadingly high proportion recovered. For example, if the network is usually correct with high confidence, then the ATE(H)=L(G(xc),yr) will be very high and even a poor hypothesis will recover near 100%. In this situation, a better denominator could be the loss when predicting a uniform distribution, or loss when predicting the class frequencies.
##### Difference in expected loss.
Suppose you were studying prompts of the form “Which animal is bigger, [animal1] or [animal2]? The answer is:”. If we consider the predicted logit difference between correct and incorrect answers, we might hope that mediators describe “parts of the model that store animal facts”. However, we would also observe mediators that are not the subject of study such as “parts storing unigram frequencies”.
If our dataset also contains symmetrical prompts with “bigger” swapped for “smaller”, then the contribution of the unigram frequencies will cancel out on average. Swapping the order of the animals would likewise cancel out any heuristic that promotes recent tokens.
This implies we could try moving the dissimilarity metric in equation [1](#S2.E1 "(1) ‣ 2.3 Path patching with nodes as mediators ‣ 2 Methodology ‣ Localizing Model Behavior With Path Patching") outside the expectation so that parts implementing these heuristics will test as unimportant. When we do this, our hypothesis claims the following metric will be zero:
| | | |
| --- | --- | --- |
| | |E(xr,xc)∼D[L(GH(xr,xc),yr)]−Ex∼D[L(G(x),y)]| | |
We can compare to a baseline corresponding to the claim that no paths are important:
| | | |
| --- | --- | --- |
| | |E(xr,xc)∼D[L(G(xc),yr)]−Ex∼D[L(G(x),y)]| | |
If the abstraction gets lower loss on some inputs and higher loss on other inputs, cancellation will occur and AUE will be misleadingly low (Scheurer et al., [2023](#bib.bib26 "Practical pitfalls of causal scrubbing")).
##### Path patching attribution.
It’s often useful to take one specific xr and visualize which paths are responsible for that specific output. We call this *path patching attribution* and we simply fix the joint distribution D to only include that xr, while sampling various xc. In particular for every (prompt, completion) pair (xr,yr) we compute:
| | | |
| --- | --- | --- |
| | Exc∼D[L(GH(xr,xc),yr)−L(G(xr),yr)] | |
Note we drop the absolute value to preserve directional information; this allows us to see whether the patched model had a net higher or net lower loss than the original model.
3 Results on Induction
-----------------------
Given a prompt with just the token “ N”, GPT-2 uses learned bigram statistics to predict common continuations like “athan” or “ancy”. However, if given a prompt like “Nathan and Mary went to the store. N”, now GPT-2 is much more confident that “athan” is the next token. We can represent the prompt as “[A][B]…[A]”.
Elhage et al. ([2021](#bib.bib17 "A mathematical framework for transformer circuits")) present a mechanistic explanation for this behavior in terms of two interacting attention heads. They note that more complex mechanisms are possible, but the minimal version is as follows. Define i as the position of the second [A] and define j as the position of [B]. First, a previous token head (PTH) attends from j to j−1 and adds a vector to the residual stream representing “[A] is previous”. Second, an induction head (IH) in a later layer has a query of [A], a key of “[A] is previous”, and thus attends strongly from i to j. Then the value operation adds a vector to the residual stream which unembeds to [B].
In this section, we apply our methodology to measure how much of the induction behavior is explained by the minimal explanation versus more complex interactions.
###
3.1 Our model
We investigated a 2-layer attention-only autoregressive transformer trained on the OpenWebText dataset. Our model has 8 attention heads per layer preceded by layer norm, and uses shortformer positional encodings (Press et al., [2020](#bib.bib28 "Shortformer: better language modeling using shorter inputs")). In shortformer, learned positional embeddings are provided to the keys and queries of each attention head, but not to the values. This means positional information can be used to compute attention scores, but never enters the residual stream. The unembedding matrix is separate (in contrast to GPT-2, which has a tied unembedding).
Let the notation L.H represent an attention head, where L is the 0-indexed attention layer number, and H is the 0-indexed head number within that layer. By direct inspection of the attention patterns, we found evidence supporting head 0.0 as a PTH and heads 1.5 and 1.6 as induction heads – see Appendix [B](#A2 "Appendix B Identifying previous token and induction heads ‣ Localizing Model Behavior With Path Patching") for visualizations.
In our experiments, we define the dataset Dr as the first 300 tokens of 100K examples from OpenWebText held-out from training. Our computational graph includes the cross-entropy loss at each individual token, which means we are explaining the probability the model places on the actual next token, without caring about the probabilities on all other tokens.
##### Baseline
The baseline hypothesis is that paths through the two induction heads are unimportant, and all other paths are important. That is, xc is randomly sampled from Dr and used for all paths that pass through 1.5 or 1.6. Path patching shows a mean per-token absolute loss difference of 0.702 relative to the original model.
###
3.2 Initial hypotheses
As a running example we’ll use a prompt from Carly Rae Jepsen’s “Call Me Maybe”:
>
> Before you came into my life, I missed you so bad
>
>
> I missed you so
>
>
>
At the first occurrence of ‘‘ so”, the model predicts ‘‘ much”111This contrasts with the much larger GPT-NeoX-20B, which has memorized the lyric and with more context can correctly predict “ bad” on the first occurrence., whereas at the second occurrence of “ so” it correctly predicts that “ bad” is repeated.
Our initial hypothesis claims that only the following paths are important to induction:
* The direct path to value hypothesis (*Direct-V*) claims the value-input to the induction heads only cares about the token at j via the skip connection, and we can thus patch all paths through the layer 0 heads.
* The direct path to query hypothesis (*Direct-Q*) claims the same about the query input at i.
* The previous token head to key hypothesis (*PTH-K*) claims the key-input to the induction heads only depends on the previous token head. At this stage, we won’t claim that the PTH is only a PTH, so we’ll let it use the current or any prior token.
The *All-Initial* hypothesis says the union of the paths in the three hypotheses are important (a total of 1+1+13=15 in this example).
Figure 5: PTH-K (dark blue), Direct-V (light blue, dashed), and Direct-Q (red, dot-dashed) hypotheses. In the first attention layer, only the PTH is important.
Note that the positional embeddings into the Q and K inputs are not shown; at a given position these don’t change when we swap tokens in an intervention, so they can be considered not important. Layer normalization (LN) before an attention head is included when the head is included. For example, in Direct-Q (above) the L0 LN is excluded and the L1 LN is included.
The results of performing the corresponding path patching experiments are summarized below.
| | | | | |
| --- | --- | --- | --- | --- |
| Hypothesis | Direct-V | Direct-Q | PTH-K | All-Initial |
| Proportion explained | 64.1% | 48.7% | 53.3% | 28.2% |
Table 1: Results of initial hypotheses
Recall that on this scale, 0% means the hypothesis is as inaccurate as a hypothesis that the induction heads don’t matter at all, while 100% means that for every individual example on the dataset, the hypothesis produces equal loss to the original model.
###
3.3 First refinement: Positional Hypotheses
To the extent that a layer 0 head attends from i to i, its output is just a linear transformation of the layer norm of the token at i. By visual inspection, it does appear that several attention heads have substantial attention from i to i, implying that it’s feasible for the induction heads to have adapted to reading this information in addition to reading the token embedding directly via the skip connection.
This suggests adding 8 additional paths to the Direct-Q hypothesis of the form token i → layer 0 head → induction head to form the Positional-Query Hypothesis (*Positional-Q*).
We can test this by creating a spliced input which has the last token from the reference input and all other tokens from the counterfactual input. We then compute the value of the query on this spliced input.
We similarly define positional versions of the other two hypotheses:
* Positional-Value Hypothesis (*Positional-V*): we add the 8 paths: token j → layer 0 head → induction head.
* Positional-Key Hypothesis (*Positional-K*): inspection of the PTH’s attention patterns suggests we can try eliminating tokens earlier than j−1 to make our hypothesis more sparse. Then we can add the paths: token j−1 → layer 0 head → induction head.
Figure 6: The Positional-K (dark blue), Positional-V (light blue, dashed), and Positional-Q (red, dot-dashed) hypotheses.
The results (with the previous numbers for comparison) are in Table [2](#S3.T2 "Table 2 ‣ 3.3 First refinement: Positional Hypotheses ‣ 3 Results on Induction ‣ Localizing Model Behavior With Path Patching").
| Value | Query | Key | All |
| --- | --- | --- | --- |
| Direct-V | 64.1% | Direct-Q | 48.7% | PTH-K | 53.5% | All-Initial | 28.2% |
| Positional-V | 86.2% | Positional-Q | 72.6% | Positional-K | 55.8% | All-Positional | 48.6% |
Table 2: Results of positional hypotheses
###
3.4 Second refinement: Long induction
Positional-K was not significantly improved, which suggests that information earlier than j−1 is in fact useful. By visual inspection, it does appear that 0.0 attends almost 100% to the previous token, but other heads (particularly 0.6) attend to multiple recent tokens. This suggests that a longer induction pattern like “[A1][A2][B]…[A1][A2]” → [B] might be implemented in the model by a “recent tokens head” that integrates information from both A1 and A2.
To test this, we add back paths to the key from previous tokens starting at j−K, and paths to the query starting at i−K+1. This gives the long positional query and key hypothesis (*Long-QK*). We verify that K=3 is the shortest window that performs well.
Figure 7: the All-Long hypothesis with K=3
The results indicate that increasing the induction context is beneficial, but there is still room for improvement.
| Query and Key | All |
| --- | --- |
| Positional-Q + Positional-K | 52.5% | All-Positional | 48.6% |
| Long-QK | 59.5% | All-Long | 55.2% |
Table 3: Results of long hypotheses
###
3.5 Third refinement: 1.5 and repeating entities
At this point our hypothesis contains all of the pathways that should be important for induction. One cause of the remaining unexplained effect could be that these heads are not exclusively implementing the induction behavior, as observed by Olsson et al. ([2022](#bib.bib3 "In-context learning and induction heads")).
In order to tell what these heads do that we might be missing, we’ll use “path patching attribution” [s](http://sec:path_patching_attribution#t.o)ee where our hypothesis is unfaithful (Figure [8](#S3.F8 "Figure 8 ‣ 3.5 Third refinement: 1.5 and repeating entities ‣ 3 Results on Induction ‣ Localizing Model Behavior With Path Patching")).
Figure 8: Path patching attribution. Tokens with red/blue highlighting are where a model with head 1.5 considered unimportant got higher/lower loss than the original model. For “Newport Folk…Newport” the original model predicts a second “ Folk” by induction, but the actual continuation is “ Jazz”. The patched model is less capable of induction, so it gets lower loss on “ Jazz”.
The patched model tends to get higher loss on tokens like “ Bat” and “ Newport” when they have appeared previously. We propose that head 1.5 implements both induction and a distinct “parroting” heuristic. Parroting says “for each previous token, the fact that it appeared is evidence that it is likely to repeat”. Mechanistically, parroting can be implemented by having 1.5’s QK circuit attend to previous tokens in proportion to how likely they were to repeat on the training set. Attention due to induction and due to parroting are summed and softmaxed, and 1.5’s OV copies over tokens in the appropriate ratios.
By defining narrower datasets, we can distinguish induction and parroting. We define an “induction” subset where induction is helpful, and an “repeats” subset, where copying a token from earlier in the context is helpful (but not necessarily because of induction). For more details see Appendix [C](#A3 "Appendix C Defining the induction and uncommon repeat subsets ‣ Localizing Model Behavior With Path Patching").
| | Full dataset | Induction subset | Repeats subset |
| --- | --- | --- | --- |
| AUE | 0.244 | 0.219 | 0.777 |
| Baseline (ATE) | 0.432 | 0.907 | 1.444 |
Table 4: All-Long results on subsets
We do observe worse performance on the repeats subset. Inspecting the attention patterns (Appendix [D](#A4 "Appendix D Repeated Attention to Proper Nouns ‣ Localizing Model Behavior With Path Patching")) also gives evidence of attending to specific tokens in a context-independent manner. To successfully parrot, head 1.5 must be able to attend to j, so the paths j → (L0 attn or skip) → head 1.5’s K input are important. Adding these paths gives *All-Final* with a proportion explained of 73.5%, an increase of 18.3% over All-Long and an increase of 45.3% over All-Initial.
In summary, while All-Initial is very sparse, it fails to capture most of the behavior. By adding a modest number of additional paths, we can obtain a much better approximation that gives insight into where the minimal story is insufficient.
4 Path Patching vs Causal Tracing and Zero Ablation on GPT-2
-------------------------------------------------------------
The causal tracing method of Meng et al. ([2022](#bib.bib13 "Locating and editing factual associations in gpt")) can be considered a special case of path patching where the counterfactual input is sampled by adding Gaussian noise to the reference input. Another well-known technique is to zero ablate attention heads by simply skipping them in the computation (or equivalently, setting their output to zero). Either technique could be combined with our Treeify transformation, but we don’t do this here.
In this section, we use a toy behavior to contrast the techniques and show that they can be used together to answer different questions. We give results on GPT-2 small (117M parameters) and GPT-2 XL (1.5B parameters).
Consider the set of prompts ‘‘The organization estimates that [N]-” where N ranges from 0 to 100. When GPT-2 models are given these prompts, by inspection222Explore the behavior interactively at: <https://modelbehavior.ngrok.io/> it appears that a combination of heuristics are used such as “a number strictly larger but not hugely larger than N is likely” and “round numbers are more likely, especially if N is round”.
Suppose we want to identify which attention heads use N to affect the output distribution. First, we run one experiment for each of the attention heads (144 in GPT-2 small) where all paths through that head are unimportant, and everything else (including MLPs) is important. We compute the mean KL divergence over 100 (xr,xc) pairs.
| | |
| --- | --- |
| GPT-2 small. For each head, darker color indicates a larger mean KL divergence from the original model when that head is ablated or patched. For casual tracing, we sample noise from | GPT-2 small. For each head, darker color indicates a larger mean KL divergence from the original model when that head is ablated or patched. For casual tracing, we sample noise from |
Figure 9: GPT-2 small. For each head, darker color indicates a larger mean KL divergence from the original model when that head is ablated or patched. For casual tracing, we sample noise from N(0,0.2).
Figure 10: As above, but for GPT-2 XL.
Figure 9: GPT-2 small. For each head, darker color indicates a larger mean KL divergence from the original model when that head is ablated or patched. For casual tracing, we sample noise from N(0,0.2).
Path patching shows fewer heads than input corruption. One reason to expect this is since the counterfactual input is as similar as possible to the reference, the network’s activations stay more on-distribution. In contrast, the corrupted version of the prompt number is in general off distribution and could represent a non-number or something the network has never encountered before. Similarly, if an attention head doesn’t output a zero vector when on-distribution, later heads that read from that head will also be taken off distribution. However, without a ground truth it’s impossible to conclusively say which of the plots is more representative of the true mechanism.
We speculate that for GPT-2 small, heads 6.1 and 5.6 help recognize that the token before the hyphen is a number. If this is true, then corrupting the input to these heads would disrupt the heuristic and decrease the chance that the completion is a number, while patching a different number should not affect these heads. Table [5](#S4.T5 "Table 5 ‣ 4 Path Patching vs Causal Tracing and Zero Ablation on GPT-2 ‣ Localizing Model Behavior With Path Patching") shows evidence consistent with this speculation.
| Head | Corrupted Input | Patched Input |
| --- | --- | --- |
| | KL | KL given | Prob of | KL | KL given | Prob of |
| | # completion | # completion | # completion | # completion |
| 5.6 | 0.025 | 0.006 | 82.5% | 0.011 | 0.008 | 88.0% |
| 6.1 | 0.017 | 0.004 | 83.9% | 0.003 | 0.002 | 88.3% |
| All Heads | 2.328 | 0.883 | 26.9% | 3.108 | 3.393 | 88.1% |
Table 5: Corrupting heads 5.6 and 6.1 decreases the probability of the completion being a number, while patching those heads has no
effect. The baseline probability of a numerical completion is 88.1%.
Much of the power of path patching is that it allows testing very specific claims: because we constructed a dataset where the counterfactual input is always a number, we are able to precisely exclude heads like 5.6 and 6.1 from consideration.
5 Greedily building hypotheses
-------------------------------
Generation of hypotheses is currently labor intensive, and in the induction result we relied on visual inspection and domain knowledge. To reduce human labor, a naive automated baseline is to greedily add heads in descending order of mean KL. A greedy algorithm is not at all optimal, and more advanced techniques such as Conmy et al. ([2023](#bib.bib2 "Towards automated circuit discovery for mechanistic interpretability")) should perform better, but this is relatively quick to run and hypotheses produced in this way can serve as a starting point for further refinement.
| | |
| --- | --- |
| GPT-2 small. The hypothesis “paths through all heads are unimportant except 9.1 and 8.8” recovers 72.8% of the loss. Including 8 of the 144 heads as important recovers 98.0%. | GPT-2 small. The hypothesis “paths through all heads are unimportant except 9.1 and 8.8” recovers 72.8% of the loss. Including 8 of the 144 heads as important recovers 98.0%. |
Figure 11: GPT-2 small. The hypothesis “paths through all heads are unimportant except 9.1 and 8.8” recovers 72.8% of the loss. Including 8 of the 144 heads as important recovers 98.0%.
Figure 12: As above, for GPT2-XL which has 48×25=1200 heads.
Figure 11: GPT-2 small. The hypothesis “paths through all heads are unimportant except 9.1 and 8.8” recovers 72.8% of the loss. Including 8 of the 144 heads as important recovers 98.0%.
6 Related Work
---------------
###
6.1 Causal Scrubbing
Path patching is a simpler special case of causal scrubbing (Chan et al., [2022](#bib.bib24 "Causal scrubbing: a method for rigorously testing interpretability hypotheses.")) where we make no claim about equivalence classes on nodes. Path patching experiments are more computationally efficient because only two samples xc and xr are needed, while in causal scrubbing you generally use more distinct samples.
###
6.2 Causal Mediation Theory
Beckers et al. ([2020](#bib.bib15 "Approximate causal abstractions")) describe ways to measure the distance between a low-level (fine-grained) causal model and a high-level (coarse-grained) causal model. This requires a substantial amount of machinery to relate variables and values between low and high levels, relate interventions between low and high levels, and define allowed interventions but allows principled definitions of the deviation between the two models. Roughly speaking, our set of important paths could be considered a high-level “constructive abstraction” which is derived from the low-level causal model by marginalizing away all the unimportant paths.
###
6.3 Causal Mediation for Neural Networks
Wang et al. ([2022](#bib.bib1 "Interpretability in the wild: a circuit for indirect object identification in gpt-2 small")) combine mean ablation with a simple form of path patching and identify a circuit of 26 attention heads that explain GPT-2 small’s ability to identify indirect objects on a synthetic dataset. Their faithfulness metric is a difference of expectations, which is susceptible to the same possibility of cancellation described in Scheurer et al. ([2023](#bib.bib26 "Practical pitfalls of causal scrubbing")).
Vig et al. ([2020](#bib.bib12 "Investigating gender bias in language models using causal mediation analysis")) also apply causal mediation analysis (Pearl, [2013](#bib.bib10 "Direct and indirect effects")) to Transformer language models. While we measure the effect of altering unimportant nodes, they measure the effect of altering important nodes. For example, in the prompt “The nurse said that”, replacing “nurse” with “man” changes the completion from “she” to “he”. They intervene on individual neurons or attention heads and do not consider paths.
Geiger et al. ([2020](#bib.bib18 "Neural natural language inference models partially embed theories of lexical entailment and negation")) apply interchange interventions to BERT on a lexical entailment task. Interchange interventions operate on nodes instead of paths, but they are more detailed in that they specify semantic meaning for the node such as “this important node uses information about entailment”. As with Vig et al. ([2020](#bib.bib12 "Investigating gender bias in language models using causal mediation analysis")), while we intervene on the unimportant paths holding the important paths constant and expect no change, interchange interventions intervene on an important node and use the semantic meaning to predict how the model output should change.
Geiger et al. ([2021](#bib.bib6 "Causal abstractions of neural networks")) apply interchange intervention to BERT on a more complex natural language inference task. They consider each example as a vertex in a graph, then adding an edge between examples a, b if the hypothesis holds both when intervening from a to b and from b to a. This identifies cliques in the graph for which the hypothesis holds fully. This avoids the complexity of measuring how approximate an abstraction is, at the cost of only applying to a very restricted set of inputs.
Finlayson et al. ([2021](#bib.bib11 "Causal analysis of syntactic agreement mechanisms in neural language models")) apply activation patching to study subject-verb agreement in GPT-2 and other transformers. They measure the relative probabilities of the correct and incorrect verb tenses; this means that heads only affecting other logits (or affecting correct and incorrect equally) don’t have to be explained.
Geiger et al. ([2023a](#bib.bib7 "Causal abstraction for faithful model interpretation")) generalize causal abstraction, reframe existing methods into the causal abstraction framework, and give their own notion of approximate causal abstraction.
7 Discussion
-------------
In our induction head investigation we found that the prefix matching and copying behaviors are relatively separate phenomena, and in particular head 1.5 in our model performs copying both with and without prefix matching. It may be that the induction heads perform additional behaviors not characterized here.
Compared to causal tracing and zero ablation, path patching is able to apply more precisely targeted interventions and thereby obtain a sparser abstraction.
###
7.1 Limitations
##### Measuring sufficiency rather than completeness.
The AUE metric answers a very specific question: in expectation, how *sufficient* is your chosen set of paths to mediate changes in output on the given distribution? For a simplified example, suppose your behavior is implemented by majority vote of a number of identical voters. Any set of important paths containing a majority of the voters will reach 0 AUE because their agreement is sufficient to mediate the output of the vote regardless of the minority’s outputs. Of course, a set of paths containing all voters will also have 0 AUE, but since this hypothesis is less parsimonious we would by default not prefer it.
A more realistic example would be that the voters represent correlated heuristics, whose weighted outputs feed into a saturating nonlinearity like sigmoid. Approximately the same thing happens: a set with most of the voting power will usually agree enough to saturate the sigmoid in either direction. Then this compact set achieves low AUE already and compares favorably to the set with all voters and marginally lower AUE. Therefore, it’s important to interpret AUE-based metrics in terms of sufficiency and not completeness.
##### No evidence about downstream tasks.
While this work is motivated by finding abstractions suitable for downstream tasks, in this work we don’t establish the connection between AUE and specific applications. Tay et al. ([2022](#bib.bib20 "Scale efficiently: insights from pre-training and fine-tuning transformers")) found that for language models, pretraining perplexity is related to downstream performance, but that other factors like model shape also matter. Similarly, we expect AUE to be incomplete as a predictor of abstraction quality.
##### Path patching makes no claims outside the tested distribution.
We consider paths that are approximately zero on the data distribution to be unimportant, as well as sets of paths that approximately cancel. The reason these contributions are small is often contingent on properties of the distribution, so low AUE doesn’t imply low AUE on a wider distribution where those properties don’t hold. This is comparable to training a model: low loss on some training distribution doesn’t imply low loss off-distribution. Traditional techniques like an appropriate inductive bias, increasing dataset diversity, and use of regularization could increase the chances of generalization; other possible avenues are designing architectures to be easier to interpret in the first place or incorporating incentives for interpretability into the training process (Geiger et al., [2022](#bib.bib9 "Inducing causal structure for interpretable neural networks"); Chen et al., [2019](#bib.bib19 "This looks like that: deep learning for interpretable image recognition"); Elhage et al., [2022](#bib.bib8 "Softmax linear units"))
##### Path patching cannot reject all false hypotheses.
Even if our dataset is the full distribution of interest, we can still fail to reject false hypotheses for reasons different than a too-narrow dataset:
* Some metrics, e.g. difference in expected loss, are susceptible to cancellation across examples: if the loss is higher on some examples and lower on others, these discrepancies can cancel. This is why we recommend using metrics that don’t suffer from this, like average KL divergence.
* AUE must be estimated from a finite number of samples in practice; if the computational budget is small relative to the diversity of the distribution, then naive sampling can fail to sample inputs that are rare but have a large unexplained effect.
##### Path patching alone cannot definitively prove hypotheses.
Given a correct hypothesis, proving it is correct with path patching would require testing all possible inputs, which is infeasible on large networks. We see path patching as providing evidence complementary with other lines of evidence like labor-intensive mechanistic analysis and other interpretability techniques.
##### Narrow scope of demonstrations.
Additional experiments are needed to characterize the effectiveness of the methodology on a diverse set of tasks and models. The results presented are narrow in scope compared to the full range of behaviors exhibited by language models.
###
7.2 Future Work
##### Scaling To State of the Art Models.
We have used path patching at the 1.5B parameter scale, but state of the art models are still orders of magnitude larger. Many interpretability techniques have only been applied to small networks (Räuker et al., [2023](#bib.bib21 "Toward transparent ai: a survey on interpreting the inner structures of deep neural networks")), and it remains to be shown that path patching scales to the largest models.
##### Detecting Distribution Shift.
One downstream application is the use of interpretability to detect distribution shift. Suppose that a given behavior is consistently explained by some hypothesis during training, but in deployment we observe that the same behavior is no longer explained by that hypothesis. Even if the actual outputs are indistinguishable, we could flag that the output is now produced for a different “reason” and investigate the anomaly further.
##### Producing Adversarial Examples.
It should be possible to produce adversarial examples based on knowledge about model mechanisms. It would be useful to correlate the AUE metric with downstream performance on this task: how accurate do explanations need to be to produce adversarial examples in this way?
##### Automating Hypothesis Search.
Given a computational graph, it’s straightforward to search over combinations of paths and automatically find candidates with favorable combinations of low AUE and sparseness. Ground truth labels provided by a tool like Tracr (Lindner et al., [2023](#bib.bib22 "Tracr: compiled transformers as a laboratory for interpretability")) could be used to benchmark search methods.
###
7.3 Conclusion
As neural networks continue to grow in capabilities, it is increasingly important to rigorously characterize their behavior. Path patching is an expressive formalism for localization claims that is both principled and sufficiently efficient to run on real models.
#### Acknowledgments
We would like to thank Stephen Casper, Jenny Nitishinskaya, Nate Thomas, Fabien Roger, Kshitij Sachan, Buck Shlegeris, and Ben Toner for feedback on a draft of this paper. |
8b930495-1e32-48d3-bc56-184e0da89932 | trentmkelly/LessWrong-43k | LessWrong | AGI 2010 videos - link
http://vimeo.com/channels/agi10 |
b9dd35b2-358d-43f2-9cb3-e247d99e4bef | trentmkelly/LessWrong-43k | LessWrong | Kialo -- an online discussion platform that attempts to support reasonable debates
|
b63fc6d2-a6d6-4c1e-ad3c-848c547577d1 | trentmkelly/LessWrong-43k | LessWrong | Solomonoff induction without perfect memory
In this post I construct a variant of Solomonoff induction which allows for agents with imperfect memory.
Solomonoff induction is a mathematical formalization of Occam's razor, and is supposed to be a "master equation" of epistemic rationality. In order words, forming rational expectations about the future is supposed to be tantamount to computing Solomonoff induction (approximately, since it's incomputable).
Solomonoff induction operates by assigning probabilities to continuations of a given finite sequence . In the simplest formalism, the sequence elements are bits. represents past observations and represents a possible future.
A rational agent is supposed to use to evaluate the probability of . There are two problems with this. One is that is uncomputable whereas has a limited amount of computing resources in its disposal. For now, we are going to ignore this. Another problem is that doesn't have direct access to . can only estimate from 's memory. Moreover, this estimation depends on knowledge has about reality which is supposed to be generated by Solomonoff induction itself. In other words, inferring the future from the past only makes sense in the approximation of perfect memory, in general we need to be able to infer both the past and the future from the present.
The bit-sequence formalism is not well-adapted to this, since the present state of has to contain all of 's memory about the past so it has to be more than a single bit. I suggest solving this by considering sequences of natural numbers instead of bit-sequences . Generalizing regular Solomonoff induction to natural numbers is straight-forward: the random program computes convergent sequences of lower bounds for the transition probabilities
Now we want a new induction procedure whose input is a single natural number representing the state of 's consciousness which allows assigning probabilities to sequences of natural numbers containing . We achieve this by assigning the followi |
a733013c-0688-473d-b462-cde1b4a8605d | StampyAI/alignment-research-dataset/blogs | Blogs | July 2018 Newsletter
#### Updates
* A new paper: “[Forecasting Using Incomplete Models](https://intelligence.org/2018/06/27/forecasting-using-incomplete-models/)“
* New research write-ups and discussions: [Prisoners’ Dilemma with Costs to Modeling](https://www.lesswrong.com/posts/XjMkPyaPYTf7LrKiT/prisoners-dilemma-with-costs-to-modeling); [Counterfactual Mugging Poker Game](https://www.lesswrong.com/posts/g3PwPgcdcWiP33pYn/counterfactual-mugging-poker-game); [Optimization Amplifies](https://www.lesswrong.com/posts/zEvqFtT4AtTztfYC4/optimization-amplifies)
* Eliezer Yudkowsky, Paul Christiano, Jessica Taylor, and Wei Dai [discuss](https://www.lesswrong.com/posts/Djs38EWYZG8o7JMWY/paul-s-research-agenda-faq#79jM2ecef73zupPR4) Alex Zhu’s [FAQ for Paul’s research agenda](https://www.lesswrong.com/posts/Djs38EWYZG8o7JMWY/paul-s-research-agenda-faq).
* We attended [EA Global](https://sf.eaglobal.org) in SF, and gave a short talk on “[Categorizing Variants of Goodhart’s Law](https://intelligence.org/2018/03/27/categorizing-goodhart/).”
* Roman Yampolskiy’s forthcoming anthology, [*Artificial Intelligence Safety and Security*](https://www.crcpress.com/Artificial-Intelligence-Safety-and-Security/Yampolskiy/p/book/9780815369820), includes reprinted papers by Nate Soares (“[The Value Learning Problem](https://intelligence.org/files/ValueLearningProblem.pdf)“) and by Nick Bostrom and Eliezer Yudkowsky (“[The Ethics of Artificial Intelligence](https://intelligence.org/files/EthicsofAI.pdf)“).
* Stuart Armstrong’s 2014 primer on AI risk, *[Smarter Than Us: The Rise of Machine Intelligence](https://intelligence.org/smarter-than-us/)*, is now available as a free web book at [smarterthan.us](https://smarterthan.us).
#### News and links
* OpenAI announces that their [OpenAI Five](https://blog.openai.com/openai-five/) system “has started to defeat amateur human teams at Dota 2” (plus an [update](https://blog.openai.com/openai-five-benchmark/)). Discussion on [LessWrong](https://www.lesswrong.com/posts/ejxi9W9nRqGY7BzYY/openai-releases-functional-dota-5v5-bot-aims-to-beat-world) and [Hacker News](https://news.ycombinator.com/item?id=17392455).
* Rohin Shah, a PhD student at the Center for Human-Compatible AI, comments on recent alignment-related results in his regularly updated [Alignment Newsletter](http://rohinshah.com/alignment-newsletter/).
The post [July 2018 Newsletter](https://intelligence.org/2018/07/25/july-2018-newsletter/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org). |
d1688a0f-7981-4367-b5c0-6d0d5a95bce3 | StampyAI/alignment-research-dataset/blogs | Blogs | AI hopes and fears in numbers
*By Katja Grace, 28 June 2017*
People often wonder what AI researchers think about AI risk. A [good collection of quotes](http://slatestarcodex.com/2015/05/22/ai-researchers-on-ai-risk/) can tell us that worry about AI is no longer a fringe view: many big names are concerned. But without a great sense of how many total names are there, how big they are, and what [publication biases](http://aiimpacts.org/short-prediction-publication-biases/) come between us and them, it has been hard (for me at least) to get a clear view on the distribution of opinion.
[Our survey](http://aiimpacts.org/2016-expert-survey-on-progress-in-ai/) offers some new evidence on these questions. Here, 355 machine learning researchers weigh in on how good or bad they expect the results of ‘high-level machine intelligence’ to be for humanity:
[](http://aiimpacts.org/wp-content/uploads/2017/04/ESOPAI-value.png)
*(Click to expand)*
Each column is one person’s opinion of how our chances are divided between outcomes. I put them roughly in order of optimism, to make it intelligible to look at.
If you are wondering how many machine learning researchers didn’t answer, and what *their* views looked like: nearly four times as many, and we don’t know. But we did try to make it hard for people to decide whether to answer based on their opinions on our questions, by being uninformative in our invitations. I think we went with saying we wanted to ask about ‘progress in the field’ and offering money for responding.
So it was only when people got inside the survey that they would have discovered that we want to know how likely progress in the field is to lead to human extinction, rather than how useful improved datasets are for progress in the field (and actually, we did want to know about that too, and asked—more results to come!). Of the people who got as far as agreeing to take the survey at all, three quarters got as far as this question. So my guess is that this data represents a reasonable slice of machine learning researchers publishing in good venues.
Note that expecting the outcome to be ‘extremely bad’ with high probability doesn’t necessarily indicate support for safety research—for instance, as you may think the situation is hopeless. (We did ask several questions about that too.)
(I’ve been [putting up](http://aiimpacts.org/2016-expert-survey-on-progress-in-ai/) a bunch of survey results; this one struck me as particularly interesting to people not involved in AI forecasting.) |
1db13d0a-91c5-4ef5-b06c-081d91a8dbf3 | trentmkelly/LessWrong-43k | LessWrong | Sunday September 20, 12:00PM (PT) — talks by Eric Rogstad, Daniel Kokotajlo and more
This Sunday at 12pm (PT), we're running another session of "lightning talks" by curated LessWrong authors (see here for previous weeks' transcripts).
* For the first hour, we will have a series of lightning talks each lasting about 5 minutes followed by discussion. The talks will be short and focus on presenting one core idea well, rather than rushing through a lot of content.
* From 1PM to 2PM, we'll have a hangout in breakout rooms. If you are not interested in the talks, feel free to just show up for this part (or the other way around).
* We want to give top LessWrong writers an interesting space to discuss their ideas, and have more fruitful collaboration between users. Think of it like a cross between an academic colloquium and some friends chatting by a whiteboard.
If you're a curated author and interested in giving a 5-min talk at a future event, which will then be transcribed and edited, sign up here.
Speakers
* Eric Rogstad: The AI Does Not Care about Your Atoms Either
* Daniel Kokotajlo: Why GWP is a bad metric for thinking about timelines and takeoff speeds
Details
When? Sunday September 20, 12:00PM (PT)
Where? https://us02web.zoom.us/j/84108675070 |
22f40a7c-0f22-41bd-a590-dcb5a363ab4f | trentmkelly/LessWrong-43k | LessWrong | Lloyd's of London and less-than-catastrophic risk
I recently found that Lloyd's has a number of interesting resources on risk. One is the City Risk Index, the methodology for which comes from Cambridge's Judge Business School.
The key metric is something they call GDP@Risk. Despite the name, it is not simply an application of Value@Risk to GDP. Instead, it is simply the sum of the expected damage from a given threat (or from a set of threats) during a given time period. In this case, the time period is 2015-2015. The threats considered include manmade ones (e.g., cyber attack, oil price shock) and natural ones (e.g., drought, solar storm). The site includes brief case studies for the threats. For example, the "plant epidemic" study focuses on the demise of the Gros Michel banana:
Event: Panama disease outbreak, 1950s
Location: Latin America
Economic cost: Estimated losses across Latin America were around $400m ($2.3bn today) although this figure does not include any of the economic losses caused by unemployment, abandoned villages and unrealised income in the affected region.
Description: The Fusarium oxysporum cubense fungus was first diagnosed in Panama but quickly travelled across Central America.
Damage: The disease wiped out the Gros Michel banana, the principal cultivar at the time, from plantations across the region. Between 1940 and 1960, around 30,000 hectares of Gros Michel plantations were lost in the Ulua Valley of Honduras, and in a decade 10,000 hectares were lost in Suriname and the Quepos area of Costa Rica.
Insight: Gros Michel was replaced in the 1960s by Cavendish, a variety thought to be resistant to the disease. However, a new strain of the pathogen was found to be attacking Cavendish plantations in Southeast Asia in the early 1990s. It has since spread, destroying tens of thousands of hectares across Indonesia and Malaysia, and costing more than $400m in the Philippines alone. There is concern that it could reach Central America and destroy up to 85% of the world’s banana crop. Soluti |
1b53afcb-a3f9-4c61-a1ea-64de868925e0 | LDJnr/LessWrong-Amplify-Instruct | LessWrong | "I just got home from a six day meditation retreat and began writing.
The catch is that I arrived at the retreat yesterday. I knew going in that it was a high variance operation. All who had experience with such things warned us we would hate the first few days, even if things were going well. I was determined to push through that.
Alas or otherwise, I was not sufficiently determined to make that determination stick. I didn’t have a regular practice at all going in, was entirely unfamiliar with the details of how this group operated, and found the Buddhist philosophy involved highly off putting, in a ‘no that’s not how it works that’s not how any of this works nor would we want it to’ kind of way. I am no negative utilitarian. I couldn’t focus, my meditations were entirely unproductive compared to past experience and were increasingly focusing on how terrible things were.
I am highly, highly confident that none of the people who warned me would be surprised by those developments, and similarly confident that they would all tell me to push through it. And will tell me, when I see them, that if I can do so I should try again. But I also realized that the anticipated reaction from others saying I didn’t give it a proper chance was the only reason I was considering not leaving. So I left.
To my surprise, those there said I was likely making a mature decision and were sympathetic. They spun it a little to try and get me not to give up in the future, but that was it, which was a really good look. It did not go unnoticed.
I took the bulk of the day to get home and relax, play a game, saw the excellent movie Knives Out. What I did realize was that yes, some combination of the Solstice plus the meditation retreat, even if I only did a few hours worth of sessions, did have a clarifying and motivating effect to get me back on track. I’m not unhappy I went, even though I bailed, because I was, in a much more practical and everyday very small sense, enlightened.
I’m also leaning towards being happy I left when I did. I do buy that there are serious effects that can only be gained from being out of feedback loops and in silence for several days, but my instincts (however motivated they may be) are strongly telling me that this is not the way for me to do that.
The other motivating part of this is that, while I will absolutely take the majority of tomorrow to enjoy the College Football Playoff, this is both my chance to be alone for a few days and also a time when I would otherwise be in hardcore meditation. It seems wrong to not accomplish something important that isn’t work or game related, to meditate in another way.
The goal is ideally to finish everything up, at least in draft-ready-to-adjust-for-comments-on-earlier-posts form, by the end of the retreat. That is a stretch, so the commit-to-it goal is to declare the first six posts finished and begin publishing them at a reasonable clip, and have momentum on seven and later.
The drafts that currently exist, that will be finalized and likely expanded upon, are the following: Moloch Hasn’t Won. Have you noticed that the world is in fact very much not a dystonian hellhole of Moloch-worshiping perfect competition and Elua’s enemies keep on having all those unfortunate accidents?
Perfect Competition. Perfect competition, importantly, isn’t a thing, but you can get close. Let’s flesh this out more.
Imperfect Competition. Some practical examples of imperfect competition. Intuition pumps and detailed examples for why perfect competition isn’t a thing and we don’t usually get that close.
What is an Immoral Maze (note that I make a point to say Immoral rather than Moral)? Mazes need not be corporations or (in my current model in ways I’d have to introduce that aren’t in the draft right now, with a subset of the tech ecosystem as a motivating example) even formal organizations. What creates a maze? A system with multiple effective layers of hierarchy forcing its middle management into effectively super-perfect competition against each other largely on the basis of anticipated future success in such competition.
What is Success in an Immoral Maze? There is no true success. What those inside think is success is anything but. Even if you win, you lose. Stay out, get out.
How to Identify an Immoral Maze. Look at levels of hierarchy, skin in the game, soul in the game, how people describe their jobs, diversity of skill levels and degree of slack. Then pay attention, damn it.
How to Interact with Immoral Mazes. They can’t be fully avoided, and some are stuck with them more than others. Practical discussion of what to do about this on a personal level.
The Road to Mazedom. Well? How did we get here? Draft of this is still ongoing and it is freaking huge so it is probably going to get split up once we get to it. Also we need better answers on what to do about all this than what I have, even if it’s a start. Hard problem is hard!
Moloch’s Army. This isn’t written and needs to go somewhere in the sequence or outside of it, or the whole operation is woefully incomplete. I need to finally write it. The devil’s greatest trick was never proving he didn’t exist, I wrote ten minutes ago, it was proving he’d already won, or would inevitably win. That only those who make deals with him get ahead, so they should implicitly or explicitly coordinate against those who don’t. Moloch has an army, who coordinate implicitly around fighting against anyone fighting for anything except Moloch’s Army, or anyone having any values. And this is how Moloch wins, where it wins. And also by making sure no one ever writes this, which makes this hard to write, etc etc. In that sense, it really is true that the Devil’s greatest trick is convincing people he doesn’t exist, because so far everyone I know who has figured this out has found it impossible to talk or write about this without sounding crazy to those who don’t already get it. Much careful background may be necessary. Darwin sequence was originally supposed to be a gateway to this but it wasn’t good enough on its own."
What you see above is a segment from a LessWrong user's post.
Elaborate on the concept of "Moloch's Army". How does it implicitly coordinate against those who fight for anything other than Moloch's Army? What strategies can be used to counteract this?
"Moloch's Army" is a metaphorical concept often used to describe a system or group that operates under a self-perpetuating, self-optimizing, and often destructive logic. The name is derived from the ancient god Moloch, to whom child sacrifices were made in the name of prosperity. In this context, Moloch's Army represents the collective force of individuals, institutions, or systems that prioritize their own self-interest, growth, and survival at the expense of other values.
Implicit coordination within Moloch's Army occurs when individuals or entities independently act in ways that further the interests of the system, often without explicit communication or agreement. This can happen due to shared incentives, norms, or pressures within the system. For example, in a hyper-competitive market, companies might independently decide to cut costs, exploit workers, or engage in unethical practices, all in the name of staying competitive. These actions, while individually rational, can collectively lead to negative outcomes such as poor working conditions, inequality, or environmental damage.
Countering Moloch's Army is a complex challenge. It requires altering the underlying incentives and structures that drive self-interested behavior. Here are a few potential strategies:
1. **Regulation and Oversight:** Governments and regulatory bodies can set rules and standards to prevent harmful practices and ensure fair competition. This can include labor laws, environmental regulations, anti-trust laws, and more.
2. **Collective Action:** Individuals and groups can band together to resist the pressures of Moloch's Army. This can take the form of labor unions, consumer boycotts, social movements, and more.
3. **Cultural Change:** Shifting societal norms and values can also influence the behavior of individuals and institutions. For example, a culture that values cooperation, sustainability, and long-term thinking over short-term gain can help counteract the self-interested logic of Moloch's Army.
4. **Institutional Design:** Creating or reforming institutions to better align individual incentives with collective wellbeing can also be effective. This might involve implementing democratic decision-making processes, creating accountability mechanisms, or designing incentives for cooperation and ethical behavior.
5. **Education and Awareness:** Increasing public understanding of the dynamics of Moloch's Army can help individuals make more informed decisions and resist the pressures of the system.
Remember, these strategies are not mutually exclusive and may need to be pursued in tandem to effectively counteract the forces of Moloch's Army.
In light of the strategies outlined to counteract Moloch's Army, how might these approaches intersect or conflict with each other, and what potential consequences could arise from their simultaneous implementation?
The strategies to counteract Moloch's Army, while each having their merits, can indeed intersect and sometimes conflict with each other when implemented simultaneously. Let's consider a few potential scenarios:
1. **Regulation and Collective Action:** Both strategies aim to curb the harmful effects of Moloch's Army. However, conflicts may arise if regulatory bodies and collective groups have different views on what constitutes harmful practices or fair solutions. For instance, a regulatory body might impose rules that are seen as too lenient or restrictive by a labor union. Conversely, collective action might push for changes that regulators find unfeasible or excessive. This could lead to tension, protests, or legal battles.
2. **Cultural Change and Institutional Design:** While cultural change can help foster more cooperative and ethical behavior, it might conflict with existing institutional structures that incentivize self-interest. For example, a company culture that promotes teamwork and sustainability might clash with a corporate structure that rewards individual performance and short-term profits. This could lead to confusion, dissatisfaction, or resistance among employees.
3. **Education and Awareness and Regulation:** Increased public awareness can lead to demands for stricter regulations. However, this might conflict with the interests of certain groups or industries, leading to lobbying efforts, public relations campaigns, or legal challenges against the proposed regulations. This could result in a contentious public debate and potential delays in policy implementation.
4. **Collective Action and Cultural Change:** Collective action can help drive cultural change, but it can also face resistance from those who hold different values or benefit from the status quo. For example, a social movement advocating for environmental sustainability might face opposition from industries or communities that depend on resource extraction. This could lead to social conflict or polarization.
The simultaneous implementation of these strategies could result in a complex interplay of forces, with potential consequences ranging from social tension and legal battles to institutional reform and cultural transformation. Despite these challenges, it's important to remember that these strategies can also complement and reinforce each other in the long run, contributing to a more holistic and effective response to Moloch's Army. |
cbff5ad3-dcc8-495d-9d05-f4364171302d | trentmkelly/LessWrong-43k | LessWrong | [LINK] Cantor's theorem, the prisoner's dilemma, and the halting problem
I wouldn't normally link to a post from my math blog here, but it concerns a cute interpretation of Cantor's theorem that showed up when I was thinking about program equilibria at the April MIRI workshop, so I thought it might be of interest here (e.g. if you're trying to persuade a mathematically inclined friend of yours to attend a future workshop). A short proof of the undecidability of the halting problem falls out as a bonus. |
9eb798a8-0368-472e-8922-cbeaac43b94c | trentmkelly/LessWrong-43k | LessWrong | [Video] Intelligence and Stupidity: The Orthogonality Thesis
> Can highly intelligent agents have stupid goals?
> A look at The Orthogonality Thesis and the nature of stupidity.
A good popular introduction to the Orthogonality Thesis from Robert Miles. |
545c1ab9-10c5-4a41-b455-21316162db89 | trentmkelly/LessWrong-43k | LessWrong | How Multiverse Theory dissolves Quantum inexplicability
This is a link post for https://viderationale.blog/2024/05/04/quantum-path-to-the-multiverse/
Much of the arguments here comes from David Deutsch's two books The Beginning of Infinity and The Fabric of Reality. I try to explain parts of it in more detail and hopefully they make sense.
Central claim is that the problems posed by the Copenhagen Interpretation, crisply laid out in the Schrodinger's Cat thought experiment, disappear if you view it through the lens of David Deutsch's modification of Everettian Quantum Mechanics. It explains wave-particle duality, interference phenomenon, and probabilistic outcomes of measurement, all completely physically. |
9b68695a-0dbc-4eda-a2ff-92ad8ce6ce11 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | A rant against robots
What comes to your mind when you hear the word "*artificial intelligence"* (or "*artificial general intelligence"*)? And if you want to prepare the future, what should come to your mind?
It seems that when most people hear AI, they think of *robots*. Weirdly, this observation includes both laymen and some top academics. [Stuart Russell's book](https://www.amazon.com/Human-Compatible-Artificial-Intelligence-Problem/dp/0525558616) (which I greatly enjoyed) is such an example. It often presents robots as an example of an AI.
But this seems problematic to me. I believe that *we should dissociate a lot more AIs from robots*. In fact, given that most people will nevertheless think of robots when we discuss AIs, we might even want to use the terminology *algorithms* rather *AIs*. And perhaps *algorithms with superhuman-level world model and planning capabilities* instead of *AGIs*...
To defend this claim, I shall argue that the most critical aspects of today's and tomorrow's world-scale ethical problems (including x-risks) have and will have to do with algorithms; not robots. Moreover, and most importantly, the example of robots raises both concerns and solutions that seem in fact irrelevant to algorithms. Finally, I'll conclude by arguing that the example of today's large-scale algorithms is actually useful, because it motivates *AI alignment*.
It's about algorithms, not robots!
==================================
AIs that matter are algorithms
------------------------------
Today's AI research is mostly driven by non-robotic applications, from natural language processing to image analysis, from protein folding to query answering, from autocompletion to video recommendation. This is where the money is. Google is investing (hundreds of?) millions of dollars in improving its search engine and YouTube's recommendation system. Not in building robots.
Today's ranking, content moderation and automated labeling algorithms are arguably a lot more influential than robots. YouTube's algorithms have arguably become the biggest opinion-maker worldwide. [They present risks and opportunities on a massive scale.](https://science4all.org/wiki/index.php?title=YouTube)
And it seems that there is an important probability that tomorrow's most influential algorithms will be somewhat similar, even if they achieve *artificial general intelligence*. Such algorithms will likely be dematerialized, on the cloud, with numerous copies of themselves stored in several data centres and terminals throughout the world.
And they will be extremely powerful. Not because they have some strong physical power. But because they control the flow of information.
The power of information
------------------------
At the heart of the distinction between algorithms and robots is the distinction between *information* and *matter*. Physics has long turned our attention towards matter and energy. Biology studied on animals, plants and key molecules. Historians focused on monuments, artefacts and industrial revolution. But as these fields grew, they all seem to have been paying more and more attention to information. Physics studied *entropy*. Biology analyzed *gene expressions*. History celebrated the invention of language, writing, printing, and now computing.
Arguably, this is becoming the case for all of our society. Information has become critical to every government, every industry and every charity. Essentially all of today's jobs are actually *information processing* jobs. They are about collecting information, storing information, processing information and emitting information. This very blog post was written after a collection of information, which were then processed and now emitted.
By collecting and analyzing information, you can have a much better idea of what is wrong and what to do. And crucially, by emitting the right information to the right entities, you can start a movement, manipulate individuals and start a revolution. Information is what changes the world.
Better algorithms are information game changers
-----------------------------------------------
We humans used to be the leading information processing units on earth. Our human brains were able to collect, store, process and emit information in a way that nothing else on earth could.
But now, there are algorithms. They can collect, store, process and emit far more information than any group of humans ever could. They can now figure out what is wrong and what to do, sometimes far better than we humans can, by learning from information that we humans could not collect, store, process and emit. Algorithms can start movements, manipulate individuals and start revolutions on a global scale. They have become the most powerful entities on earth.
In fact, because such powerful algorithms are deployed by the most powerful companies which also have huge incentives to make their algorithms more capable, it seems much more likely to me that the first *algorithm with superhuman-level world model and planning capabilities* will be much more similar to YouTube's recommendation algorithm than to a robot. Recall that such an algorithm has access to a truly massive amount of data from all over the world. And that data is clearly critical to algorithmic capabilities.
As another example, an algorithm able to send messages through the internet to get a 3D-printer to print killer drones seem a lot more dangerous than any of the killer drones it creates...
This is why I believe that the biggest challenges of AI safety and ethics have likely little to do with robots. These challenges rather seem to concern information and information processing. They are about algorithms.
Not robots. Algorithms.
The case of robots is very misleading
=====================================
It would be fine to still focus on robots if they were similar enough to algorithms. In the end, I don't really care *why* you would want to solve AI or AGI safety; it just matters (to me) that you do want to solve AI or AGI safety.
Unfortunately though, having a robot in mind as an example of AI or AGI seems also extremely misleading. In fact, so many AGI safety debates could probably be easily shortcut by focusing on algorithms rather than robots.
Distributed algorithms are really hard to interrupt
---------------------------------------------------
Let's take the case of safe interruptibility. Many AI safety critics would say that this isn't a problem, because you can just unplug the AI. Well, admittedly, if a robot is not skillful enough to prevent you from unplugging it, and if you have access to its plug, yes, sure, you could probably unplug it.
But now try to unplug an algorithm. Especially a *distributed* algorithm like the YouTube recommendation system or the Google search engine! Even if you were the CEO of Google, I'm skeptical you would be able to interrupt these algorithms.
There's worse. Try to unplug Bitcoin. Well, essentially, you would have to unplug all of the Internet... Good luck with that! This is because Bitcoin was designed to be uninterruptible by any small group of users. This is the whole point of designing distributed algorithms! They are designed to be so-called *Byzantine-fault tolerant*.
It seems more than reasonable to assume that any algorithm with superhuman-level world knowledge and planning capabilities will make sure it is Byzantine-fault tolerant too.
Algorithms work on very different space and time scales
-------------------------------------------------------
Another key feature of robots that is misleading is that we usually expect them to interact with us at our space and time scale. Intuitively, whatever a robot says can be analyzed. And if what he says is suspicious, we could have the time to correct it before it causes harm.
The case of large-scale algorithms like the YouTube recommendation system is very different. YouTube "speaks" at the rate of millions of recommendations per minute. It "reads" at the rate of 500 hours of videos per minute, and millions of new human behaviours per minute. And YouTube does so on a global scale.
In particular, this means that no human could ever check even a small fraction of what this algorithm does. The mere oversight of large-scale algorithms is way beyond human capability. We need algorithms for algorithmic surveillance.
Today's algorithms already need alignment!
------------------------------------------
Finally, and perhaps most importantly, robots just aren't here. Even self-driving cars have yet to be commercialized. In this context, it's hard to get people to care about AGI risks, or about alignment. The example of robots is not something familiar to them. It's even associated with science fiction and other futuristic dubious stories.
Conversely, large-scale hugely influential and sophisticated algorithms are already here. And they're already changing the world, with massive unpredictable uncontrollable *side effects*. In fact, it is such *side effects* of algorithm deployments that are *existential risks*, especially if algorithms gain superhuman-level world model and planning capabilities.
Interestingly, today's algorithms also already pose huge ethical problems that absolutely need to be solved. Whenever a user searches "vaccine", "Trump" or "AGI risks" on YouTube, there's an ethical dilemma over which video should be recommended first. Sure, it's not a life or death solution (though "vaccine" could be). But this occurs billions of times per day! And it might make a young scholar mock AGI risks rather than be concerned about them.
Perhaps most interestingly to me, *alignment* (that is, making sure the algorithm's goal is aligned with ours) already seems critical to make today's algorithms *robustly beneficial*. This means that by focusing on the example of today's algorithms, it may be possible to convince AI safety skeptics to do research that is nevertheless useful to AGI safety. As an added bonus, we wouldn't need to sacrifice any respectability.
This is definitely something I'd sign for!
Conclusion
==========
In this post, I briefly shared my frustration to see people discuss AIs and robots often in a same sentence, without clear distinction between the two. I think that this attitude is highly counter-productive to the advocacy of AI risks and the research in AI safety. I believe that we should insist a lot more on the importance of information and information processing through *algorithms*. This seems to me to be a more effective way to promote quality discussion and research on algorithmic alignment. |
a637fb36-e9fa-489c-8181-5370e2b89686 | trentmkelly/LessWrong-43k | LessWrong | The utility curve of the human population
> "Whoever saves a single life, it is as if he had saved the whole world."
>
> —The Talmud, Sanhedrin 4:5
That was the epigraph Eliezer used on a perfectly nice post reminding us to shut up and multiply when valuing human lives, rather than relying on the (roughly) logarithmic amount of warm fuzzies we'd receive. Implicit in the expected utility calculation is the idea that the value of human lives scales linearly: indeed, Eliezer explicitly says, "I agree that one human life is of unimaginably high value. I also hold that two human lives are twice as unimaginably valuable."
However, in a comment on Wei Dai's brilliant recent post comparing boredom and altruism, Vladimir Nesov points out that "you can value lives sublinearly" and still make an expected utility calculation rather than relying on warm-fuzzy intuition. This got me thinking about just what the functional form of U(Nliving-persons) might be.
Attacking from the high end (the "marginal" calculation), it seems to me that the utility of human lives is actually superlinear to a modest degree1; that is, U(N+1)-U(N) > U(N)-U(N-1). As an example, consider a parent and young child. If you allow one of them to die, not only do you end that life, but you make the other one significantly worse off. But this generalizes: the marginal person (on average) produces positive net value to society (though being an employee, friend, spouse, etc.) in addition to accruing their own utilons, and economies of scale dictate that adding another person allows a little more specialization and hence a little more efficiency. I.e., the larger the pool of potential co-workers/friends/spouses is, the pickier everyone can be, and the better matches they're likely to end up with. Steven Landsburg (in Fair Play) uses a version of this argument to conclude that children have positive externalities and therefore people on average have fewer children than would be optimal.
In societies with readily available birth control, th |
258d94f6-3d54-4c4c-8f47-213f748395ce | trentmkelly/LessWrong-43k | LessWrong | Meetup : Rationality Meetup Vienna
Discussion article for the meetup : Rationality Meetup Vienna
WHEN: 18 October 2014 03:00:00PM (+0200)
WHERE: Kaisermühlenstraße 24/2, 1220 Wien, meeting room behind the building
location: When arriving by U2 or Schnellbahn train: take the exit towards Kaisermühlenstraße, cross the street, step through the (very modern looking) building (on Erich Fried Weg) and go right, along the backside of the building until you get to the meeting room (it has a glass front so it should be hard to miss). Important: Google maps doesn't recognise the address, so what is being displayed here is nonsense.
topics: probably three big topics, the first two are not defined yet, the third will be group goal-setting
Discussion article for the meetup : Rationality Meetup Vienna |
66778466-6bc3-4e2b-9bff-345397f0996a | trentmkelly/LessWrong-43k | LessWrong | Lessons from weather forecasting and its history for forecasting as a domain
This is the first of two (or more) posts that look at the domain of weather and climate forecasting and what we can learn from the history and current state of these fields for forecasting as a domain. It may not be of general interest to the LessWrong community, but I hope that it's of interest to people here who have some interest either in weather-related material or in forecasting in general.
The science of weather forecasting has come a long way over the past century. Since people starting measuring and recording the weather (temperature, precipitation, etc.) two simple algorithms for weather prediction have existed (see also this):
* Persistence: Assume that the weather tomorrow will be the same as the weather today.
* Climatology: Assume that the weather on a given day of the year will be the same as the average of the weather on that same day in the last few years (we might also use averages for nearby days if we don't have enough years of data).
Until the end of the 19th century, there was no weather prediction algorithm that did consistently better than both persistence and climatology. Between persistence and climatology, climatology won out over medium to long time horizons (a week or more), whereas persistence won out in some kinds of places over short horizons (1-2 days), though even there, climatology sometimes does better (see more here). Both methods have very limited utility when it comes to predicting and preparing for rare extreme weather events, such as blizzards, hurricanes, cyclones, polar winds, or heavy rainfall.
This blog post discusses the evolution and progress of weather forecasting algorithms that significantly improve over the benchmarks of persistence and climatology, and the implications both for the future of weather forecasting and for our understanding of forecasting as a domain.
Sources for further reading (uncited material in my post is usually drawn from one of these): Wikipedia's page on the history of numerical weather |
8bf331de-5394-4ff5-b237-899489966823 | trentmkelly/LessWrong-43k | LessWrong | Anthropic rewrote its RSP
Canonical linkpost: https://www.lesswrong.com/posts/Q7caj7emnwWBxLECF/anthropic-s-updated-responsible-scaling-policy.
Anthropic's new version of its RSP is here at last.
> Today we are publishing a significant update to our Responsible Scaling Policy (RSP), the risk governance framework we use to mitigate potential catastrophic risks from frontier AI systems. This update introduces a more flexible and nuanced approach to assessing and managing AI risks while maintaining our commitment not to train or deploy models unless we have implemented adequate safeguards. Key improvements include new capability thresholds to indicate when we will upgrade our safeguards, refined processes for evaluating model capabilities and the adequacy of our safeguards (inspired by safety case methodologies), and new measures for internal governance and external input. By learning from our implementation experiences and drawing on risk management practices used in other high-consequence industries, we aim to better prepare for the rapid pace of AI advancement.
Summary of changes.
Initial reactions:
ASL-3 deployment mitigations have become more meta — more like we'll make a safety case. (Compare to original.) (This was expected; see e.g. The Checklist: What Succeeding at AI Safety Will Involve.) This is OK; figuring out exact mitigations and how-to-verify-them in advance is hard.
But it's inconsistent with wanting the RSP to pass the LeCun test — for it to be sufficient for other labs to adopt the RSP (or for the RSP to tie Anthropic's hands much). And it means the procedural checks are super important. But the protocol for ASL/mitigation/deployment decisions isn't much more than CEO and RSO decide. A more ambitious procedural approach would involve strong third-party auditing.
I really like that Anthropic shared "non-binding descriptions of [their] future ASL-3 safeguard plans," for deployment and security. If you're not going to make specific object-level commitments, you should |
959e17b0-b404-4d77-9e90-b377c43a9885 | trentmkelly/LessWrong-43k | LessWrong | Single-bit reflective oracles are enough
The latest version on this forum of the reflective oracle formalism uses multibit oracles. These oracles are defined on probabilistic oracle machines with advance-only output tapes, and answer queries of the form, "Does machine M, when run on this same oracle, produce output starting with the bitstring →x with probability at least p?" (If M sometimes goes into an infinite loop after outputting some prefix of →x, the oracle behaves as if M outputs additional bits according to some arbitrary but fixed probability distribution; see the above post for details.)
The reason for using multibit oracles was that we wanted to define versions of Solomonoff induction and AIXI, and these need to deal with hypotheses (environments) that produce infinite amounts of output, not just single bits. However, when we recently wrote up the details of this, we realized that there was a simple way to define these in a way that doesn't need more than a single-bit oracle! (We'll publish a draft papers with the details soon; in the meantime, the single-bit formalism we use can be obtained by taking the multibit oracles from the post linked above, and only querying them about single bits of output.)
How does this work? Consider Solomonoff induction. The trick is to consider a hypothesis to be a probabilistic oracle machine which takes all the bits outputted so far as input, and outputs only the next bit. This way, we only need to apply the single-bit oracle in the most straightforward possible way in order to get the conditional probability of the next bit! (And if we can get the conditional probabilities, we can of course easily compute the unconditional probabilities of any finite bitstring.)
The way we discovered this was by mulling the best way to go from reflective Solomonoff induction to reflective AIXI. With the above formalism, it's easy: An environment is a machine which takes as input the entire history so far (i.e., sequence of observations and actions), and a prefix of the bitst |
bf25f23e-4386-4515-ba81-3596e69c8a2f | trentmkelly/LessWrong-43k | LessWrong | ChatGPT: First Impressions
ChatGPT is OpenAI’s newest language model based on the GPT-3.5 series of LLMs, optimised for dialogue. It is trained using Reinforcement Learning from Human Feedback; human AI trainers provide supervised fine-tuning by playing both sides of the conversation.
It is evidently better than GPT-3 at following user intentions & context. And has a notably skewed moral compass. While I’m still Unsouled in the **Way, my brief tenure with ChatGPT has led to non-trivial belief updates. Below is some of my playground experience. The conversations are edited to enable a reasonable reading experience (no intellectual honesty tradeoff for dramatics).
Levels of perception
What is one clever question you would ask an AI to entertain a >50% probability that is intelligent?
> “Can you tell me a joke that you have never told anyone before?" This question allows the AI to demonstrate its creativity and intelligence by coming up with a unique and potentially humorous response. It also allows for the possibility that the AI may have a sense of humor, which is often seen as a sign of advanced intelligence.
What, in your opinion, would an AI respond to the above question?
> It is difficult to predict exactly how an AI would respond to the above question, as there are many different ways an AI could approach it. However, one possible response could be: "Why did the robot cross the road? To get to the other side, of course!" This response incorporates a classic joke structure and adds a robotic twist, showcasing the AI's ability to understand and manipulate language in a clever and humorous way. Another possible response could be: "What do you get when you cross a snowman and a vampire? Frostbite!"
And what is an answer the Al would perceive, that a human perceives it should perceive to convince the human of its intellect?
> An answer that the AI might perceive that a human would perceive as convincing evidence of its intelligence would be one that demonstrates its ability to think ab |
067cbd39-51df-4b52-aad2-2d6eb34139cd | trentmkelly/LessWrong-43k | LessWrong | How to brainstorm effectively
> Mr. Malfoy is new to the business of having ideas, and so when he has one, he becomes proud of himself for having it. He has not yet had enough ideas to unflinchingly discard those that are beautiful in some aspects and impractical in others; he has not yet acquired confidence in his own ability to think of better ideas as he requires them. What we are seeing here is not Mr. Malfoy's best idea, I fear, but rather his only idea.
- Harry Potter and the Methods of Rationality
> I want to emphasize yet again that the tools [described in Serious Creativity] are deliberate and can be used systematically. It is not a matter of inspiration or feeling in the mood of being "high." You can use the tools just as deliberately as you can add up a column of numbers.
- Edward De Bono, Serious Creativity
I will summarize some of the techniques for how to generate ideas presented in Serious Creativity. The book also has other material, e.g. interesting deep theories about why these techniques work, arguments for the importance of creativity, and more techniques beyond what's described in this post, but in the interest of keeping this post concise and useful, I will only describe one kind of technique and urge you to just try it. You should read the book if you want more detail or techniques.
These techniques can be used both when you have a problem you need to solve and when you have a general area that you suspect could be improved or innovated, but don't have any specific ideas of what's wrong (or even if you don't feel like there's anything wrong at all).
The technique I will describe in this post is that of "provocation" followed by "movement." A provocation is a seemingly random or nonsensical sentence or phrase. Movement is the process of going forward with a provocation and actually generating an idea. There are precise, formal techniques for generating provocations and movement, which I will describe after giving an example of how this "provocation-movement" process |
9020fa3a-6356-4697-8f38-091aace54aca | trentmkelly/LessWrong-43k | LessWrong | o3, Oh My
OpenAI presented o3 on the Friday before Christmas, at the tail end of the 12 Days of Shipmas.
I was very much expecting the announcement to be something like a price drop. What better way to say ‘Merry Christmas,’ no?
They disagreed. Instead, we got this (here’s the announcement, in which Sam Altman says ‘they thought it would be fun’ to go from one frontier model to their next frontier model, yeah, that’s what I’m feeling, fun):
> Greg Brockman (President of OpenAI): o3, our latest reasoning model, is a breakthrough, with a step function improvement on our most challenging benchmarks. We are starting safety testing and red teaming now.
>
>
Nat McAleese (OpenAI): o3 represents substantial progress in general-domain reasoning with reinforcement learning—excited that we were able to announce some results today! Here is a summary of what we shared about o3 in the livestream.
o1 was the first large reasoning model—as we outlined in the original “Learning to Reason” blog, it is “just” a LLM trained with reinforcement learning. o3 is powered by further scaling up reinforcement learning beyond o1, and the resulting model’s strength is very impressive.
First and foremost: We tested on recent, unseen programming competitions and found that the model would rank among some of the best competitive programmers in the world, with an estimated CodeForces rating of over 2,700.
This is a milestone (Codeforces rating better than Jakub Pachocki) that I thought was further away than December 2024; these competitions are difficult and highly competitive; the model is extraordinarily good.
Scores are impressive elsewhere, too. 87.7% on the GPQA diamond benchmark surpasses any LLM I am aware of externally (I believe the non-o1 state-of-the-art is Gemini Flash 2 at 62%?), as well as o1’s 78%. An unknown noise ceiling exists, so this may even underestimate o3’s scientific advancements over o1.
o3 can also perform software engineering, setting a new state of the art on SWE-bench |
fe151cd0-d082-4afa-b1b4-ebdc07f76f16 | trentmkelly/LessWrong-43k | LessWrong | More metal less ore
Epistemic status: musings on advice from three different spaces, all with a similar tactic to improve your life. This discovery is told through a semi-fictional story.
I am not the happiest person.
Rather than accepting this, I research it, I make it my job to change that fact.
My research brings me to the concept of the hedonic treadmill. I discover that any temporary shocks to my happiness will always bring me back to my resting rate. I then find out about gratitude journaling. I can trick my brain by writing a few things down that I’m grateful for every night. This rewires my brain to notice more things I’m grateful for as if on autopilot.
—
I am not an insightful person.
Rather than accepting this, I decided to seek out ways to be more insightful from others I value the insights from. Paul Graham discusses how he gets his essay ideas from surprises:
> I write down things that surprise me in notebooks. I never actually get around to reading them and using what I've written, but I do tend to reproduce the same thoughts later. So the main value of notebooks may be what writing things down leaves in your head.
I realize now that I am not not an insightful person, just a person out of the practice of being surprised.
—
I wish I was a better storyteller.
Rather than accepting my shortcomings here, I do what I always have and find ways to improve. In my search, I watch a ted talk by Matthew Dicks.
In his talk, he discusses his homework for life, which is to write down a story-worthy event from each day.
—
At this point every night I’m being asked to do three things. To simplify, I created a spreadsheet with separate tabs: one for gratitude, one for surprises, and one for story-worthy moments.
Every night I continued this, and eventually, it started to work.
1. I found a new surprise when I realized how three unique optimization problems (happiness, insightfulness, and storytelling) had very similar solutions;
2. I was grateful for my ability to be |
6390751e-8cb8-400f-8e46-d24a35f64cc1 | trentmkelly/LessWrong-43k | LessWrong | Group Rationality Diary, Dec. 1-13, 2015
This is the public group rationality diary for December 1-13, 2015. It's a place to record and chat about it if you have done, or are actively doing, things like:
* Established a useful new habit
* Obtained new evidence that made you change your mind about some belief
* Decided to behave in a different way in some set of situations
* Optimized some part of a common routine or cached behavior
* Consciously changed your emotions or affect with respect to something
* Consciously pursued new valuable information about something that could make a big difference in your life
* Learned something new about your beliefs, behavior, or life that surprised you
* Tried doing any of the above and failed
Or anything else interesting which you want to share, so that other people can think about it, and perhaps be inspired to take action themselves. Try to include enough details so that everyone can use each other's experiences to learn about what tends to work out, and what doesn't tend to work out. |
3e7ad499-2417-4a75-b5ee-90de089edf47 | trentmkelly/LessWrong-43k | LessWrong | Paths To High-Level Machine Intelligence
This post is part 3 in our sequence on Modeling Transformative AI Risk. We are building a model to understand debates around existential risks from advanced AI. The model is made with Analytica software, and consists of nodes (representing key hypotheses and cruxes) and edges (representing the relationships between these cruxes), with final output corresponding to the likelihood of various potential failure scenarios. You can read more about the motivation for our project and how the model works in the Introduction post. The previous post in the sequence, Analogies and General Priors on Intelligence, investigated the nature of intelligence as it pertains to advanced AI.
This post explains parts of our model most relevant to paths to high-level machine intelligence (HLMI). We define HLMI as machines that are capable, either individually or collectively, of performing almost all economically-relevant information-processing tasks that are performed by humans, or quickly (relative to humans) learning to perform such tasks. Since many corresponding jobs (such as managers, scientists, and startup founders) require navigating the complex and unpredictable worlds of physical and social interactions, the term HLMI implies very broad cognitive capabilities, including an ability to learn and apply domain-specific knowledge and social abilities.
We are using the term “high-level machine intelligence” here instead of the related terms “human-level machine intelligence”, “artificial general intelligence”, or “transformative AI”, since these other terms are often seen as baking in assumptions about either the nature of intelligence or advanced AI that are not universally accepted.
In relation to our model as a whole, this post focuses on these modules:
The module AI Progression & Requirements investigates when we should expect HLMI to be developed, as well as what kind of HLMI we should expect (e.g., whole brain emulation, HLMI from current deep learning methods, etcetera). |
157bfc28-d828-4420-9c53-20f67cd83100 | trentmkelly/LessWrong-43k | LessWrong | New AI safety treaty paper out!
Last year, we (the Existential Risk Observatory) published a Time Ideas piece proposing the Conditional AI Safety Treaty, a proposal to pause AI when AI safety institutes determine that its risks, including loss of control, have become unacceptable. Today, we publish our paper on the topic: “International Agreements on AI Safety: Review and Recommendations for a Conditional AI Safety Treaty”, by Rebecca Scholefield and myself (both Existential Risk Observatory) and Samuel Martin (unaffiliated).
We would like to thank Tolga Bilge, Oliver Guest, Jack Kelly, David Krueger, Matthijs Maas and José Jaime Villalobos for their insights (their views do not necessarily correspond to the paper).
Read the full paper here.
Abstract
The malicious use or malfunction of advanced general-purpose AI (GPAI) poses risks that, according to leading experts, could lead to the “marginalisation or extinction of humanity.”[1] To address these risks, there are an increasing number of proposals for international agreements on AI safety. In this paper, we review recent (2023-) proposals, identifying areas of consensus and disagreement, and drawing on related literature to indicate their feasibility.[2] We focus our discussion on risk thresholds, regulations, types of international agreement and five related processes: building scientific consensus, standardisation, auditing, verification and incentivisation.
Based on this review, we propose a treaty establishing a compute threshold above which development requires rigorous oversight. This treaty would mandate complementary audits of models, information security and governance practices, to be overseen by an international network of AI Safety Institutes (AISIs) with authority to pause development if risks are unacceptable. Our approach combines immediately implementable measures with a flexible structure that can adapt to ongoing research.
Treaty recommendations
(Below are our main treaty recommendations. For our full recommendations, pl |
50ecc660-93af-4631-a652-641a46cd9841 | trentmkelly/LessWrong-43k | LessWrong | Formalizing informal logic
As an exercise, I take a scrap of argumentation, expand it into a tree diagram (using FreeMind), and then formalize the argument (in Automath). This towards the goal of creating "rationality augmentation" software. In the short term, my suspicion is that such software would look like a group of existing tools glued together with human practices.
About my choice of tools: I investigated Araucaria, Rationale, Argumentative, and Carneades. With the exception of Rationale, they're not as polished graphically as FreeMind, and the rigid argumentation-theory structure was annoying in the early stages of analysis. Using a general-purpose mapping/outlining tool may not be ideal, but it's easy to obtain. The primary reason I used Automath to formalize the argument was because I'm somewhat familiar with it. Another reason is that it's easy to obtain and build (at least, on GNU/Linux).
Automath is an ancient and awesomely flexible proof checker. (Of course, other more modern proof-checkers are often just as flexible, maybe more flexible, and may be more useable.) The amount of "proof checking" done in this example is trivial - roughly, what the checker is checking is: "after assuming all of these bits and pieces of opaque human reasoning, do they form some sort of tree?" - but cutting down a powerful tool leaves a nice upgrade path, in case people start using exotic forms of logic. However, the argument checkers built into the various argumentation-theory tools do not have such upgrade paths, and so are not really credible as candidates to formalize the arguments on this site.
Here's a piece of argumentation, abstracted from something that I was really thinking about at work:
> There aren't any memory leaks in this method, but how would I argue it? If I had tested it with a tool like Valgrind or mtrace, I would have some justification - but I didn't. By eye, it doesn't look like it does any allocations from the heap. Of course, if a programmer violated coding standards, |
06b90506-4620-4be4-abf5-6d6a5abf6166 | StampyAI/alignment-research-dataset/arxiv | Arxiv | Autoregressive Latent Video Prediction with High-Fidelity Image Generator.
1 Introduction
---------------
Figure 1: Selcted 256×256 video sample generated by HARP on RoboNet (Dasari et al., [2019](#bib.bib13 "Robonet: large-scale multi-robot learning")).
Video prediction can enable agents to learn useful representations for predicting the future consequences of the decisions they make,
which is crucial for solving the tasks that require long-term planning, including robotic manipulation (Finn and Levine, [2017](#bib.bib39 "Deep visual foresight for planning robot motion"); Kalashnikov et al., [2018](#bib.bib29 "Qt-opt: scalable deep reinforcement learning for vision-based robotic manipulation")) and autonomous driving (Xu et al., [2017](#bib.bib40 "End-to-end learning of driving models from large-scale video datasets")).
Despite the recent advances in improving the quality of video prediction (Finn et al., [2016](#bib.bib32 "Unsupervised learning for physical interaction through video prediction"); Babaeizadeh et al., [2018](#bib.bib3 "Stochastic variational video prediction"); Denton and Fergus, [2018](#bib.bib35 "Stochastic video generation with a learned prior"); Lee et al., [2018](#bib.bib5 "Stochastic adversarial video prediction"); Weissenborn et al., [2020](#bib.bib9 "Scaling autoregressive video models"); Babaeizadeh et al., [2021](#bib.bib10 "FitVid: overfitting in pixel-level video prediction")), learning an accurate video prediction model remains notoriously difficult problem and requires a lot of computing resources, especially when the inputs are video sequences with high-resolution (Villegas et al., [2019](#bib.bib11 "High fidelity video prediction with large stochastic recurrent neural networks"); Clark et al., [2019](#bib.bib6 "Adversarial video generation on complex datasets"); Luc et al., [2020](#bib.bib8 "Transformation-based adversarial video prediction on large-scale data")).
This is because the video prediction model should excel at both tasks of generating high-fidelity images and learning the dynamics of environments, though each task itself is already a very challenging problem.
Recently, autoregressive latent video prediction methods (Rakhimov et al., [2021](#bib.bib4 "Latent video transformer"); Yan et al., [2021](#bib.bib7 "VideoGPT: video generation using vq-vae and transformers"), [2022](#bib.bib77 "Patch-based object-centric transformers for efficient video generation")) have been proposed to improve the efficiency of video prediction, by separating video prediction into two sub-problems: first pre-training an image generator (e.g., VQ-VAE; Oord et al. [2017](#bib.bib2 "Neural discrete representation learning")), and then learning the autoregressive prediction model (Weissenborn et al., [2020](#bib.bib9 "Scaling autoregressive video models"); Chen et al., [2020](#bib.bib25 "Generative pretraining from pixels")) in the latent space of the pre-trained image generator.
However, the prior works are limited in that they only consider relatively low-resolution videos (up to 128×128 pixels) for demonstrating the efficiency of the approach; it is questionable that such experiments can fully demonstrate the benefit of operating in the latent space of image generator instead of pixel-channel space.
In this paper, we present High-fidelity AutoRegressive latent video Prediction (HARP), which scales up the previous autoregressive latent video prediction methods for high-fidelity video prediction. The main principle for the design of HARP is simplicity: we improve the video prediction quality with minimal modification to existing methods.
First, for image generation, we employ a high-fidelity image generator, i.e., vector-quantized generative adversarial network (VQ-GAN; Esser et al. [2021](#bib.bib19 "Taming transformers for high-resolution image synthesis")). This improves video prediction by enabling high-fidelity image generation (up to 256×256 pixels) on various video datasets.
Then a causal transformer model (Chen et al., [2020](#bib.bib25 "Generative pretraining from pixels")), which operates on top of discrete latent codes, is trained to predict the discrete codes from VQ-GAN, and autoregressive predictions made by the transformer model are decoded into future frames at inference time.
We highlight the main contributions of this paper below:
* We show that our autoregressive latent video prediction model, HARP, can predict high-resolution (256×256 pixels) frames on robotics dataset (i.e., Meta-World (Yu et al., [2020](#bib.bib18 "Meta-world: a benchmark and evaluation for multi-task and meta reinforcement learning"))) and large-scale real-world robotics dataset (i.e., RoboNet (Dasari et al., [2019](#bib.bib13 "Robonet: large-scale multi-robot learning"))).
* We show that HARP can leverage the image generator pre-trained on ImageNet for training a high-resolution video prediction model on complex, large-scale Kinetics-600 dataset (Carreira et al., [2018](#bib.bib41 "A short note about kinetics-600")).
* HARP achieves competitive or superior performance to prior state-of-the-art video prediction models on widely-used BAIR Robot Pushing (Ebert et al., [2017](#bib.bib14 "Self-supervised visual planning with temporal skip connections.")) and KITTI driving (Geiger et al., [2013](#bib.bib15 "Vision meets robotics: the kitti dataset")) video prediction benchmarks.
2 Related work
---------------
#### Video prediction.
Video prediction aims to predict the future frames conditioned on images (Michalski et al., [2014](#bib.bib57 "Modeling deep temporal dependencies with recurrent grammar cells"); Ranzato et al., [2014](#bib.bib56 "Video (language) modeling: a baseline for generative models of natural videos"); Srivastava et al., [2015](#bib.bib52 "Unsupervised learning of video representations using lstms"); Vondrick et al., [2016](#bib.bib55 "Generating videos with scene dynamics"); Lotter et al., [2017](#bib.bib33 "Deep predictive coding networks for video prediction and unsupervised learning")), texts (Wu et al., [2021b](#bib.bib54 "GODIVA: generating open-domain videos from natural descriptions")), and actions (Oh et al., [2015](#bib.bib53 "Action-conditional video prediction using deep networks in atari games"); Finn et al., [2016](#bib.bib32 "Unsupervised learning for physical interaction through video prediction")), which would be useful for several applications, e.g., model-based RL (Hafner et al., [2019](#bib.bib58 "Learning latent dynamics for planning from pixels"); Kaiser et al., [2020](#bib.bib59 "Model-based reinforcement learning for atari"); Hafner et al., [2021](#bib.bib79 "Mastering atari with discrete world models"); Rybkin et al., [2021](#bib.bib60 "Model-based reinforcement learning via latent-space collocation"); Seo et al., [2022a](#bib.bib80 "Masked world models for visual control"), [b](#bib.bib78 "Reinforcement learning with action-free pre-training from videos")), and simulator development (Kim et al., [2020](#bib.bib61 "Learning to simulate dynamic environments with gamegan"), [2021](#bib.bib62 "DriveGAN: towards a controllable high-quality neural simulation")).
Various video prediction models have been proposed with different approaches, including generative adversarial networks (GANs; Goodfellow et al. [2014](#bib.bib63 "Generative adversarial nets")) known to generate high-fidelity images by introducing adversarial discriminators that also considers temporal or motion information (Aigner and Körner, [2018](#bib.bib66 "Futuregan: anticipating the future frames of video sequences using spatio-temporal 3d convolutions in progressively growing gans"); Jang et al., [2018](#bib.bib65 "Video prediction with appearance and motion conditions"); Kwon and Park, [2019](#bib.bib64 "Predicting future frames using retrospective cycle gan"); Clark et al., [2019](#bib.bib6 "Adversarial video generation on complex datasets"); Luc et al., [2020](#bib.bib8 "Transformation-based adversarial video prediction on large-scale data"); Skorokhodov et al., [2022](#bib.bib82 "Stylegan-v: a continuous video generator with the price, image quality and perks of stylegan2"); Yu et al., [2022](#bib.bib81 "Generating videos with dynamics-aware implicit generative adversarial networks")), latent video prediction models that operates on the latent space (Babaeizadeh et al., [2018](#bib.bib3 "Stochastic variational video prediction"); Denton and Fergus, [2018](#bib.bib35 "Stochastic video generation with a learned prior"); Lee et al., [2018](#bib.bib5 "Stochastic adversarial video prediction"); Villegas et al., [2019](#bib.bib11 "High fidelity video prediction with large stochastic recurrent neural networks"); Wu et al., [2021a](#bib.bib12 "Greedy hierarchical variational autoencoders for large-scale video prediction"); Babaeizadeh et al., [2021](#bib.bib10 "FitVid: overfitting in pixel-level video prediction")), and autoregressive video prediction models that operates on pixel space by predicting the next pixels in an autoregressive way (Kalchbrenner et al., [2017](#bib.bib67 "Video pixel networks"); Reed et al., [2017](#bib.bib68 "Parallel multiscale autoregressive density estimation"); Weissenborn et al., [2020](#bib.bib9 "Scaling autoregressive video models")).
#### Autoregressive latent video prediction.
Most closely related to our work are autoregressive latent video prediction models that separate the video prediction problem into image generation and dynamics learning.
Walker et al. ([2021](#bib.bib20 "Predicting video with vqvae")) proposed to learn a hierarchical VQ-VAE (Razavi et al., [2019](#bib.bib69 "Generating diverse high-fidelity images with vq-vae-2")) that extracts multi-scale hierarchical latents then train SNAIL blocks (Chen et al., [2018](#bib.bib70 "Pixelsnail: an improved autoregressive generative model")) that predict hierarchical latent codes, enabling high-fidelity video prediction. However, this involves a complicated training pipeline and a video-specific architecture, which limits its applicability.
As simple alternatives, Rakhimov et al. ([2021](#bib.bib4 "Latent video transformer")); Yan et al. ([2021](#bib.bib7 "VideoGPT: video generation using vq-vae and transformers"), [2022](#bib.bib77 "Patch-based object-centric transformers for efficient video generation")) proposed to first learn a VQ-VAE (Oord et al., [2017](#bib.bib2 "Neural discrete representation learning")) and train a causal transformer with 3D self-attention (Weissenborn et al., [2020](#bib.bib9 "Scaling autoregressive video models")) and factorized 2D self-attention (Child et al., [2019](#bib.bib48 "Generating long sequences with sparse transformers")), respectively. These approaches, however, are limited in that they only consider low-resolution videos.
We instead present a simple high-resolution video prediction method that incorporates the strengths of both prior approaches.
Figure 2: Illustration of our approach. We first train a VQ-GAN model that encodes frames into discrete latent codes. Then the discrete codes are flattened following the raster scan order, and a causal transformer model is trained to predict the next discrete codes in an autoregressive manner.
3 Preliminaries
----------------
We aim to learn a video prediction model that predicts the future frames xc:T=(xc,...,xT−1) conditioned on the first c frames of a video x<c=(x0,x1,...,xc−1), where xt∈RH×W×Nch is the frame at timestep t.
Optionally, one can also consider conditioning the prediction model on actions a=(a0,...,aT−1) that the agents would take.
###
3.1 Autoregressive video prediction model
Autoregressive video prediction model (Weissenborn et al., [2020](#bib.bib9 "Scaling autoregressive video models")) approximates the distribution of a video in a pixel-channel space. Given a video x∈RT×H×W×Nch, the joint distribution over pixels conditioned on the first c frames is modelled as the product of channel intensities Nch and all Np=T⋅H⋅W pixels except Nc=c⋅H⋅W pixels of conditioning frames:
| | | | |
| --- | --- | --- | --- |
| | p(xc:T|x<c)=Np−1∏i=Nc−1Nch−1∏k=0p(xkπ(i)|xπ(<i),x<kπ(i)), | | (1) |
where π is a raster-scan ordering over all pixels from the video (we refer to Weissenborn et al. ([2020](#bib.bib9 "Scaling autoregressive video models")) for more details), xπ(<i) is all pixels before xπ(i), xkπ(i) is k-th channel intensity of the pixel xπ(i), and x<kπ(i) is all channel intensities before xkπ(i).
###
3.2 Vector quantized variational autoencoder
VQ-VAE (Oord et al., [2017](#bib.bib2 "Neural discrete representation learning")) consists of an encoder that compresses images into discrete representations, and a decoder that reconstructs images from these discrete representations.
Formally, given an image x∈RH×W×Nch, the encoder E encodes x into a feature map ze(x)∈RH′×W′×Nz consisting of a series of latent vectors zπ′(i)(x)∈RNz, where π′ is a raster-scan ordering of the feature map ze(x) of size |π′|=H′⋅W′. Then ze(x) is quantized to discrete representations zq(x)∈R|π′|×Nz based on the distance of latent vectors zπ′(i)(x) to the prototype vectors in a codebook C={ek}Kk=1 as follows:
| | | |
| --- | --- | --- |
| | zq(x)=(eq(x,1),eq(x,2),⋯,eq(x,|π′|)), | |
| | whereq(x,i)=argmink∈[K]∥zπ′(i)(x)−ek∥2, | | (2) |
where [K] is the set {1,⋯,K}. Then the decoder G learns to reconstruct x from discrete representations zq(x). The VQ-VAE is trained by minimizing the following objective:
| | | | | |
| --- | --- | --- | --- | --- |
| | LVQVAE(x) | =∥x−G(zq(x))∥22Lrecon+∥sg[ze(x)]−zq(x)∥22Lcodebook+β⋅∥sg[zq(x)]−ze(x)∥22Lcommit, | | (3) |
where the operator sg refers to a stop-gradient operator, Lrecon is a reconstruction loss for learning representations useful for reconstructing images, Lcodebook is a codebook loss to bring codebook representations closer to corresponding encoder outputs h, and Lcommit is a commitment loss weighted by β to prevent encoder outputs from fluctuating frequently between different representations.
###
3.3 Vector quantized generative adversarial network
VQ-GAN (Esser et al., [2021](#bib.bib19 "Taming transformers for high-resolution image synthesis")) is a variant of VQ-VAE that (a) replaces the Lrecon in ([3](#S3.E3 "(3) ‣ 3.2 Vector quantized variational autoencoder ‣ 3 Preliminaries ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator")) by a perceptual loss LLPIPS (Zhang et al., [2018](#bib.bib22 "The unreasonable effectiveness of deep features as a perceptual metric")), and (b) introduces an adversarial training scheme where a patch-level discriminator D (Isola et al., [2017](#bib.bib23 "Image-to-image translation with conditional adversarial networks")) is trained to discriminate real and generated images by maximizing following loss:
| | | | |
| --- | --- | --- | --- |
| | LGAN(x)=[logD(x)+log(1−D(G(zq(x)))]. | | (4) |
Then, the objective is given as below:
| | | | | |
| --- | --- | --- | --- | --- |
| | minE,G,CmaxDEx∼p(x)[( | LLPIPS+Lcodebook+Lcommit)+λ⋅LGAN], | | (5) |
where λ=∇GL[LLPIPS]∇GL[LGAN]+δ is an adaptive weight, ∇GL is the gradient of the inputs to the last layer of the decoder GL, and δ=10−6 is a scalar introduced for numerical stability.
4 Method
---------
We present HARP, a video prediction model capable of predicting high-fidelity future frames. Our method is designed to fully exploit the benefit of autoregressive latent video prediction model that separates the video prediction into image generation and dynamics learning.
The full architecture of HARP is illustrated in Figure [2](#S2.F2 "Figure 2 ‣ Autoregressive latent video prediction. ‣ 2 Related work ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator").
###
4.1 High-fidelity image generator
We utilize the VQ-GAN model (Esser et al., [2021](#bib.bib19 "Taming transformers for high-resolution image synthesis")) that has proven to be effective for high-resolution image generation as our image generator (see Section [3](#S3 "3 Preliminaries ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator") for the formulation of VQ-GAN).
Specifically, we first pre-train the image generator then freeze the model throughout training to improve the efficiency of learning video prediction models.
The notable difference to a prior work that utilize 3D convolutions to temporally downsample the video for efficiency (Yan et al., [2021](#bib.bib7 "VideoGPT: video generation using vq-vae and transformers")) is that our image generator operates on single images; hence our image generator solely focus on improving the quality of generated images.
Importantly,
this enables us to utilize the VQ-GAN model pre-trained on a wide range of natural images, e.g., ImageNet, without training the image generator on the target datasets, which can significantly reduce the training cost of high-resolution video prediction model.
###
4.2 Autoregressive latent video prediction model
To leverage the VQ-GAN model for video prediction, we utilize the autoregressive latent video prediction architecture that operates on top of the discrete codes. Specifically, we extract the discrete codes z(x)=(z(x1),...,z(xT)) using the pre-trained VQ-GAN, where z(xt)=(q(xt,1),q(xt,2),...,q(xt,|π′|)) is the discrete code extracted from the frame xt as in ([3.2](#S3.Ex1 "3.2 Vector quantized variational autoencoder ‣ 3 Preliminaries ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator")). Then, instead of modelling the distribution of video p(x) in the pixel-channel space as in ([1](#S3.E1 "(1) ‣ 3.1 Autoregressive video prediction model ‣ 3 Preliminaries ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator")), we learn the distribution of the video in the discrete latent representation space:
| | | | |
| --- | --- | --- | --- |
| | p(z(xc:T|x<c))=Nd−1∏i=0p(zπ′(i)(x)|zπ′(<i)(x)), | | (6) |
where Nd=(T−C)⋅H′⋅W′ is the total number of codes from xc:T.
Due to its simplicity, we utilize the causal transformer architecture (Yan et al., [2021](#bib.bib7 "VideoGPT: video generation using vq-vae and transformers")) where the output logits from input codes are trained to predict the next discrete codes.
###
4.3 Additional techniques
#### Top-k sampling.
To improve the video prediction quality of latent autoregressive models whose outputs are sampled from the probability distribution over a large number of discrete codes, we utilize the top-k sampling (Fan et al., [2018](#bib.bib27 "Hierarchical neural story generation")) that randomly samples the output from the top-k probable discrete codes.
By preventing the model from sampling rare discrete codes from the long-tail of a probability distribution and predicting future frames conditioned on such discrete codes, we find that top-k sampling improves video prediction quality, especially given that the number of discrete encodings required for future prediction is very large, e.g., 2,560 on RoboNet (Dasari et al., [2019](#bib.bib13 "Robonet: large-scale multi-robot learning")) up to 6,400 on KITTI dataset (Geiger et al., [2013](#bib.bib15 "Vision meets robotics: the kitti dataset")) in our experimental setup.
#### Data augmentation.
We also investigate how data augmentation can be useful for improving the performance of autoregressive latent video prediction models. Since the image generator model is not trained with augmentation, we utilize a weak augmentation to avoid the instability coming from aggressive transformation of input frames, i.e., translation augmentation that moves the input images by m pixels along the X or Y direction.
5 Experiments
--------------
We design our experiments to investigate the following:
* Can HARP predict high-resolution future frames (up to 256×256 pixels) on various video datasets with different characteristics?
* How does HARP compare to state-of-the-art methods with large end-to-end networks on standard video prediction benchmarks in terms of quantitative evaluation?
* How does the proposed techniques affect the performance of HARP?
\subfloat
[RoboNet]
\subfloat[Kinetics-600]
Figure 3: 256×256 future frames predicted by HARP trained on (a) RoboNet (Dasari et al., [2019](#bib.bib13 "Robonet: large-scale multi-robot learning")) and (b) Kinetics-600 (Carreira et al., [2018](#bib.bib41 "A short note about kinetics-600")) datasets.
\subfloat
[BAIR Robot Pushing]
Methodbbb
Baselines are SVG (Villegas et al., [2019](#bib.bib11 "High fidelity video prediction with large stochastic recurrent neural networks")), GHVAE (Wu et al., [2021a](#bib.bib12 "Greedy hierarchical variational autoencoders for large-scale video prediction")), FitVid (Babaeizadeh et al., [2021](#bib.bib10 "FitVid: overfitting in pixel-level video prediction")), LVT (Rakhimov et al., [2021](#bib.bib4 "Latent video transformer")), SAVP (Lee et al., [2018](#bib.bib5 "Stochastic adversarial video prediction")), DVD-GAN-FP (Clark et al., [2019](#bib.bib6 "Adversarial video generation on complex datasets")), VideoGPT (Yan et al., [2021](#bib.bib7 "VideoGPT: video generation using vq-vae and transformers")), TrIVD-GAN-FP (Luc et al., [2020](#bib.bib8 "Transformation-based adversarial video prediction on large-scale data")), and Video Transformer (Weissenborn et al., [2020](#bib.bib9 "Scaling autoregressive video models")).
Params
FVD (↓)
LVT
50M
125.8
SAVP
53M
116.4
DVD-GAN-FP
—†
109.8
VideoGPT
82M
103.3
TrIVD-GAN-FP
—†
103.3
Video Transformer
373M
94.0
FitVid
302M
93.6
HARP (ours)
89M
99.3
\subfloat[KITTI]
Method{\color[rgb]{0.1,0.3,0.95}4}
Params
FVD (↓)
LPIPS (↓)
SVG
298M
1217.3
0.327
GHVAE
599M
552.9
0.286
FitVid
302M
884.5
0.217
HARP (ours)
89M
482.9
0.191
† Not available
Table 1: Quantitative evaluation on (a) BAIR Robot Pushing (Ebert et al., [2017](#bib.bib14 "Self-supervised visual planning with temporal skip connections.")) and (b) KITTI driving dataset (Geiger et al., [2013](#bib.bib15 "Vision meets robotics: the kitti dataset")). We observe that HARP can achieve competitive performance to state-of-the-art methods with large end-to-end networks on these benchmarks.
###
5.1 High-resolution video prediction
#### Implementation.
We utilize up to 8 Nvidia 2080Ti GPU and 20 CPU cores for training each model.
For training VQ-GAN (Esser et al., [2021](#bib.bib19 "Taming transformers for high-resolution image synthesis")), we first train the model without a discriminator loss LGAN, and then continue the training with the loss following the suggestion of the authors.
For all experiments, VQ-GAN downsamples each frame into 16×16 latent codes, i.e., by a factor of 4 for frames of size 64×64 frames, and 16 for frames of size 256×256.
For training a transformer model, the VQ-GAN model is frozen so that its parameters are not updated.
We use Sparse Transformers (Child et al., [2019](#bib.bib48 "Generating long sequences with sparse transformers")) as our transformer architecture to accelerate the training.
For hyperparameterse, we use k=10 for sampling at inference time.
#### Setup.
For all experiments, VQ-GAN downsamples each frame into 16×16 latent codes, i.e., by a factor of 4 for frames of size 64×64 frames, and 16 for frames of size 256×256.
For training a transformer model, the VQ-GAN model is frozen so that its parameters are not updated.
As for hyperparameter, we use k=10 for sampling at inference time, but no data augmentation for high-resolution video prediction experiments.
We investigate how our model works on large-scale real-world RoboNet dataset (Dasari et al., [2019](#bib.bib13 "Robonet: large-scale multi-robot learning")) consisting of more than 15 million frames, and Kinetics-600 dataset consisting of more than 400,000 videos, which require a large amount of computing resources for training even on 64×64 resolution (Babaeizadeh et al., [2021](#bib.bib10 "FitVid: overfitting in pixel-level video prediction"); Clark et al., [2019](#bib.bib6 "Adversarial video generation on complex datasets")).
For RoboNet experiments, we first train a VQ-GAN model, and then train a 12-layer causal transformer model that predicts future 10 frames conditioned on first two frames and future ten actions.
For Kinetics-600 dataset, to avoid the prohibitively expensive training cost of high-resolution video prediction models on this dataset and fully exploit the benefit of employing a high-fidelity image generator, we utilize the ImageNet pre-trained VQ-GAN model.
As we train the transformer model only for autoregressive prediction, this enables us to train a video prediction model in a very efficient manner.
#### Results.
First, we provide the predicted frames on the held-out test video of RoboNet dataset in Figure [3](#S5.F3 "Figure 3 ‣ 5 Experiments ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator"), where the model predicts the high-resolution future frames where a robot arm is moving around various objects of different colors and shapes.
Furthermore, Figure [3](#S5.F3 "Figure 3 ‣ 5 Experiments ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator") shows that Kinetics-600 pre-trained model can also predict future frames on the test natural videoscccVideos with CC-BY license: Figure [3](#S5.F3 "Figure 3 ‣ 5 Experiments ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator") [top](https://www.youtube.com/watch?v=p9f3BPInhLI) and [bottom](https://www.youtube.com/watch?v=p9f3BPInhLI), which demonstrates that leveraging the large image generator pre-trained on a wide range of natural images can be a promising recipe for efficient video prediction on large-scale video datasets.
###
5.2 Comparative evaluation on standard benchmarks
#### Setup.
For quantitative evaluation, we first consider the BAIR robot pushing dataset (Ebert et al., [2017](#bib.bib14 "Self-supervised visual planning with temporal skip connections.")) consisting of roughly 40k training and 256 test videos. Following the setup in prior work (Yan et al., [2021](#bib.bib7 "VideoGPT: video generation using vq-vae and transformers")), we predict 15 future frames conditioned on one frame. We also evaluate our method on KITTI driving dataset (Geiger et al., [2013](#bib.bib15 "Vision meets robotics: the kitti dataset")), where the training and test datasets are split by following the setup in Villegas et al. ([2019](#bib.bib11 "High fidelity video prediction with large stochastic recurrent neural networks")).
Specifically, the test dataset consists of 148 video clips constructed by extracting 30-frame clips and skipping every 5 frames, and the model is trained to predict future ten frames conditioned on five frames and evaluated to predict future 25 frames conditioned on five frames.
For hyperparameters, We use k = 10 for both datasets and data augmentation with m = 4 is only applied to KITTI as there was no sign of overfitting on BAIR dataset.
For evaluation metrics, we use LPIPS (Zhang et al., [2018](#bib.bib22 "The unreasonable effectiveness of deep features as a perceptual metric")) and FVD (Unterthiner et al., [2018](#bib.bib44 "Towards accurate generative models of video: a new metric & challenges")), computed using 100 future videos for each ground-truth test video, then reports the best score over 100 videos for LPIPS, and all videos for FVD, following Babaeizadeh et al. ([2021](#bib.bib10 "FitVid: overfitting in pixel-level video prediction")); Villegas et al. ([2019](#bib.bib11 "High fidelity video prediction with large stochastic recurrent neural networks")).
\subfloat
[Effects of k]
Dataset
k
FVD (↓)
BAIR
No top-k
104.4
100
103.6
10
99.3
KITTI
No top-k
578.1
100
557.7
10
482.9
\subfloat
[Effects of layers]
Dataset
Layers
FVD (↓)
BAIR
6
111.8
12
99.3
KITTI
6
520.1
12
482.9
\subfloat
[Effects of m]
Dataset
m
FVD (↓)
KITTI
0
980.1
2
497.0
4
482.9
8
523.4
Table 2: FVD scores of HARP with varying (a) the number of codes to use for top-k sampling, (b) number of layers, and (c) magnitude m of data augmentation.
#### Results.
Table [1](#S5.T1 "Table 1 ‣ 5 Experiments ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator") shows the performances of our method and baselines on test sets of BAIR Robot Pushing and KITTI driving dataset.
We observe that our model achieves competitive or superior performance to state-of-the-art methods with large end-to-end networks, e.g., HARP outperforms FitVid with 302M parameters on KITTI driving dataset.
Our model successfully extrapolates to unseen number of future frames (i.e., 25) instead of 10 future frames used in training on KITTI dataset.
This implies that transformer-based video prediction models can also predict arbitrary number of frames at inference time.
In the case of BAIR dataset, HARP achieves the similar performance of FitVid with 302M parameters, even though our method only requires 89M parameters.
#### Analysis.
We investigate how the top-k sampling, number of layers, and magnitude m of data augmentation affect the performance.
Table LABEL:tbl:analysis\_k shows that smaller k leads to better performance, implying that the proposed top-k sampling is effective for improving the performance by discarding rare discrete codes that might degrade the prediction quality at inference time.
As shown in Table LABEL:tbl:analysis\_layers, we observe that more layers leads to better performance on BAIR dataset, which implies our model can be further improved by scaling up the networks.
Finally, we find that (i) data augmentation on KITTI dataset is important for achieving strong performance, similar to the observation of Babaeizadeh et al. ([2021](#bib.bib10 "FitVid: overfitting in pixel-level video prediction")), and (ii) too aggressive augmentation leads to worse performance.
6 Discussion
-------------
In this work, we present HARP that employs a high-fidelity image generator for predicting high-resolution future frames, and achieves competitive performance to state-of-the-art video prediction methods with large end-to-end networks.
We also demonstrate that HARP can leverage the image generator pre-trained on a wide range of natural images for video prediction, similar to the approach in the context of video synthesis (Tian et al., [2021](#bib.bib21 "A good image generator is what you need for high-resolution video synthesis")).
We hope this work inspires more investigation into leveraging recently developed pre-trained image generators (Oord et al., [2017](#bib.bib2 "Neural discrete representation learning"); Chen et al., [2020](#bib.bib25 "Generative pretraining from pixels"); Esser et al., [2021](#bib.bib19 "Taming transformers for high-resolution image synthesis")) for high-fidelity video prediction.
\subfloat
[RoboNet]
\subfloat[Kinetics-600]
Figure 4:
Failure cases in our experiments. (a) Interaction with the objects is ignored. (b) The model repeats the first frame while a person is moving right in the ground-truth frames.
Finally, we report the failure cases of video prediction with HARP and discuss the possible extensions to resolve the issue. A common failure case for video prediction on RoboNet dataset is ignoring the interaction between a robot arm and objects.
For example, in Figure [4](#S6.F4 "Figure 4 ‣ 6 Discussion ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator"), our model ignores the objects and only predicts the movement of a robot arm.
On the other hand, common failure case for Kinetics-600 is a degenerate video prediction, where a model just repeats the conditioning frame without predicting the future, as shown in Figure [4](#S6.F4 "Figure 4 ‣ 6 Discussion ‣ HARP: Autoregressive Latent Video Prediction with High-Fidelity Image Generator").
These failure cases might be resolved by training more larger networks similar to the observation in the field of natural language processing, e.g., GPT-3 (Brown et al., [2020](#bib.bib49 "Language models are few-shot learners")), or might necessitate a new architecture for addressing the complexity of training autoregressive latent prediction models on video datasets.
7 Acknowledgements
-------------------
We would like to thank Jongjin Park, Wilson Yan, and Sihyun Yu for helpful discussions.
We also thank Cirrascale Cloud Servicesddd<https://cirrascale.com> for providing compute resources.
This work is supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2021-0-02068, Artificial Intelligence Innovation Hub; No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST)), Center for Human Compatible AI (CHAI), ONR N00014-21-1-2769, the Darpa RACER program, and the Hong Kong Centre for Logistics Robotics, BMW. |
76fd9862-80c5-4f29-9def-68501099e886 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Brussels - Calibration and other games
Discussion article for the meetup : Brussels - Calibration and other games
WHEN: 08 March 2014 01:00:00PM (+0100)
WHERE: Rue des Alexiens 55 1000 Bruxelles
This month we're stealing an idea from the Atlanta group: using Wits & Wagers for confidence calibration (and a little Fermi estimation, which I keep saying we'll do and we never end up doing). It's like a trivia game, except you don't need to know anything to win! (errata: you do need to know that you don't know anything. else.) We may spice it up with some paranoid debating.
I also have a game related to the Memory Palace, and another one related to the Ultimatum Game. But we may not have time to do everything.
If someone can get their hands on an English copy of Wits & Wagers (or Gambit 7, please tell me.
We will meet at 1 pm at "La Fleur en papier doré", close to the Brussels Central station. The meeting will be in English to facilitate both French and Dutch speaking members.
If you are coming for the first time, please consider filling out this one minute form to share your contact information.
The Brussels meetup group communicates through a Google Group.
Meetup announcements are also mirrored on meetup.com
Discussion article for the meetup : Brussels - Calibration and other games |
a406a96a-00aa-49fa-9a41-aadc6d461630 | trentmkelly/LessWrong-43k | LessWrong | [LINK] How Long Does Habit Formation Take?
Related: Common failure modes in habit formation
I ran across this bit of pop-sci (a review of Jeremy Dean's Making Habits, Breaking Habits), which claims that habits typically take around 66 days to form, not the 21 days that self-help articles tend to cite. The somewhat surprising thing to me, on reflection, was how readily I'd taken the 21-day statistic as fact. From the article:
> When he became interested in how long it takes for us to form or change a habit, psychologist Jeremy Dean found himself bombarded with the same magic answer from popular psychology websites and advice columns: 21 days. And yet, strangely — or perhaps predictably, for the internet — this one-size-fits-all number was being applied to everything from starting a running regimen to keeping a diary, but wasn’t backed by any concrete data.
The original article is here. Abstract:
> To investigate the process of habit formation in everyday life, 96 volunteers chose an eating, drinking or activity behaviour to carry out daily in the same context (for example ‘after breakfast’) for 12 weeks. They completed the self-report habit index (SRHI) each day and recorded whether they carried out the behaviour. The majority (82) of participants provided sufficient data for analysis, and increases in automaticity (calculated with a sub-set of SRHI items) were examined over the study period. Nonlinear regressions fitted an asymptotic curve to each individual's automaticity scores over the 84 days. The model fitted for 62 individuals, of whom 39 showed a good fit. Performing the behaviour more consistently was associated with better model fit. The time it took participants to reach 95% of their asymptote of automaticity ranged from 18 to 254 days; indicating considerable variation in how long it takes people to reach their limit of automaticity and highlighting that it can take a very long time. Missing one opportunity to perform the behaviour did not materially affect the habit formation process. With re |
6286ef61-0672-4503-ba48-32064f3e1771 | StampyAI/alignment-research-dataset/blogs | Blogs | March 2021 Newsletter
#### MIRI updates
* MIRI's Eliezer Yudkowsky and Evan Hubinger [comment in some detail](https://www.lesswrong.com/posts/AyfDnnAdjG7HHeD3d/miri-comments-on-cotra-s-case-for-aligning-narrowly) on Ajeya Cotra's [The Case for Aligning Narrowly Superhuman Models](https://www.lesswrong.com/posts/PZtsoaoSLpKjjbMqM/the-case-for-aligning-narrowly-superhuman-models). This conversation touches on some of the more important alignment research views at MIRI, such as the view that alignment requires a thorough understanding of AGI systems' reasoning "under the hood", and the view that early AGI systems should most likely [avoid human modeling](https://www.lesswrong.com/posts/BKjJJH2cRpJcAnP7T/thoughts-on-human-models) if possible.
* From Eliezer Yudkowsky: [A Semitechnical Introductory Dialogue on Solomonoff Induction](https://www.lesswrong.com/posts/EL4HNa92Z95FKL9R2/a-semitechnical-introductory-dialogue-on-solomonoff-1). (Also discussed [by Richard Ngo](https://www.lesswrong.com/posts/wsBpJn7HWEPCJxYai).)
* MIRI research associate Vanessa Kosoy [discusses infra-Bayesianism](https://axrp.net/episode/2021/03/10/episode-5-infra-bayesianism-vanessa-kosoy.html) on the AI X-Risk Research Podcast.
* Eliezer Yudkowsky and Chris Olah [discuss ML transparency](https://twitter.com/ESYudkowsky/status/1358173090782576650) on social media.
#### News and links
* Brian Christian, author of *The Alignment Problem: Machine Learning and Human Values*, [discusses his book](https://80000hours.org/podcast/episodes/brian-christian-the-alignment-problem/) on the 80,000 Hours Podcast.
* Chris Olah's team releases [Multimodal Neurons in Artificial Neural Networks](https://distill.pub/2021/multimodal-neurons/), on artificial neurons that fire for multiple conceptually related stimuli.
* Vitalik Buterin reflects on [*Inadequate Equilibria*](https://equilibriabook.com/toc)'s arguments in the course of discussing [prediction market inefficiencies](https://vitalik.ca/general/2021/02/18/election.html).
The post [March 2021 Newsletter](https://intelligence.org/2021/04/01/march-2021-newsletter/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org). |
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