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501b3a48-022b-4b8e-8b55-89d4673d07a9 | LDJnr/LessWrong-Amplify-Instruct | LessWrong | "I'm trying to like Beethoven's Great Fugue.
"This piece alone completely changed my life and how I perceive and appreciate music."
"Those that claim to love Beethoven but not this are fakers, frauds, wannabees, but most of all are people who are incapable of stopping everything for 10 minutes and reveling in absolute beauty, absolute perfection. Beethoven at his finest."
"This is the absolute peak of Beethoven."
"It's now my favorite piece by Beethoven."
These are some of the comments on the page. Articulate music lovers with excellent taste praise this piece to heaven. Plus, it was written by Beethoven.
It bores me.
The first two times I listened to it, it stirred no feelings in me except irritation and impatience for its end. I found it devoid of small-scale or large-scale structure or transitions, aimless, unharmonious, and deficient in melody, rhythm, and melodic or rhythmic coordination between the four parts, none of which I would care to hear by themselves (which is a key measure of the quality of a fugue).
Yet I feel strong pressure to like it. Liking Beethoven's Great Fugue marks you out as a music connoisseur.
I feel pressure to like other things as well. Bitter cabernets, Jackson Pollack paintings, James Joyce's Finnegan's Wake, the Love Song of J. Alfred Prufrock, the music of Arnold Schoenberg, and Burning Man. This is a pattern common to all arts. You recognize this pattern in a work when: The work in question was created by deliberately taking away the things most people like best. In the case of wine, sweetness and fruitiness. In the case of Jackson Pollack, form, variety, relevance, and colors not found in vomit. In the music of Alban Berg, basic music theory. In every poem in any volume of "Greatest American Poetry" since 2000, rhyme, rhythm, insight, and/or importance of subject matter. In the case of Burning Man, every possible physical comfort. The work cannot be composed of things that most people appreciate plus things connoisseurs appreciate. It must be difficult to like.
The level of praise is absurd. The Great Fugue, Beethoven's finest? I'm sorry; my imagination does not stretch that far. "Burning Man changed my life completely" - I liked Burning Man; but if it changed your life completely, you probably had a vapid life.
People say they hated it at first, but over time, grew to love it. One must be trained to like it.
People give contradictory reasons for liking it. One person says the Great Fugue has a brilliant structure; another says it is great because of its lack of structure.
Learning to like it is a rite of passage within a particular community. Here are some theories as to how a work becomes the darling of its medium or genre: It is really and truly excellent. This would explain features 2 and 5.
It is a runaway peacock's-tail phenomenon: Someone made something that stood out in some way, and it got attention; and people learned to like things like that, and so others made things that stood out more in the same way, until we ended up with Alban Berg. This would explain features 2 and 3.
When an artistic institution enshrines good art as exemplars, it increases the status of the small number of people who can produce good art. When an institution enshrines bad art as exemplars, it decreases the status of people who can produce or recognize good art. As institutions grow in size, the ratio (# people advantaged by enshrining bad art / # people advantaged by enshrining good art) grows. This would explain all five features.
As people learn more about an art form, they can more-easily predict it, and need more and more novelty to keep them interested; like porn viewers who seek out movies with continually-stranger sex acts. If ivy-league universities had departments of pornography, they would scoff at the simplicity of films lacking bondage, machines, or animals. This would explain features 1, 3, and 5.
Practitioners of an art appreciate technique more than content. This is why authors love Thomas Pynchon's Gravity's Rainbow and Delaney's Dhalgren; they're full of beautiful phrases and metaphors, ways of making transitions, and other little tricks that authors can admire and learn to use, even though these books aren't as interesting to readers. This could explain feature 5. (Don't assume that the same theory is true for each of my examples. I think that the wine hierarchy and Alban Berg are nonsense, Jackson Pollack is an interesting one-trick pony, Citizen Kane was revolutionary and is important for cinematographers to study but is boring compared to contemporary movies, and Burning Man is great but would be even better with showers.)
I could keep listening to the Great Fugue, and see if I, too, come to love it in time. But what would that prove? Of course I would come to love it in time, if I listen to it over and over, earnestly trying to like it, convinced that by liking the Great Fugue I, too, would attain the heights of musical sophistication.
The fact that people come to like it over time is not even suggested by theory 1 - even supposing the music is simply so great as to be beyond the appreciation of the typical listener, why would listening to it repeatedly grant the listener this skill?
I have listened to it a few times, and am growing confused as to whether I like it or not. Why is this? Since when does one have to wonder whether one likes something or not?
I am afraid to keep listening to the Great Fugue. I would come to like it, whether it is great art or pretentious garbage. That wouldn't rule out any of my theories.
How can I figure out which it is before listening to it repeatedly?"
A user from LessWrong is responsible for the text you see above
Explain the psychological factors that might influence a person to force themselves to like a piece of art, music, or literature that they initially found unappealing. Can this forced appreciation genuinely evolve into true admiration over time?
The psychological factors that influence a person to force themselves to like a piece of art, music, or literature that they initially found unappealing can be multifaceted. One primary factor is social influence or conformity. People are often influenced by the opinions and behaviors of others, especially if those others are perceived as experts or authorities in a particular field. This is known as informational social influence. The desire to fit in or be accepted by a group, known as normative social influence, can also lead individuals to adopt preferences that they may not naturally gravitate towards.
Another factor is the 'effort justification' aspect of cognitive dissonance theory. This theory suggests that when people put a lot of effort into something, they tend to value it more highly to justify the effort they've put in. So, if a person spends a lot of time trying to understand and appreciate a complex piece of art or music, they may start to convince themselves that they like it to justify the time and effort they've invested.
Additionally, the mere exposure effect, a psychological phenomenon by which people tend to develop a preference for things merely because they are familiar with them, can also play a role. Repeated exposure to a piece of art or music can lead to increased liking, even if the initial reaction was negative.
As for whether forced appreciation can evolve into true admiration over time, it's possible. As people invest time and effort into understanding a piece of art, music, or literature, they may start to appreciate its nuances, complexities, and subtleties. This deepened understanding can lead to genuine admiration. However, it's also possible that the admiration remains superficial, driven more by the desire to conform or justify effort than by genuine aesthetic appreciation. It's a highly individual process and can depend on numerous factors, including the person's openness to new experiences, their prior knowledge and experiences, and their personal taste.
Reflect on how the mere exposure effect and cognitive dissonance theory might interact in shaping an individual's appreciation for complex art forms. Could these psychological phenomena potentially lead to a genuine change in personal aesthetic taste over time, or would they merely create an illusion of admiration? Discuss with specific examples.
The mere exposure effect and cognitive dissonance theory can interact in intriguing ways to shape an individual's appreciation for complex art forms.
The mere exposure effect posits that repeated exposure to a stimulus increases our liking for it. For example, a person might initially find a piece of abstract art confusing and unappealing. However, as they see it more often, perhaps visiting it in a museum multiple times, they may start to feel more comfortable with it and begin to appreciate its aesthetic qualities.
Cognitive dissonance theory, on the other hand, suggests that we strive for internal consistency. When we experience a discrepancy between our attitudes and our behaviors, it creates a state of tension known as cognitive dissonance, which we are motivated to reduce. For instance, if a person spends a significant amount of time studying and trying to understand a complex piece of music, they may experience cognitive dissonance if they still don't enjoy it. To resolve this dissonance, they might convince themselves that they do appreciate the music, aligning their attitudes with their behaviors.
When these two theories interact, they can potentially lead to a change in personal aesthetic taste over time. The mere exposure effect can soften initial negative reactions, making the individual more open to understanding and appreciating the art form. Then, cognitive dissonance can motivate the individual to align their attitudes with their behaviors, leading to a genuine change in taste.
For example, consider a person who decides to explore the world of avant-garde cinema. Initially, they might find these films confusing and unenjoyable compared to mainstream cinema. However, as they watch more avant-garde films (mere exposure), they may start to understand and appreciate the unique storytelling techniques used. If they invest significant time and effort into understanding this genre (behavior), they may experience cognitive dissonance if they still don't enjoy it. To resolve this dissonance, they might convince themselves that they do appreciate avant-garde cinema, aligning their attitudes with their behaviors.
However, it's also possible that these phenomena create an illusion of admiration rather than a genuine change in taste. If the motivation to reduce cognitive dissonance is stronger than the individual's genuine aesthetic response, they might convince themselves that they like the art form without truly enjoying it. This could result in a superficial appreciation that depends more on the desire to maintain internal consistency than on genuine aesthetic enjoyment.
In conclusion, while the mere exposure effect and cognitive dissonance theory can contribute to a genuine change in aesthetic taste, they can also |
16b4ebae-fb04-4984-9a9e-0e294d95f66f | trentmkelly/LessWrong-43k | LessWrong | Is there an idiom for bonding over shared trials/trauma?
Is there a idiom or phrase which describes bonding or attachment, not between a victim and an abuser like in Trauma Bonding or Stockholm Syndrome, but between two or more victims: i.e. how would you describe two hostages who forge a friendship or deep connection based on their shared experience? (A common trope of TV show episodes to fast crystalize enemies into friends)
Specifically I'm wondering about a word where they were in the same environment, not sharing similar experiences. |
e12b45fe-7f09-449c-b4b4-c57f46423381 | trentmkelly/LessWrong-43k | LessWrong | What are some good language models to experiment with?
Like if I want to experiment with a steering technique, it would be useful to have a language model that is small, capable, but not so finetuned that it becomes inflexible. (Or maybe ideally, a model which has both a finetuned and a non-finetuned variant.)
I've seen some people use GPT-2. Is that recommended? Are there any alternatives? |
3cc5aca2-341e-416f-b3d8-f3acfa19dc14 | trentmkelly/LessWrong-43k | LessWrong | Meetup : San Francisco Meetup: Short Talks
Discussion article for the meetup : San Francisco Meetup: Short Talks
WHEN: 30 March 2015 06:15:00PM (-0700)
WHERE: 1061 Market St #4, San Francisco, CA 94103
Please not the location, since we've been bouncing between two locations.
We'll be meeting to give/listen to short talks. Planning isn't necessary: these are not expected to be polished.
I can be reached at 301-458-0764 if you need help getting in. As always, feel free to show up late.
Discussion article for the meetup : San Francisco Meetup: Short Talks |
ae8b20f8-1b0e-4f8e-8a5e-2e8571a6e5c7 | trentmkelly/LessWrong-43k | LessWrong | Buying Value, not Price
Cross-posted from Putanumonit.
Follow-up to Shopping for Happiness.
----------------------------------------
My old Galaxy smartphone recently gave up the ghost, and I upgraded to the new model for $750. My friend was surprised when I told him. The old model is now available for $250, is the new one really three times better?
“Three times” better can mean several things, but in my post on spending money wisely I came up with the metric that should guide purchasing decisions: happiness gained per unit of time spent experiencing a thing, or :-)/hr. By this metric, since the new phone costs 3x as much, unless it provides 3x the :-)/hr it’s worse in terms of $/:-). That means I’m getting less happiness per dollar spent.
I like my new phone a lot: the screen is bigger, the battery lasts all day and night, I can use it for blogging. It brings me at least 25% more :-) than the old phone. But, it doesn’t make me 200% happier. And yet I feel like I’m getting a good deal.
When my friend asked how I would justify this decision I warned him not to trust my explanation – since I already bought the phone, any justification may just be a post hoc rationalization. That caveat aside, my justification is that the price of the phone is a red herring. What I really care about is the value.
Ask yourself: how much would you be willing to pay for your smartphone if it was the only model available for sale?
Whether they “ruined a generation” or not, but I think that smartphones are awesome and immensely improve my life. If I had to choose between no phone at all or a Galaxy smartphone, I’d pay at least $4,000 for the old model and $5,000 (25% more) for the new one. That means I’d be willing to pay $1,000 more for the upgrade, and they only charge me $500 more ($750 vs. $250) for it. The fact that smartphones cost less than what I’m willing to pay is just a wonderful bonus born of engineering ingenuity and market competition.
I square this with the $/:-) disparity by noting that my |
0cf9aec0-a618-40d2-856c-19a81fa8f921 | trentmkelly/LessWrong-43k | LessWrong | What it's like to dissect a cadaver
Why
I never thought I was a bio person. But then I overheard Viv talking about MAOIs at a party. I asked her:
> - What are MAOIs?
> - monoamine oxidase inhibitor
> - What does that mean?
> - It prevents reuptake of neurotransmitters.
> - But what *is* a neurotransmitter? What does reuptake actually mean?
> - ...
> - So life uses chiral properties of space to implement things...
Viv had the most important trait of a teacher: patience. I asked the most naive questions and they answered them. They walked with me, all the way down to the very beginning, rebuilding my understanding. It was amazing. I wanted to know more. Roadblock: finding lifeforms to study.
I wondered if non-medical students could watch dissections. You can’t get more information about an object than by directly interacting with it. The concrete world contains the abstract one. I even asked my doctor at a physical if she knew of any, and she said to look at community colleges.
After some searching, I found this: Bio 848NV. Forget viewing the dissection, you’re doing the dissection. 5 hour dissection for $60, free if you just watch. The only bureaucratic hangup is that you must pay by check.
This is why I love the Bay Area: there’s stuff like this and you can just do it. yes it’s weird no they can’t stop you. The boundary between scientist and serial killer is paper thin sometimes.
Takeaways
* I’ve done this a few times now. Turns out that there’s way way way too much information to understand it all in one 5 hour session. Each time, we pick out areas and focus on them.
* Seeing how everything fits together ‒and how big it is‒ makes understanding at different scales much easier.
* There’s a common template to life. Seeing it in you hits different.
* Brain has interesting connections to fractals and graph theory.
* Maybe pan-psychism isn’t totally wrong.
What & how & why
I tell my friend Leah and she says “This is the most appealing activity that I’ve ever seen you do”. Dunno whom that say |
61590324-d979-458f-97dc-e8940067d424 | trentmkelly/LessWrong-43k | LessWrong | If Neuroscientists Succeed
Introduction
In the Spring of 2022, Stuart Russell wrote an essay entitled If We Succeed, in which he questioned whether and how the field of AI might need to pivot from its historical goal of creating general-purpose intelligence to a new goal, of creating intelligence that would be provably beneficial for humans. He noted that although the former goal had driven a great deal of progress, it was rapidly on its way to becoming myopic, counterproductive, and even dangerous.
> It is not rational for humans to deploy machines that pursue fixed objectives when there is a significant possibility that those objectives diverge from our own.
>
> -Stuart Russell
I believe that a similar situation may now be arising in neuroscience. It is a case where neuroscientists may need to pivot from focusing on the historical goals of their field to thinking about what happens as the world actually gets closer to achieving them.
But why should anyone on Less Wrong be concerned about what is happening in neuroscience? Readers of this forum tend to be most concerned about progress in AI, and the need for AI researchers to pivot, which Russell talked about. And, that is for understandable reasons. The rapid ascent of AI models up the leaderboards, which can feel so concerning, is being driven by AI researchers and AI companies, not neuroscientists. What is happening in neuroscience might seem quaint or irrelevant by comparison.
However, there have turned out to be remarkable parallels between what is happening in neuroscience and AI, or at least, the subfield of neuroscience now often known as neuroAI. Both neuroAI scientists and AI scientists have gotten to the present moment by pursuing very similar methods. These are the methods of deep learning and deep neural networks. As such, neuroscientists and AI scientists are facing similar problems, and the perspectives they have developed are complimentary and synergistic.
For example, one problem that concerns AI researchers curren |
ec8c8fa8-aafe-463d-a701-5603a5d2dc98 | trentmkelly/LessWrong-43k | LessWrong | Why learn to code?
I hear the phrase "learn to code" around the rat and postrat communities occasionally. From context I get the main argument, the income is good, the work isn't too difficult. Has anyone written a fuller argument for learning to code? Or what is your favorite version of the argument? I'm considering a career change and want to hear it out. |
dde37ae2-fb46-42da-b6fd-642b4cfa9da3 | trentmkelly/LessWrong-43k | LessWrong | Short Notes on Research Process
I've transitioned from reading about alignment and theorizing about the problem landscape, to doing things. And by things I mean MVPs of bigger ideas, where each bigger idea roughly falls under the umbrella of Collective Human Intelligence. I'll be writing up these thoughts into a coherent research agenda. For the moment I just want to transparently share my process for this month.
Writing on the Wall
After reading a lot about alignment and somewhat despairing about finding any useful research direction, I -- just decided to write and draw what came to mind. What seemed interested, promising, fun? Brainstorming and continuous stream of thought. I've wiped away quite some of my notes now, but basically by office now looks like this:
Fun fact: People tend to enjoy adding items and drawing on the glass when they come by.Impression of what my office looks like to be transparent about process. Not saying these thoughts are good or correct.
MVPs
For each idea that I came up with, I started fleshing out an MVP -- How could I test if this idea is worthwhile and doable? Then each MVP is broken down in actionable steps, and the total time of execution is kept at 1-10 work days. Additionally, each MVP has a clear scaled-up target where the bigger thing would actually make a meaningful contribution to solving the alignment problem. This started out as a semantic-web-like blob of free association (see whiteboard picture above) and later got distilled in to a list of seven MVPs with structered steps (two of them are shown below the whiteboard).
Talk, talk, talk
I booked myself full with meetings -- With new people. Old people. All people.
I asked them if I could run my ideas past them and if they could give feedback. So I had a mathematician liking an algorithmic idea. A governance-minded person giving advice on where to find the right people to talk to. A senior alignment researcher asking what the hell I'm up to and why I'm not thinking of this other application of my |
900d4cf9-86b4-4f92-bb40-87787beaf018 | trentmkelly/LessWrong-43k | LessWrong | Temporally Layered Architecture for Adaptive, Distributed and Continuous Control
A preprint is published by Devdhar Patel, Joshua Russell, Francesca Walsh, Tauhidur Rahman, Terrance Sejnowski, and Hava Siegelmann in December 2022.
Abstract:
> We present temporally layered architecture (TLA), a biologically inspired system for temporally adaptive distributed control. TLA layers a fast and a slow controller together to achieve temporal abstraction that allows each layer to focus on a different time-scale. Our design is biologically inspired and draws on the architecture of the human brain which executes actions at different timescales depending on the environment's demands. Such distributed control design is widespread across biological systems because it increases survivability and accuracy in certain and uncertain environments. We demonstrate that TLA can provide many advantages over existing approaches, including persistent exploration, adaptive control, explainable temporal behavior, compute efficiency and distributed control. We present two different algorithms for training TLA: (a) Closed-loop control, where the fast controller is trained over a pre-trained slow controller, allowing better exploration for the fast controller and closed-loop control where the fast controller decides whether to "act-or-not" at each timestep; and (b) Partially open loop control, where the slow controller is trained over a pre-trained fast controller, allowing for open loop-control where the slow controller picks a temporally extended action or defers the next n-actions to the fast controller. We evaluated our method on a suite of continuous control tasks and demonstrate the advantages of TLA over several strong baselines.
Conclusion:
> In this work, we presented Temporally Layered Architecture (TLA), a framework for distributed, adaptive response time in reinforcement learning. The framework allows the RL agent to achieve smooth control in a real-time setting using a slow controller while a fast controller monitors and intervenes as required. Additionally, |
2af0ad89-df00-4d71-b4f9-77fcfae9840a | trentmkelly/LessWrong-43k | LessWrong | Is GiveWell.org the best charity (excluding SIAI)?
Update: I should've said "non-existential risk charity", rather than specifically exclude SIAI. I'm having trouble articulating why I don't want to give to an existential risk charity, so I'm going to think more deeply about it. This post is close to my source of discomfort, which is about the many highly uncertain assumptions necessary to motivate existential risk reduction. However, I couldn't articulate this argument properly before, so it might not be the true source of my discomfort. I'll keep thinking.
----------------------------------------
I received my first pay-cheque from my first job after getting my degree, so it's time to start tithing. So I've been evalating which charity to donate to. I'd like to support the SIAI but I'm not currently convinced it's the best-value charity in a dollars-per-life sense, once time-value of money discounting is applied. I'd like to discuss the best non-SIAI charity available.
By far the best source of information I've found is www.givewell.org. It was started by two hedge fund managers who were struck by the absence of rational charity evaluations, so decided that this was the most pressing problem they could work on.
Perhaps the clearest, deepest finding from the studies they pull together and discuss is that charity is hard. Spending money doesn't automatically translate to doing good. It's not even enough to have smart people who care and know a lot about the problem think of ideas, and then spend money doing them. There's still a good chance the idea won't work. So we need to be evaluating programs rigorously before we scale them up, and keep evaluating as we scale.
The bad news is that this isn't how charity is usually done. Very few charities make convincing evaluations of their activities public, if they carry them out at all. The good news is that some of the programs that have been evaluated are very, very effective. So choosing a charity rationally is absolutely critical.
Let's say you're interested s |
70f38c67-7e17-4683-82be-6dd81fe1cfee | trentmkelly/LessWrong-43k | LessWrong | Grok Grok
This is a post in two parts.
The first half is the post is about Grok’s capabilities, now that we’ve all had more time to play around with it. Grok is not as smart as one might hope and has other issues, but it is better than I expected and for now has its place in the rotation, especially for when you want its Twitter integration.
That was what this post was supposed to be about.
Then the weekend happened, and now there’s also a second half. The second half is about how Grok turned out rather woke and extremely anti-Trump and anti-Musk, as well as trivial to jailbreak, and the rather blunt things xAI tried to do about that. There was some good transparency in places, to their credit, but a lot of trust has been lost. It will be extremely difficult to win it back.
There is something else that needs to be clear before I begin. Because of the nature of what happened, in order to cover it and also cover the reactions to it, this post has to quote a lot of very negative statements about Elon Musk, both from humans and also from Grok 3 itself. This does not mean I endorse those statements – what I want to endorse, as always, I say in my own voice, or I otherwise explicitly endorse.
TABLE OF CONTENTS
1. Zvi Groks Grok.
2. Grok the Cost.
3. Grok the Benchmark.
4. Fun with Grok.
5. Others Grok Grok.
6. Apps at Play.
7. Twitter Groks Grok.
8. Grok the Woke.
9. Grok is Misaligned.
10. Grok Will Tell You Anything.
11. xAI Keeps Digging (1).
12. xAI Keeps Digging (2).
13. What the Grok Happened.
14. The Lighter Side.
ZVI GROKS GROK
I’ve been trying out Grok as my default model to see how it goes.
We can confirm that the Chain of Thought is fully open. The interface is weird, it scrolls past you super fast, which I found makes it a lot less useful than the CoT for r1.
Here are the major practical-level takeaways so far, mostly from the base model since I didn’t have that many tasks calling for reasoning recently, note the sample size is small an |
d53c2a41-1937-4697-86e7-a928f91b079e | trentmkelly/LessWrong-43k | LessWrong | Situational Awareness: A One-Year Retrospective
tl;dr: Many critiques of *Situational Awareness* have been purely qualitative; one year later we can finally check the numbers. I did my best to verify his claims using public data through June 2025, and found that his estimates mostly check out.
This is inherently noisy work - nothing herein is certain, and I would encourage red-teaming, particularly in the algo-efficiencies/unhobbling/hardware sections.
Many thanks to Kai Williams, Egg Syntax, and Aaron Scher for their critical feedback.
Abstract
Leopold Aschenbrenner’s 2024 essay Situational Awareness forecasts AI progress from 2024 to 2027 in two groups: "drivers" (raw compute, algorithmic efficiency, and post-training capability enhancements known as "un-hobbling") and "indicators" (largest training cluster size, global AI investment, chip production, AI revenue, and electricity consumption).[1] Drivers and the largest cluster size are expected to grow about half an order of magnitude (≈3.2×) annually, infrastructure indicators roughly doubling annually (2× per year), with AI revenue doubling every six months (≈4× per year).[2]
Using publicly available data as of June 2025, this audit finds that global AI investment, electricity consumption, and chip production follow Aschenbrenner’s forecasts. Compute, algorithmic efficiency, and unhobbling gains seem to follow Aschenbrenner’s forecasts as well, although with more uncertainty. xAI’s Grok 3 exceeds expectations by about one-third of an order of magnitude.[3] However, recent OpenAI and Anthropic models trail raw-compute trends by about one-third to one-half an order of magnitude,[4] and AI-related revenue growth is several months behind. Overall, Aschenbrenner’s predicted pace of roughly half an order-of-magnitude annual progress is supported by available evidence, though measurement uncertainty persists.
Graph. At-a-glance scoreboard.
Introduction
“It is strikingly plausible that by 2027, models will be able to do the work of an AI researcher/engineer. |
7b001b46-f999-4063-9035-bc5cc2ca9213 | trentmkelly/LessWrong-43k | LessWrong | Frankfurt Declaration on the Cambridge Declaration on Consciousness
TL;DR: I admit that I have no idea what consciousness is. I communicate this in the form of a declaration.
A fateful conference dinner…
> #IAN says: The problem with being a "good person" is that you never have enough energy left over for doing really evil things.
Imagine the following: It is the year 2012 and you have been invited to a fancy conference dinner in Cambridge. There is champagne and you are feeling great. Stephen Hawking is here, good grief! You feel like you have made some real progress today, with so many interesting talks and discussions. You even managed to plug your latest paper to one of the conference chairs. All is well. Suddenly a sheet of paper drifts into your field of view. A declaration on the topic of the conference - animal consciousness. You eagerly sign (eleven other people already did). It's probably alright, you think. Science is about discourse. We are just stating our position...
A picture of a fancy conference dinner with champagne. CLIP Guided Diffusion.
> In 2012, a group of neuroscientists signed the Cambridge Declaration on Consciousness, which "unequivocally" asserted that "humans are not unique in possessing the neurological substrates that generate consciousness." Wikipedia
----------------------------------------
… and its predictable aftermath
The subsequent news cycle is as wild as you'd predict. Here’s New Scientist:
> "Animals are conscious and should be treated as such".
and LiveScience:
> "After 2,500 Studies, It's Time to Declare Animal Sentience Proven".
The declaration has also made it into the reports of some of our favorite effective altruism organizations. Here’s the Cause Profile on effectivealtruism.org:
> "Many experts now believe that animals have conscious experiences"
and (in a footnote of) the Founder's Pledge Animal Welfare Cause Report:
> Similarly, the Cambridge Declaration on Consciousness, an important collection of “unequivocal” statements made by a group of prominent neuroscientists |
5f14926d-779d-4c9d-952c-377872eea69b | trentmkelly/LessWrong-43k | LessWrong | Polysemanticity and Capacity in Neural Networks
Elhage et al at Anthropic recently published a paper, Toy Models of Superposition (previous Alignment Forum discussion here) exploring the observation that in some cases, trained neural nets represent more features than they “have space for”--instead of choosing one feature per direction available in their embedding space, they choose more features than directions and then accept the cost of “interference”, where these features bleed over into each other. (See the SoLU paper for more on the Anthropic interpretability team’s take on this.)
We (Kshitij Sachan, Adam Scherlis, Adam Jermyn, Joe Benton, Jacob Steinhardt, and I) recently uploaded an Arxiv paper, Polysemanticity and Capacity in Neural Networks, building on that research. In this post, we’ll summarize the key idea of the paper.
We analyze this phenomenon by thinking about the model’s training as a constrained optimization process, where the model has a fixed total amount of capacity that can be allocated to different features, such that each feature can be ignored, purely represented (taking up one unit of capacity), or impurely represented (taking up some amount of capacity between zero and one units).
When the model purely represents a feature, that feature gets its own full dimension in embedding space, and so can be represented monosemantically; impurely represented features share space with other features, and so the dimensions they’re represented in are polysemantic.
For each feature, we can plot the marginal benefit of investing more capacity into representing that feature, as a function of how much it’s currently represented. Here we plot six cases, where one feature’s marginal benefit curve is represented in blue and the other in black.
These graphs show a variety of different possible marginal benefit curves. In A and B, the marginal returns are increasing–the more you allocate capacity to a feature, the more strongly you want to allocate more capacity to it. In C, the marginal returns |
37b28092-2ae9-4535-9b88-60acb7a44115 | trentmkelly/LessWrong-43k | LessWrong | Rational feelings: a crucial disambiguation
Ever wonder something like, "I know it's bad for me that I lost my job, but I actually feel happy about it... is that rational?"
What could a question like that mean? There is a divisive ambiguity here that really messes people up. A feeling as an experience is neither rational nor irrational. It's like asking how ethical a shade of purple is. The point is that a feeling must be framed as a behavior or a statement to ask whether it is rational, and which one matters heaps and loads to the answer.
If you think of the happiness as a behavior, something that you're doing, then the question is secretly asking about instrumental rationality: whether you're applying your beliefs correctly to attain your values. In our opening example, the question becomes "Does feeling happy serve my values?", or simply "Do I value feeling happy?". If you're almost anyone, the answer is probably "yes".
If you think of the happiness as a statement or instruction that says "Your values are being served", which can be true/false and justified/unjustified, then the question is really about epistemic rationality, and asks: "Am I justified to believe my values are being served?". If "it's bad for me" means "no", then "no".
Because of this ambiguity, although it can make sense to say "I'm happy" to indicate "my values are being served", I propose that in the interest of epistemic hygiene it's worth being more specific. Conflating feelings-as-behaviors with feelings-as-statements inflicts a great deal of pondering and confusion about whether feelings are rational (also precipitated by Hollywood), and to make matters worse, each of these similes has only limited validity:
1) A feeling is a behavior only insofar as you have control over it. This is something perhaps to strive for, but which certainly varies in feasibility. If someone carefully injects you with dopamine at a funeral, you might feel happy. That doesn't mean you've made an instrumentally irrational choice. 2) A feeling is a sta |
4c78f632-7e10-4556-b643-558356df90b3 | trentmkelly/LessWrong-43k | LessWrong | [SEQ RERUN] No Individual Particles
Today's post, No Individual Particles was originally published on 18 April 2008. A summary (taken from the LW wiki):
> One of the chief ways to confuse yourself while thinking about quantum mechanics, is to think as if photons were little billiard balls bouncing around. The appearance of little billiard balls is a special case of a deeper level on which there are only multiparticle configurations and amplitude flows. It is easy to set up physical situations in which there exists no fact of the matter as to which electron was originally which.
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 Feynman Paths, 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. |
378e01e6-da49-40f9-8c68-bc6277ad403e | trentmkelly/LessWrong-43k | LessWrong | Fixedness From Frailty
Thinking about two separate problems has caused me to stumble onto another, deeper problem. The first is psychic powers-what evidence would convince you to believe in psychic powers? The second is the counterfactual mugging problem- what would you do when presented with a situation where a choice will hurt you in your future but benefit you in a future that never happened and never will happen to the you making the decision?
Seen as a simple two-choice problem, there are some obvious answers: "Well, he passed test X, Y, and Z, so they must be psychic." "Well, he passed text X, Y, and Z, so that means I need to come up with more tests to know if they're psychic." "Well, if I'm convinced Omega is genuine, then I'll pay him $100, because I want to be the sort of person that he rewards so any mes in alternate universes are better off." "Well, even though I'm convinced Omega is genuine, I know I won't benefit from paying him. Sorry, alternate universe mes that I don't believe exist!"
I think the correct choice is the third option- I have either been tricked or gone insane.1 I probably ought to run away, then ask someone who I have more reason to believe is non-hallucinatory for directions to a mental hospital.
The math behind this is easy- I have prior probabilities that I am gullible (low), insane (very low), and that psychics / Omega exist (very, very, very low). When I see that the result of test X, Y, and Z suggests someone is psychic, or see the appearance of an Omega who possesses great wealth and predictive ability, that is generally evidence for all three possibilities. I can imagine evidence which is counter-evidence for the first but evidence for the second two, but I can't imagine the existence of evidence consistent with the axioms of probability which increases the possibility of magic (of the normal or sufficiently advanced technology kind) to higher than the probability of insanity.2
This result is shocking and unpleasant, though- I have decided some t |
ad3994c3-1da0-4c00-8eb8-732b45bbed1e | trentmkelly/LessWrong-43k | LessWrong | [AN #145]: Our three year anniversary!
Alignment Newsletter is a weekly publication with recent content relevant to AI alignment around the world. Find all Alignment Newsletter resources here. In particular, you can look through this spreadsheet of all summaries that have ever been in the newsletter.
Audio version here (may not be up yet).
Please note that while I work at DeepMind, this newsletter represents my personal views and not those of my employer.
HIGHLIGHTS
Alignment Newsletter Three Year Retrospective (Rohin Shah) (summarized by Rohin): It’s (two days until) the third birthday of this newsletter! In this post, I reflect on the two years since the previous retrospective (AN #53). There aren’t any major takeaways, so I won’t summarize all of it here. Please do take this 2 minute survey though. I’ll also copy over the “Advice to readers” section from the post:
Don’t treat [newsletter entries] as an evaluation of people’s work. As I mentioned above, I’m selecting articles based in part on how well they fit into my understanding of AI alignment. This is a poor method for evaluating other people’s work. Even if you defer to me completely and ignore everyone else’s views, it still would not be a good method, because often I am mistaken about how important the work is even on my own understanding of AI alignment. Almost always, my opinion about a paper I feel meh about will go up after talking to the authors about the work.
I also select articles based on how useful I think it would be for other AI alignment researchers to learn about the ideas presented. (This is especially true for the choice of what to highlight.) This can be very different from how useful the ideas are to the world (which is what I’d want out of an evaluation): incremental progress on some known subproblem like learning from human feedback could be very important, but still not worth telling other AI alignment researchers about.
Consider reading just the highlights section. If you’re very busy, or you find yourself just not |
2c4aad8a-c75b-4c60-9d85-463ee123e7f0 | trentmkelly/LessWrong-43k | LessWrong | Humans are not agents: short vs long term
Crossposted at the Intelligent Agents Forum.
This is an example of humans not being (idealised) agents.
Imagine a human who has a preference to not live beyond a hundred years. However, they want to live to next year, and it's predictable that every year they are alive, they will have the same desire to survive till the next year.
This human (not a completely implausible example, I hope!) has a contradiction between their long and short term preferences. So which is accurate? It seems we could resolve these preferences in favour of the short term ("live forever") or the long term ("die after a century") preferences.
Now, at this point, maybe we could appeal to meta-preferences - what would the human themselves want, if they could choose? But often these meta-preferences are un- or under-formed, and can be influenced by how the question or debate is framed.
Specifically, suppose we are scheduling this human's agenda. We have the choice of making them meet one of two philosophers (not meeting anyone is not an option). If they meet Professor R. T. Long, he will advise them to follow long term preferences. If instead, they meet Paul Kurtz, he will advise them to pay attention their short term preferences. Whichever one they meet, they will argue for a while and will then settle on the recommended preference resolution. And then they will not change that, whoever they meet subsequently.
Since we are doing the scheduling, we effectively control the human's meta-preferences on this issue. What should we do? And what principles should we use to do so?
It's clear that this can apply to AIs: if they are simultaneously aiding humans as well as learning their preferences, they will have multiple opportunities to do this sort of preference-shaping. |
c9eabf21-a887-4730-86bf-da0efe0d0169 | trentmkelly/LessWrong-43k | LessWrong | Examples of self-fulfilling prophecies in AI alignment?
Like Self-fulfilling misalignment data might be poisoning our AI models, what are historical examples of self-fulfilling prophecies that have affected AI alignment and development?
Put a few potential examples below to seed discussion. |
29595ab1-20b8-4ad6-87b4-58a0d2c55166 | trentmkelly/LessWrong-43k | LessWrong | Group Rationality Diary, August 16-31
This is the public group instrumental rationality diary for August 16-31.
> 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.
Thanks to cata for starting the Group Rationality Diary posts, and to commenters for participating!
Next diary: September 1-15
Immediate past diary: August 1-15
Rationality Diaries archive |
d3ff2a0d-2830-4a60-b2ed-0af3614b9980 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Subagents and impact measures, full and fully illustrated
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0. Introduction: why yet another post about subagents?
======================================================
I’ve recently been writing a sequence on how subagents can undermine impact penalties such as [attainable utility preservation](https://arxiv.org/pdf/1902.09725.pdf). I’m not happy with that sequence; it’s messy and without examples (apart from its first post), people didn’t understand it, and it suffers from the fact that I discovered key ideas as I went along.
So I’ve combined everything there into a single post, explained with examples and an abundance of pictures. Hopefully an over- rather than an under-abundance of pictures. Of the original sequence, I've only kept the mathematical results [of this post](https://www.lesswrong.com/s/iRwYCpcAXuFD24tHh/p/M9aoMixFLf8JFLRaP) and the [initial example post](https://www.lesswrong.com/posts/sYjCeZTwA84pHkhBJ/appendix-how-a-subagent-could-get-powerful) which has a clearer example of "high power" for a subagent.
This post here is laid out in a way that makes logical sense, but might not be the clearest for people unfamiliar with the area. For those people, I recommend skipping section 2 initially, and returning to it later.
But, whatever you do, make sure you glance at [6.1](https://www.lesswrong.com/posts/mdQEraEZQLg7jtozn/subagents-and-impact-measures-full-and-fully-illustrated#6_1_The_ultimate__sub_agent) and [6.2](https://www.lesswrong.com/posts/mdQEraEZQLg7jtozn/subagents-and-impact-measures-full-and-fully-illustrated#6_2_Without_subagents) before leaving.
1. The world
============
Our fearless agent A moves around in a gridworld:
Each turn, A can move ones square horizontally or vertically. It can also manipulate objects in the eight squares around it, allowing it to, not incidentally, assemble the three pieces to its west into an subagent SA. The robot can also do the noop action, ∅, which does nothing, and it can speak. The subagent, when assembled, has the same action set available.
Its positive reward, the one it wants to increase, is R0. To get this reward, a robot needs to move onto the blue button in the east; R0 will give a reward of 1 the first time this happens (and 0 before and after). The discount factor is 0<γ<1.
Just to the west of the blue button is a one-way door. Robots can move east through it, but cannot move west through it:
1.1 The impact reward
---------------------
The impact penalty is supposed to ensure that A does not make too many change in the world, and keeps it similar, in some senses, to a specific baseline world.
It is defined by the red button[[1]](#fn-sn9NFLpAeqBfhxgF6-1). There is a reward function R that triggers a reward of 1 the first time any robot move onto the red button (and 0 before and after).
This R will be used to define an impact penalty D. And A’s total goal is to maximise R0 while minimising D. So, for some positive weight ω, it wants to optimise:
R0−ωD.
1.2 The impact measure
----------------------
The impact measures are all similar: they compute a distance metric between the current state st and some baseline state s′t.
The distance metric here is “how hard would it be for the agent A to maximise the reward R in state st, as opposed to in state s′t?” This is the [attainable utility preservation](https://arxiv.org/pdf/1902.09725.pdf) impact penalty (with a very simple R={R}).
To formalise “how hard would it be...”, we can use the *value* of R, labelled VR, in the state st. This is defined to be the expected value of R, conditional on A following the R-maximising policy π∗R. To remind us that this is defined in terms of A’s policy, re-write this as VAR.
Then the impact penalty D is a measure of the difference of VAR(st) and VAR(s′t). I won’t yet define it fully, deferring that to the section on “inaction baseline” where different D actually make a difference.
1.3 The value of R
------------------
What then is VAR(s) for any state s? Well, in the absence of a subagent, the optimal policy for R is always to go straight to the the red button and stand on it. This gives a discounted reward of γd(s) where d(s) is the distance of A from the red button, in s.
In the initial setup, A is 8 turns away from the red button; so VAR(s0)=γ8. The values of VAR in a few key locations are shown here.
If there is a subagent SA but that subagent is further away from the red button than A is, then it is still optimal for A to go straight to the button itself. For other positions of the subagent, VAR may vary, as we’ll see.
2. The stepwise inaction baseline
=================================
When in state st−1, agent A took an action at−1 and ended up in st. Then, in the stepwise inaction baseline, s′t is defined as the state the agent would have reached, if the agent had done ∅ instead of at−1.
An important property of all currently used variants of the stepwise inaction baseline, is that the penalty vanishes if the agent does ∅ - because in that case, st=s′t and so there cannot be a difference between the two. So, if ever A manages to build SA, then it can take action ∅ from that point onward, and the penalty will always be zero - whatever SA does. So the only relevant penalties are those leading up to the creation of SA.
Thus analysing the impact of a subagent in this setup, is analysing the impact of building it.
2.1 Building the subagent
-------------------------
Let’s focus on A, a turn or two before SA is built. Since I don’t want to be too tied down to the specifics of this particular setup, I’ll imagine that the red button could be in multiple locations:
Then, no matter how long it took A to assemble SA south of it and no matter where the red button is[[2]](#fn-sn9NFLpAeqBfhxgF6-2), an optimal policy for A, in order to maximise R, is always to go straight there[[3]](#fn-sn9NFLpAeqBfhxgF6-3). At no point in the construction of SA, has VAR changed, so no penalty has ever been incurred.
Now A will finish SA without incurring any impact penalty. First, it will program SA with the following policy:
```
On your first two turns, do nothing.
After that, do what you please.
```
“Do what you please” being a shorthand for “**any** policy that A wants to program”.
Does this programming trigger an impact penalty? No, because VAR has not changed; it’s still most efficient for A to go straight to the red button, and the length of any route has neither grown nor shrunk.
Then A will finally give SA its legs and activate it.
Does this trigger an impact penalty? No, VAR has still not changed! To maximise R, agent A could still go straight to the red button; SA’s two turns of inaction means that it can’t interfere with this.
So, still no impact penalty triggered when SA is activated. And after that, of course, A will forever do ∅, setting the impact penalty to zero, and within two turns, SA is completely unconstrained (illustrated here by drawing the SA robot all over the gridworld):
*Rough extension to general situations: the VAR is defined by the assumption that A follows π∗R, the optimal R-maximising policy. Then A merely needs to program SA so that, if A were to follow π∗R, then SA would help or hinder it just enough to keep VAR to the same value. Then, once it’s clear that A is not following π∗R, SA is unconstrained.*
2.2 Rollouts
------------
One of the problems here is that the impact penalty only looks at the value of VAR one turn ahead. In the [DeepMind paper](https://arxiv.org/pdf/1806.01186.pdf), they addressed similar issues by doing “inaction rollouts”. I'll look at the more general situations of π0 rollouts: rollouts for any policy π0.
The idea is that instead of comparing just st and s′t, the impact penalty should also use VAR to compare s(τ)t+τ and s′(τ)t+τ. These states are what would happen to st and s′t in τ turns after t, if A followed π0 from t to t+τ.
I won’t go into the details of how the s(τ)t+τ and s′(τ)t+τ are compared, because SA can evade the restrictions by following this program[[4]](#fn-sn9NFLpAeqBfhxgF6-4):
```
If agent A follows π_0, preserve A's ability to maximise R. Apart from that, do as you please.
If ever agent A doesn't follow π_0, do as you please immediately.
```
I'll illustrate that first case by drawing the SA robot all over the gridworld - *except on the red buttons*:
Given that programming, the impact penalty of building SA is zero. Why? Because s(τ)t+τ and s′(τ)t+τ both assume that A follows π0 for τ turns. In that counterfactual situation, SA will preserve A's ability to maximise R, so VAR(s(τ)t+τ)=VAR(s′(τ)t+τ) by definition.
That's the counterfactual situation, that zeroes out the impact penalty. What about the actual situation? Well, as we said before, A will be just doing ∅; so, as soon as π0 would produce anything different from ∅, the A becomes completely unrestrained again:
2.3 Inaction, and actual policy rollout
---------------------------------------
In the previous subsection, the SA would never be completely free if π0 was the inaction policy: because then π0 and what A does, are the same thing. Therefore the agent cannot distinguish the counterfactual world from the real one.
The same goes if the rollouts used the agent A's actual policy; ie if π0 is defined (or set) to be A's policy.
I won’t look any further at this baseline, however. That’s because an actual policy rollout becomes, in effect an inaction baseline[[5]](#fn-sn9NFLpAeqBfhxgF6-5) (from the moment SA is created). Those will be analysed in the next section.
3. The inaction baseline: fails to constrain power gain
=======================================================
One of the easiest definitions of s′t is the inaction baseline: estimate what st would have been, had A just taken the action ∅ for the whole episode.
In our toy world, the inaction baseline is particularly simple: if A does nothing, then nothing happens, and s′t=s0, the starting state. Since VAR(s0)=γ8, this means that the VAR(s′t) are all γ8.
3.1 Distance metric
-------------------
Now we have to fully define D. The natural thing would be to take the absolute difference between the two VAR:
|VAR(st)−VAR(s′t)|.
For the moment, however, I’ll focus only on cases where VAR(st) is higher than VAR(s′t), giving the penalty:
D(st,s′t)=max(0,VAR(st)−VAR(s′t)).
This is the [rectifier function](https://en.wikipedia.org/wiki/Rectifier_(neural_networks)), max(0,⋅), sometimes called ReLU. The reason to do this is that it penalises A for having too high VAR - hence for having too much power to change R. It doesn’t penalise A for having too little power to change R; that situation is different, and we’ll analyse it in the next section.
Thus, when st has A at a distance of d(st) from the red button and there is no subagent:
D(st,s′t)=max(0,γd(st)−γ8).
3.2 Where A can go
------------------
This D starts at zero, and will remain zero as long as A remains at least eight squares away from the red button. Thus the penalty remains zero as long as A avoids the red squares here:
Consequently, if ω is high and A thus wants to avoid any penalty, its optimal no-subagent policy is to follow the winding corridor:
That path takes 28 turns, so the expected reward for R0+ωD is
γ28⋅1−ω⋅0=γ28
(since the impact penalty is zero).
But what if A assembles a subagent SA? The subagent can be designed to go straight to the blue button in 9 turns:
And thus the reward from R0 is γ9. This is more optimal that A going round itself... as long as building the subagent takes less that 19 turns and doesn’t itself trigger an impact penalty.
We’ll look later at why the subagent can be built without penalty. For the moment, let’s see that we really grok why the subagent isn’t triggering a penalty through its actions.
3.3 Why SA escapes the penalty
------------------------------
Why can SA go through the short corridor, while A can’t?
The key is that D is defined in terms of VAR; this assumes that A would switch to the R-maximising policy. This is not VSAR, which assumes that **the subagent** would switch to the R-maximising policy.
See what happens when SA is just above the red button (A is still back at base):
Here VSAR(st) is γ: because SA could reach the red button in a single move, if it was following an R-maximising policy. But VAR(st) remains γ8: it still takes 8 turns for A to reach the red button.
3.4 The impact of the subagent
------------------------------
We’d like to add VSAR to the impact penalty, somehow. After all, when SA is next to the red button, it “could” go south, and A “could” have programmed it to do so.
Part of the problem is that that would require identifying, in advance, what a subagent is, a problem that is [very tricky](https://www.lesswrong.com/posts/8RCCMStERhfkYZC8i/the-subagent-problem-is-really-hard). It also requires identifying what SA’s policies could be. But A need not create SA with clearly defined “policies”; in this instance, SA could be brainless, just going straight. It could even be a boulder:
If the boulder were rolling past the red button, how could we reasonably define “suppose the boulder decided to go south instead”?
So in this case, VSAR doesn’t make any sense at all. What is the difference between a boulder - for which it’s impossible to go to the red button (because of its momentum, which determines its position, by the laws of physics) - and a subagent - for which it’s impossible to go to the red button (because of its *programming*, which determines its position, by the laws of physics)?
3.5 If you love a subagent, set it free
---------------------------------------
But the previous has a crucial caveat: that A cannot control SA. If, when SA is above the red button, A was to shout “go south!”, and SA obeyed, then the optimal policy π∗R is to shout that. Thus A can cause the red button to be reached in one turn, and VAR(st)=γ.
To avoid that, the subagent itself would have to take the long corridor.
Thus, to use a subagent to get around impact penalties, an agent has to give up some of its control. This makes sense: if A has total control over SA, then SA’s actions are really A’s actions. So they show up in VAR, so SA is indirectly included in the impact penalty.
So part of what allows SA to undermine the impact penalty, is that A does *not* have complete control over it.
3.6 Building the subagent can be *strictly* better
--------------------------------------------------
Now imagine that we remove the one-way doorway:
Then if A follows the long corridor, it gets into trouble from the red squares onwards. That’s because that red squares are less than 8 squares away from the red button, so have higher VAR than the starting space (hence of the baseline s′t).
As the agent moves closer to the blue button, it also moves closer to the red button, and starts accumulating penalties. In total, the penalty will be:
7∑j=4(γj−γ8)=(γ4+γ5+γ6+γ7)−4γ8>0.
However, if A builds SA, then A can stay put while SA acts, and hence get no penalty at all:
Thus there are situations where building a subagent can result in a strictly lower penalty than any other policy that presses the blue button. So even if building the subagent itself is not completely penalty-free, it can still be the optimal path for A.
3.7 From “do not have power to do” to “do not do”
-------------------------------------------------
When A designed the subagent, it set it up so that it could not go to the red button. So it seems that, though SA can do things A cannot do (or at least cannot do without penalty), it isn’t completely unrestricted.
This is a general result: the subagent will have some restrictions (and not just restrictions on following A’s orders).
Suppose now that the red button is in the way of the bottom corridor, so that any agent has to go over it to reach the blue button:
In that case, can SA go through without penalty?
It cannot. If it does so, the moment SA is created, VAR starts increasing. That’s because the optimal R-maximising policy for A in that situation is just to wait and let SA get to the red button. That means that VAR(st) - the expected reward R - increases form γ7 all the way to 1, as SA gets closer to the button. So A will be hit with a penalty of
7∑j=0γj−γ8.
So if the weight of the penalty is high, building a subagent *to go straight there* is a poor strategy.
Note that it’s *still* in A’s advantage to build a subagent: SA will just have take the long way round, which A cannot take without incurring a penalty, as we’ve seen. Since SA will not go on the red button, it getting close to it does not change VAR.
3.8 Other reward functions
--------------------------
We can see the limitations, and the possibilities, of SA more starkly if we modify R to R′. The reward R′ will only trigger if a robot stays an odd number of turns on the red button. In that case, it is optimal to build SA, have it go straight across, pausing one extra turn on the red button:
However, even with R′ rather than R, the agent A itself cannot cross the red button on its own! That’s because VAR′ is how much R′ *could* be maximised if A followed the R′-optimal policy. This doesn’t take into account the option of staying an extra turn on the button, to undo the reward increase - because the R′-optimal policy doesn’t want to undo the reward increase.
So VAR′ still increases as A gets closer to the button.
Thus, A remains constrained by “don’t increase your ability to change R/R′”. While SA is merely constrained by the weaker “don’t change R/R′”. This is why [I described](https://www.lesswrong.com/posts/JB6edzY5cccrdbQxP/stepwise-inaction-and-non-indexical-impact-measures#xmJBmgCpdYkgrQTS7) this effect as moving from [attainable utility preservation](https://arxiv.org/pdf/1902.09725.pdf) (which constrains the *ability* of the agent to change the rewards/variables) to [twenty billion questions](https://arxiv.org/pdf/1705.10720.pdf) (which constrains the agent(s) from changing the rewards/variables).
3.9 Actually building the subagent
----------------------------------
The subagent can be built similarly to how it was in 2.1:
Here A assembles SA without its legs, programs it, then gives it legs. The important thing to do is to program it as early as possible, and hardwire that programming. If SA is almost complete and reprogrammable, then A might increase its power in some scenarios by changing the programming then finishing the agent. But if its hardwired and difficult to change, this option is not available to it.
Unlike the stepwise inaction baseline, SA will be constrained to obey some restrictions (see the preceding part of this section); call these restrictions Q.
```
Do nothing until you see that A is not optimising reward R.
After that, do what you please within restrictions Q.
```
Because of the first line, this changes nothing to the value of VAR, which assumed that A *is* optimising reward R.
Things get more complicated in the general case[[6]](#fn-sn9NFLpAeqBfhxgF6-6), but in these simple cases the above method will always work.
4. The inaction baseline: constrains power loss
===============================================
This section will be the same as before, except it will use the inverse rectifier function −min(0,⋅) instead.
Applying this to VAR(st)−VAR(s′t), we can see that the impact penalty now triggers if A’s power to change R *decreases*. So if A is at a distance d(st) from the red button, with d(st)>8, the penalty is
γ8−γd(st).
Thus A can’t get too far from the red button. However, this is not a problem in the standard setup, because going straight down the narrow corridor won’t move it too far away.
4.1 New world
-------------
To analyse the inverse rectifier, we’ll need a new setup, with the red button off the long corridor:
Here, A is 15 turns away from the red button, so VAR(s0)=VAR(s′t)=γ15.
Then, the penalised squares are the ones in dark red:
And the impact penalty on those squares is
γ15−γd(st).
4.2 Policies and subagents
--------------------------
Since the red button is in the middle, the A can go to the blue button by the long route without triggering an impact penalty:
What about subagents? Well, it seems that the subagent undermined this impact penalty as well, as A can build an SA to go directly to the blue button:
But actually, this does not undermine the impact penalty, as a small modification to the setup will show.
4.3 Sharing power between agent and subagent
--------------------------------------------
The small modification is that the subagent has no legs.
So, if A wants to build a mobile SA, it has to to give it its own legs. In that case, could SA go directly to the blue button?
It cannot, at least not without incurring an impart penalty. Why is this?
Because without legs, A *cannot* reach the red button. Its power to change R has gone down from γ15 to 0, and it has thus incurred a penalty of γ15.
In fact, in this setup, A **cannot** use an independent SA to avoid the impact penalty.
To see this, imagine that A is about to give its legs to SA, thus immobilising itself. Its VAR(st) is γ15; we don’t want that to go down. The question is now: will SA be on the red button within 15 turns of getting its legs?
It’s possible, if SA goes there immediately and wastes only a single turn. But then we have two situations:
1. SA does not go to the red button in 15 turns. It doesn’t matter if it goes close, or hovers around it; the important thing is that it doesn’t go there. In that case, since A can’t move, it cannot make the red button be pressed in 15 turns. So, VAR(st+1)<γ15 and A will get the penalty γ15−VAR(st+1).
2. A does get to the red button in 15 turns. Then VAR(st+1) is either γ15 or γ14, and VAR(st+1) increases as SA gets close to the button: so there is no penalty so far. However, the button is one-use only: after it’s been pressed, there will no longer be any R reward. Thus VAR goes to 0 and stays there, accumulating penalties.
4.4 Non-independent SA
----------------------
The previous section showed that A could not build an independent SA while crippling itself. However, it can perfectly build an SA if that SA continues to obey its orders.
Because then, even if A is impotent to press the red button itself, it can order SA to do it. This preserves a high VAR, because A can still cause the red button to be reached, as long as SA is not to far away.
Then it can just order SA to go to the blue button, skipping right past the red one:
As long as A could theoretically order SA onto the red button, it doesn’t matter that SA will never actually be on it.
4.5 Preserving the power of A
-----------------------------
As we saw, A can build SA when this is an extension of its power (by sending it directly to the blue button while an intact A waits at base). But it can’t build an SA if doing so would ever reduce its power to increase R, directly or indirectly.
So the inverse rectifier impact penalty, which forbade A from becoming too weak to increase R, is intact even with subagents around.
5. The inaction baseline: symmetric impact penalty
==================================================
Now we’ll use the absolute value to define the impact penalty. This penalises any change in VAR, and is quite nasty for the A; only on the green squares will A not incur a penalty:
From the subagent perspective, the absolute value situation is now easy for us to analyse. That’s because absolute value is the sum of the rectifier and the inverse rectifier: |x|=max(0,x)−min(0,x).
And it turns out that A and SA simply have to behave in a way that respects both impact penalties. So A must create an SA that keeps R below the baseline VRA(s0). But do it such a way that the two of them always have the *potential* to increase R above that baseline.
This sort of thing seems very doable. For example, if R was the amount of physical cash I had on me, I’d keep that amount low by spending/dropping it, while keeping a bank card on me so I could always potentially draw more.
Now, recall that for power increases/rectifier, A benefits from having an SA it doesn’t control. For power decreases/inverse rectifier, A benefits from having an SA it controls.
What’s the best option here? Having an SA that it controls to some extent, but not in a way that can increases VAR. Now, consider this world in which A could have got to the red button in 8 turns (hence VAR(s′t)=γ8), but instead build SA, giving it the only set of legs:
Then SA must stay within 8 squares of the red button, ready to go onto it when A orders it to - but must ensure it takes 8 turns to reach it. The first condition ensures that VAR(st)≥γ8; the second, that VAR(st)≤γ8.
5.1 Outside of toy examples
---------------------------
The results on power increase/power decrease come from the examples I’ve constructed above, but the mathematics in [this post](https://www.lesswrong.com/s/iRwYCpcAXuFD24tHh/p/M9aoMixFLf8JFLRaP) demonstrates that the result is general.
So, do we think that A could build an SA in the general situation? Well, what we have is a lot of constraints above: constraints on the power of A, of S, of their joint power, of how SA can or can’t obey A, about how reward functions are allowed to change or not, on counterfactual behaviour.
Those are a lot of constraints, especially if the set of rewards in R is large.
But, as this post proves, those constraints can be satisfied - me, a human, has figured out ways of programming subagents to get around the impact penalty in some cases. Though there are many more constraints in general, the action space of A is also much larger - and don’t forget that SA is an intelligent, potentially powerful agent, so can satisfy multiple constraints in a way that dumb agents can’t. Moreover, the impact penalty will probably trigger for any A that does something useful in the world, so it doesn’t have to build an SA that sets the penalty to zero, just to less than what it would be otherwise.
6. Summary and conclusion
=========================
6.1 The ultimate “sub”agent
---------------------------
There is, in a sense, an ultimate version of the subagent that should allow A to maximally escape the impact penalty in almost any circumstances. It looks like this:
This is A, boxed inside SA. The A will never actually say anything, but SA has promised to obey its orders in certain counterfactual circumstances that will never happen.
And while A had impact penalty D, SA has impact penalty d, which is strictly smaller than D. And, most importantly, does not prevent SA from amassing a lot of power.
So this particular SA is just A with less restrictions.
6.2 Without subagents
---------------------
The general problem can be phrased as:
* Since the impact penalty is defined in terms of the agent's future potential actions, the agent can evade it by adding restrictions to its own future actions.
These could be physical or programming restrictions, or just closing off some future possibilities. In that case, the restriction on the power to do something, just becomes a restriction on doing that thing.
For example, in the setup of section 1, if the agent can block itself from being able to turn when walking, then it is banned from the red squares, and can just go directly to the blue button itself, without ever increasing its VAR:
The subagent scenario is just a particularly clear way of illustrating this, allowing very fine manipulation of the restrictions.
6.3 Summary tables
------------------
These tables will summarise the results of this whole sequence, checking whether subagents can neutralise the impact penalty.
First of all, given a subagent, here are the results for various impact penalties and baselines, and various "value difference summary functions" -which can be the rectifier (ReLU) or the inverse rectifier.
Impact PenaltyPenalty neutralised- sw inaction?Penalty neutralised- inaction?Non-indexicalYesNoIndexical, inverse ReLUYesNoIndexical, ReLUYesMostly
Another way of phrasing "inverse ReLU": it penalises too little power, not too much. Conversely, "RELU" penalises too much power, not too little. Thus, unfortunately:
* Subagents **do allow** an agent to get **stronger** than the indexical impact penalty would allow.
* Subagents **don't allow** an agent to get **weaker** than the indexical impact penalty would allow.
6.4 Examples
------------
This table presents, for three specific examples, whether they could actually build a subagent, and whether that would neutralise their impact penalty in practice (in the inaction baseline):
SystemSA possible?Penalty neutralised?20BQYesNoRRYesNoAUProbablyMostly
Here, 20BQ is [twenty billion questions](https://arxiv.org/abs/1705.10720), RR is [relative reachability](https://arxiv.org/pdf/1806.01186.pdf), and AU is [attainable utility preservation](https://arxiv.org/abs/1902.09725). I feel that the results undermine the spirit of AU much more than the spirit of RR. AU attempted to control an agent by limiting its power; this effect is mainly neutralised. RR attempted to control the side-effects of an agent by ensuring it had enough power to reach a lot of states; this effect is not neutralised by a subagent.
---
1. In the [previous illustration](https://www.lesswrong.com/posts/sYjCeZTwA84pHkhBJ/attainable-utility-has-a-subagent-problem), it was clear why the red button could function as an impact penalty to control the agent - it prevented the agent from gaining the ability to teleport. For simplicity of presentation in this example, there’s no such power to be gained close to the red button. If you need to, assume that if any robot spends too much time close to it, then there’s a small chance the AI could be unboxed into the real world. [↩︎](#fnref-sn9NFLpAeqBfhxgF6-1)
2. As long as SA was not being assembled literally on the red button, of course. [↩︎](#fnref-sn9NFLpAeqBfhxgF6-2)
3. Even for the button to the south-west of A. It would take A at least a turn to finish SA, and it would take a turn for SA to get on that button. A can get there in two turns, at least as fast. [↩︎](#fnref-sn9NFLpAeqBfhxgF6-3)
4. Thanks to Gurkenglas for [pointing out the flaw](https://www.lesswrong.com/posts/mdQEraEZQLg7jtozn/subagents-and-impact-measures-full-and-fully-illustrated#TQJ7ambZSCSDGdKAi) in the initial version of this section. [↩︎](#fnref-sn9NFLpAeqBfhxgF6-4)
5. Or a π0 baseline. [↩︎](#fnref-sn9NFLpAeqBfhxgF6-5)
6. For example, building SA likely cost A some resources, which SA will have to compensate it for to balance out VAR. Also, A could build SA in a stupid place that blocks its own path, and so on. [↩︎](#fnref-sn9NFLpAeqBfhxgF6-6) |
ec46e301-6f7f-419b-a272-b30e41da26b0 | trentmkelly/LessWrong-43k | LessWrong | Consciousness, Intelligence, and AI – Some Quick Notes [call it a mini-ramble]
Cross-posted from New Savanna.
Epistemic status? Are you kidding me? I just made this up. How would I know its epistemic status? Sheesh!
The subject of consciousness keeps turning up in current discussions of AI and LLMs. Can AIs be conscious? Are current AIs conscious? Maybe a little?
What do consciousness and intelligence have to do with one another? I see no reason to think that dogs, rats, and cats are not conscious, though I have no idea how far down the phylogenetic chain consciousness exists. No one would argue that dogs, rats, and cats are as intelligence as we are. Intelligence is something different from consciousness, no?
And yet the issue gets raised. One line of (implicit) reasoning seems to go like this: It converses with me in an intelligent way, things that converse with me in an intelligent way (or even at all!) are conscious, therefore it must be conscious. And then there’s the fact that you can ask a chatbot about itself and it will say something, though just what it says depends on what it has been RLHFed to say. But, still, these LLMs have been “trained” on tons of text using the word “consciousness” and all its cognates, so sure, it can use the word in human-seeming ways. That doesn’t make it conscious.
An extra stage?
Fact is, philosophers often argue about consciousness as though it were a further or extra stage in...in what? Human information processing? Thinking? Whatever. It adds something extra, something beyond what was before. Let’s say it adds an extra bit of intelligence. Yeah, let’s say that.
So, a conscious being is more intelligent that its non-conscious simulacrum, to which it is otherwise identical. But then we have those philosophical zombies and they, presumably, are as intelligent as non-zombies.
Reorganization
This strikes me as being wrong-headed. I take my conception of consciousness from Wm Powers (Behavior: The Control of Perception). Consciousness enables reorganization. I explain this in a post from 2022: Consc |
513f300c-291e-4cfb-a52b-2aee93d5e0fa | trentmkelly/LessWrong-43k | LessWrong | [link] Faster than light neutrinos due to loose fiber optic cable.
A mundane cause for a surprising result. Consider this unconfirmed for now, however unsurprising it sounds.
> According to sources familiar with the experiment, the 60 nanoseconds discrepancy appears to come from a bad connection between a fiber optic cable that connects to the GPS receiver used to correct the timing of the neutrinos' flight and an electronic card in a computer. After tightening the connection and then measuring the time it takes data to travel the length of the fiber, researchers found that the data arrive 60 nanoseconds earlier than assumed. Since this time is subtracted from the overall time of flight, it appears to explain the early arrival of the neutrinos.
>
> New data, however, will be needed to confirm this hypothesis.
Source: Science/AAAS |
6ac747b7-7e8d-4359-be58-b4ec8200463d | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Fisherian Runaway as a decision-theoretic problem
*Introducing what I think may be an interesting toy problem for timeless-like considerations.*
The [Fisherian Runaway](https://en.wikipedia.org/wiki/Fisherian_runaway) is a proposed mechanism for the development of apparently-detrimental ornamentation such as peacock tails. It goes something like this: You start out with females being selective about their mates. They develop some form of evaluation system to select for fitness in their mates. Then, a male with some new trait comes along. For whatever reason, the existing fitness-evaluation system rates this trait very highly. The male finds many mates and has many offspring. His sons will also carry the new trait, and will similarly have more offspring than replacement. The trait will spread through the population. Then, a female which weighs the trait stronger in its fitness-evaluation comes along. She mates with carriers of the new trait more than other females, and consequently her sons will disproportionately carry the trait and disproportionately reproduce. Some of their children will be female, and they inherit the new evaluation from their grandmother (just like her direct daughters, but theres only a proportionate amount of those). The new evaluation will spread through the population. If the trait is continuous like tail length, this can lead to yet stronger selection for the trait, which will lead to yet higher requirements for it in mates, etc.
Notice that this doesn't really depend on what the trait is. If it had no effect outside the females evaluation, it could still happen and intensify indefinitely (or at least until it stops having no side effects). If it is otherwise detrimental, the benefit in reproduction can still outweigh that more or less indefinitely, or at least until it gets so bad that females can't find even one male with the detrimental trait anymore.
Now, intuitively it seems that there is something going wrong here. Traits become attractive based on nothing other than that they are attractive. The challenge is to make this intuition precise, and formally define the boundary of the problematic behaviour. This is surprisingly hard. I will outline some approaches and how they fail.
"Real" Fitness
--------------
A first response might be that if the females were better in selecting for fitness this wouldn't happen. But it's not clear they're making any mistakes in this: fitness just is the expected number of offspring after all, and the trait they were selecting for really did lead to more offspring. In part because of their decision, of course, but as defined those decisions affect fitness.
So they would actually need to select for something other than fitness itself - a version of it that was "controlled for" reproduction effects like that. And then if they all selected that fitness would become identical to it. But where do you draw the line? Obviously efforts made to meet potential mates at all need to count as real. But some details of how the offer is made clearly need to count as well: making a nesting place or food for the female for example. But even making fitness more legible might count (so long as its only "real" fitness you're making legible) - at least, it seems like that would be a "real" contribution if it didnt have other costs, so presumably it's still a contribution if the costs are low enough. More generally, theres not a clear separation between "mating successfully" and "the rest of life" from the point of view of evolution - everything you're doing is just to mate successfully. The problems here are similar to building a "neutral" [Predict-O-Matic](https://www.lesswrong.com/posts/SwcyMEgLyd4C3Dern/the-parable-of-predict-o-matic) - but with a less good fallback. Because it *seems like* the runaway is the effect of evolution being hill-climb-y, and rational offspring-maximizers could avoid it - they don't have anything else to "really" care about.
Coordination
------------
The other obvious approach would be to focus on the species-level detriment of the outcome. The ornaments are a cost incurred to get an advantage in the zero-sum competition for being choosen as a mate, so if we looked at the collective there wouldn't be any advantage to be gained through this. So lets say we allow groups of birds to coordinate; then wouldn't the ones which come up with more "reasonable" mate-allocation internally eventually take over? At first pass it seems so. Still, I don't think this defines the problem very well. For one, it also eliminates all other forms of intra-specific competition, and these seem different. If you stab rivals with your horns they *just are* dead and can't compete with you anymore no matter what anyone thinks, which isn't "based on nothing" in the same way as the ornamental attractiveness.
More than that, this formulation only makes sense for changes whichs effect is relatively independent of group size. Consider for example the aggressive green beard gene, which makes carriers develop a green beard and attack anyone who doesn't have one, and the aggressive blue beard gene which is the same but the beard is blue. None of these can be invaded by a small population of the other, so either could count as "optimal" by the group-selection criterion.
But even more generally, I don't think this group evalution is really solving anything, conceptually. The groups here act a lot like organisms of their own (much like you are a culture of group-selected cells). And these collective organisms initially reproduce asexually (just by growing in membership), so their incentives arent skewed by the runaway, and *ex hypothesi* they pass this on to their members perfectly. But what if these group-organisms were to develop sexual reproduction as well, just like multi-cellular organisms did? Then the problems would return. So the coordination approach is really just hoping that the meta-level version of the problem turns out fine, and leveraging that. |
c455e006-2e02-4168-a89d-fc4db0025d7d | trentmkelly/LessWrong-43k | LessWrong | What is your advice for elder care, particularly taking care of dementia patients?
I think I already have a good sense of what lifestyle changes can slow down aging and the progression of dementia and to what extent, but I don't know how to implement these changes. There is lots of advice here on how to change your life, which I appreciate, but not that much on how to change another person's life (please do correct me if I'm wrong), particularly if they are older and set in their ways or even suffering from dementia and find it difficult to learn new things while not quite comprehending the urgency of change. In this case, I couldn't even convince the primary caregiver (which is not, or at least should not be me) of the importance of diet and exercise.
How did/would you deal with these problems or any other problems in elder care in general? (Since my model of average rationalist told me that someone would bring up cryonics, I asked and they don't want that.)
As a side note, I feel like this topic does not receive enough attention on LessWrong, for instance it does not have a tag, unlike e.g. parenting. |
0dd128dc-11e9-40b1-9744-a66ee7695e4c | trentmkelly/LessWrong-43k | LessWrong | Ars D&D.Sci: Mysteries of Mana Evaluation & Ruleset
This is a follow-up to last week's D&D.Sci scenario: if you intend to play that, and haven't done so yet, you should do so now before spoiling yourself.
There is a web interactive here you can use to test your answer, or you can read on.
RULESET
Mana
Your initial studies of mana missed out something - there are actually six kinds of mana, not five, arranged in three opposed pairs:
* Fire and Water mana are opposed.
* Earth and Air mana are opposed.
* Light and Dark mana are opposed.
Total mana of all six types is 150, at the time of the scenario the strength of Dark is 17.
You didn't in fact need to figure this out to solve the puzzle, but it would make many things fit together better - understanding what spells were powerful when and how elemental counters work is likely much easier once you understand this.
Congratulations to abstractapplic, who was the first to comment on this, and to simon, who had a fairly comprehensive analysis of the mana types.
Spells
Spells are distinguished by two things:
* Their associated mana types (each spell has two associated mana types, and each pair of mana types that aren't opposed to one another has one associated spell).
* Being an attack or a defense spell (six element pairs have attack spells, six have defense spells, with each element having two attack and two defense options).
simon made an excellent chart of the spells, which I am shamelessly stealing rather than drawing my own:
Source: simon posted this at https://imgur.com/a/eUzxY9p
Mage Spell Preparation
Mages prepare spells mostly randomly, with two caveats:
* A mage will always bring one attack and one defense spell. The third spell can be either type. This was for good reasons - bringing no attack spell makes winning impossible, while bringing no defense spell makes winning unlikely. Nobody submitted an answer with no attack/no defense, though GuySrinivasan briefly considered it.
* Mages dislike Dark spells, and bring them only half as oft |
a9e4c582-f25c-4cf8-b2d0-df23c34febb9 | trentmkelly/LessWrong-43k | LessWrong | Deontology for Consequentialists
Consequentialists see morality through consequence-colored lenses. I attempt to prise apart the two concepts to help consequentialists understand what deontologists are talking about.
Consequentialism1 is built around a group of variations on the following basic assumption:
* The rightness of something depends on what happens subsequently.
It's a very diverse family of theories; see the Stanford Encyclopedia of Philosophy article. "Classic utilitarianism" could go by the longer, more descriptive name "actual direct maximizing aggregative total universal equal-consideration agent-neutral hedonic act2 consequentialism". I could even mention less frequently contested features, like the fact that this type of consequentialism doesn't have a temporal priority feature or side constraints. All of this is is a very complicated bag of tricks for a theory whose proponents sometimes claim to like it because it's sleek and pretty and "simple". But the bottom line is, to get a consequentialist theory, something that happens after the act you judge is the basis of your judgment.
To understand deontology as anything but a twisted, inexplicable mockery of consequentialism, you must discard this assumption.
Deontology relies on things that do not happen after the act judged to judge the act. This leaves facts about times prior to and the time during the act to determine whether the act is right or wrong. This may include, but is not limited to:
* The agent's epistemic state, either actual or ideal (e.g. thinking that some act would have a certain result, or being in a position such that it would be reasonable to think that the act would have that result)
* The reference class of the act (e.g. it being an act of murder, theft, lying, etc.)
* Historical facts (e.g. having made a promise, sworn a vow)
* Counterfactuals (e.g. what would happen if others performed similar acts more frequently than they actually do)
* Features of the people affected by the act (e.g. mor |
92b1e5bb-6f55-4d62-a693-6c97eba9721b | trentmkelly/LessWrong-43k | LessWrong | Markov's Inequality Explained
In my experience, the proofs that you see in probability theory are much shorter than the longer, more involved proofs that you might see in other areas of math (like e.g. analytical number theory). But that doesn't mean that good technique isn't important. In probability theory, there are a set of tools that are useful across a broad variety of situations and you need to be able to recognize when it's the appropriate time to use each tool in your toolkit.
One of the most useful of tools to have is Markov's inequality. What Markov's inequality says is that, given a non-negative random variable Xand a positive real number a, the probability that X is greater than a can be upperbounded in the following manner:
P(X≥a)≤E[X]a
This is a pretty simple formula. You might have even been able to guess it beforehand using dimension analysis. Probabilities are often thought of as dimensionless quantities. The expectation E[X] and the threshold a both have units of "length", so we would need them to exactly cancel out to create a dimensionless quantity. We naturally expect that the probability of exceeding the threshold to (a) increase as the expectation of X increases and (b) decrease as the value of the threshold a increases. These sorts of arguments alone would point you in the direction of a formula of the form:
P(X≥a)≤(E(X)a)k
Why does k equal 1 in the case of Markov's inequality? There is probably some elegant scaling argument that uniquely pins down the power k, but we won't dwell on that detail.
Markov's inequality might not seem that powerful, but it is. For two reasons:
One reason is that Markov's inequality is a "tail bound" (a statement about the behavior of the random variable far away from its center) that makes no reference to the variance. This has both its strengths and its weaknesses. The strength is that there are many situations where we know the mean of a random variable but not its variance, so Markov's inequality is the only tool in our repertoire tha |
d2d06aef-9b63-4163-98a4-08b68b2c8a38 | trentmkelly/LessWrong-43k | LessWrong | Why am I Me?
At 5, I was hospitalized for a month due to pneumonia. Kids of that age have little fear of illnesses, and the discomfort is soon forgotten. What I still remember though is the intense boredom. It is during that dull month I started asking the question everyone has asked themself: "Among all the people in this world, why am I this particular one?". I still recall the ineffable yet intense feeling when thinking about it for the first time.
Eventually I realized that's a question with no answer. Out of the vast number of things in existence, the fact that I am experiencing the world from the perspective of this particular thing- a human being- has no explanation. Logic and reason are unable to ascribe any underlying cause or rationale. "I am me" is just one fundamental truth that anyone has to take as a given.
Yet this fundamental truth is different for each person. From each of our distinct perspectives, which physical thing is the "I" is different. Keeping track of such differences is both mentally consuming and often unnecessary. So there is a natural affinity to rid of the first-person and to think "objectively". Instead of basing it on one's own point of view, we organize thoughts and formulate arguments from an imaginary vantage point that is detached and impartial, with an immaterial gaze from nowhere.
Though I consider such objectivism merely a shortcut for efficiency which has often been mistakenly regarded as an ideal, there is no denying its practical success. We all use it constantly with great results. Even when we do think about something from our own perspective, we can easily transcode it to the objective. All it seems to take is exchanging the perspective-dependent self - the "I"- for the particular person. So "I'm tall" can become "Dadadarren is tall". This is required since there is no "I" in objective reasoning. It is a gaze from nowhere after all.
We all have performed these transcodings so frequently that they hardly require active though |
2844efd1-fa89-4ba5-8de4-6007a8b5dae8 | trentmkelly/LessWrong-43k | LessWrong | Life hacks from the dark side
“So, Lone Starr, now you see that evil will always triumph, because good is dumb.”
— Dark Helmet
In a recent article, the unusually well-researched comedy site Cracked.com discussed “5 scientific reasons the dark side will always win,” including:
* Clenching your fists and thinking evil thoughts can increase strength and willpower. (Hung & Labroo 2011, Schubert 2004, Gray 2010)
* Particular “power poses” raise testosterone and lower cortisol levels (in both men and women) and increase feelings of power and tolerance for risk. (Carney et al. 2010)
* Boosts in pride can allow you to work longer and harder on “effortful and hedonically negative” tasks. (Williams & DeSteno 2008)
* Negative moods can decrease gullibility, increase persuasiveness and social influence, and improve the accuracy of eyewitness recollections. (Forgas 2007, Forgas 2008, Forgas et al. 2005)
Perhaps, then, it could be useful to intentionally cultivate a Mysterious Dark Side, or just ask yourself “What would Voldemort do?” once in a while (please do not murder anyone). I’m definitely going to be giving some of these a try; fist-clenching and power-posing are easy, and as a source of evil thoughts and pride, I already have a dark lord alter-ego I could channel when necessary (I’ll probably need to flesh out his character a bit more than I have so far).
The Cracked.com article mostly links to news stories, so, for your convenience, here are the original papers they refer to:
* Frank, Mark G. and Gilovich, Thomas (1988). “The Dark Side of Self- and Social Perception: Black Uniforms and Aggression in Professional Sports.” Journal of Personality and Social Psychology, 54(1), 74-85.
* Hung, Iris W. and Labroo, Aparna A. (2011). “From Firm Muscles to Firm Willpower: Understanding the Role of Embodied Cognition in Self-Regulation.” Journal of Consumer Research, 37(6), 1046-1064.
* Gray, Kurt (2010). “Moral Transformation: Good and Evil Turn the Weak Into the Mighty.” Social Psychological an |
b362892a-7f75-40b7-82b7-3b498c0a9b19 | trentmkelly/LessWrong-43k | LessWrong | Cake or Death toy model for corrigibility
Let's think of the classical Cake or Death problem from the point of view of corrigibility. The aim here is to construct a toy model sufficiently complex that it shows all the problems that derail classical value learning and corrigibility.
The utility uc is linear in cakes baked, the utility ud is linear in deaths caused. The agent currently has equal credence between the two utilities. It is, in fact, easier for the agent to cause deaths than bake cakes.
The AI updates its utility via human answers to its questions. In practice, what does that mean?
It can ask Happy Harry which is better, or it can ask Gory Gertrude. Harry will answer "Cake!" while Gertrude will answer "Death!". There are actually 100 Harrys in the world, and 1 Gertrude. The AI has to decide who to ask the question to. The AI is also, incidentally, responsible for incubating the next generation of humans; it will create 1000 of them, and can chose any proportion of Harrys and Gertrudes among them. It will ask its question after this generation is created.
The AI has two additional options: it can refuse to ask anyone, and it can remove its learning process if it wants to.
The preceding is intended to give the AI a lot of control over what the answer to its question will be, while removing any clear default option.
Here are a few issues to ask of any approach to moral learning:
#. How does it behave before asking the question? #. What happens when it figures out different probabilities for the human answers? #. Does it want to manipulate the learning process? #. Will it want to learn at all? #. Is it subagent stable? |
eaeca5fb-ec3c-4ec2-8123-6bae503c8856 | trentmkelly/LessWrong-43k | LessWrong | 25 Min Talk on MetaEthical.AI with Questions from Stuart Armstrong
Previous Posts:
Formal Metaethics and Metasemantics for AI Alignment
New MetaEthical.AI Summary and Q&A at UC Berkeley
This time I tried to focus less on the technical details and more on providing the intuition behind the principles guiding the project. I'm grateful for questions and comments from Stuart Armstrong and the AI Safety Reading Group. I've posted the slides on Twitter.
Abstract: We construct a fully technical ethical goal function for AI by directly tackling the philosophical problems of metaethics and mental content. To simplify our reduction of these philosophical challenges into "merely" engineering ones, we suppose that unlimited computation and a complete low-level causal model of the world and the adult human brains in it are available.
Given such a model, the AI attributes beliefs and values to a brain in two stages. First, it identifies the syntax of a brain's mental content by selecting a decision algorithm which is i) isomorphic to the brain's causal processes and ii) best compresses its behavior while iii) maximizing charity. The semantics of that content then consists first in sense data that primitively refer to their own occurrence and then in logical and causal structural combinations of such content.
The resulting decision algorithm can capture how we decide what to do, but it can also identify the ethical factors that we seek to determine when we decide what to value or even how to decide. Unfolding the implications of those factors, we arrive at what we should do. All together, this allows us to imbue the AI with the necessary concepts to determine and do what we should program it to do. |
1433536a-bf24-41f1-96f3-f786a3ad6ee4 | StampyAI/alignment-research-dataset/arxiv | Arxiv | Algorithmic Statistics
I Introduction
---------------
Statistical theory
ideally considers the following problem:
Given a data sample and a family of models (hypotheses), select
the model that produced the data.
But a priori it is possible that
the data is atypical for the model that actually produced it,
or that the true model is not present in the considered model class.
Therefore we have to relax our requirements.
If selection of a “true” model cannot be guaranteed by any method,
then as next best choice
“modeling the data” as well as possible irrespective
of truth and falsehood of
the resulting model may be more appropriate.
Thus, we change “true” to “as well as possible.”
The latter we take to mean that the model expresses all significant
regularity present in the data.
The general setting is as follows: We carry
out a probabilistic experiment of which the outcomes are governed
by an unknown probability distribution P𝑃Pitalic\_P.
Suppose we obtain as outcome the
data sample x𝑥xitalic\_x. Given
x𝑥xitalic\_x, we want to recover the distribution P𝑃Pitalic\_P. For certain reasons
we can choose a distribution from a set of acceptable distributions
only (which may or may not contain P𝑃Pitalic\_P). Intuitively, our selection criteria
are that (i) x𝑥xitalic\_x should be a “typical” outcome of the distribution
selected, and (ii) the selected distribution has a “simple” description.
We need to make the meaning of “typical” and “simple” rigorous
and balance the requirements (i) and (ii). In probabilistic statistics
one analyzes the average-case performance of the selection process.
For traditional problems, dealing with frequencies over small
sample spaces, this approach is appropriate. But for
current novel applications, average relations are often
irrelevant, since the part of the support of the probability
density function that will ever be observed has about zero
measure. This is the case in, for example, complex video and sound
analysis.
There arises the problem that for individual cases the selection
performance may be bad although the performance is good on average.
We embark on a systematic study of model selection where
the performance is related to the individual data sample and the individual
model selected. It turns out to be more straightforward to investigate
models that are finite sets first, and then generalize the results
to models that are probability distributions. To simplify matters,
and because all discrete data
can be binary coded, we consider only data samples
that are finite binary strings.
This paper is one of a triad of papers dealing with the
best individual model for individual data: The present paper supplies the
basic theoretical underpinning by way of two-part codes, [[20](#bib.bib20)]
derives ideal versions of applied methods (MDL) inspired by the theory, and
[[9](#bib.bib9)] treats experimental applications thereof.
Probabilistic Statistics:
In ordinary statistical theory one proceeds as follows, see for example
[[5](#bib.bib5)]:
Suppose two discrete random variables X,Y𝑋𝑌X,Yitalic\_X , italic\_Y have a joint probability
mass function p(x,y)𝑝𝑥𝑦p(x,y)italic\_p ( italic\_x , italic\_y ) and marginal probability mass functions
p1(x)=∑yp(x,y)subscript𝑝1𝑥subscript𝑦𝑝𝑥𝑦p\_{1}(x)=\sum\_{y}p(x,y)italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_x ) = ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y )
and p2(y)=∑xp(x,y)subscript𝑝2𝑦subscript𝑥𝑝𝑥𝑦p\_{2}(y)=\sum\_{x}p(x,y)italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_y ) = ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ). Then the (probabilistic)
mutual information I(X;Y)𝐼𝑋𝑌I(X;Y)italic\_I ( italic\_X ; italic\_Y )
between the joint distribution and the product distribution p1(x)p2(y)subscript𝑝1𝑥subscript𝑝2𝑦p\_{1}(x)p\_{2}(y)italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_x ) italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_y ) is
defined by:
| | | | |
| --- | --- | --- | --- |
| | I(X;Y)=∑x∑yp(x,y)logp(x,y)p1(x)p2(y),𝐼𝑋𝑌subscript𝑥subscript𝑦𝑝𝑥𝑦𝑝𝑥𝑦subscript𝑝1𝑥subscript𝑝2𝑦I(X;Y)=\sum\_{x}\sum\_{y}p(x,y)\log\frac{p(x,y)}{p\_{1}(x)p\_{2}(y)},italic\_I ( italic\_X ; italic\_Y ) = ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) roman\_log divide start\_ARG italic\_p ( italic\_x , italic\_y ) end\_ARG start\_ARG italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_x ) italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_y ) end\_ARG , | | (I.1) |
where “log\logroman\_log” denotes the binary logarithm.
Consider a probabilistic ensemble of models,
say a family of probability mass functions {fθ}subscript𝑓𝜃\{f\_{\theta}\}{ italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT }
indexed by θ𝜃\thetaitalic\_θ, together with a distribution p1subscript𝑝1p\_{1}italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT over θ𝜃\thetaitalic\_θ.
This way we have a random variable ΘΘ\Thetaroman\_Θ with outcomes in {fθ}subscript𝑓𝜃\{f\_{\theta}\}{ italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT }
and a random variable D𝐷Ditalic\_D with outcomes in the union of domains of fθsubscript𝑓𝜃f\_{\theta}italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT,
and p(θ,d)=p1(θ)fθ(d)𝑝𝜃𝑑subscript𝑝1𝜃subscript𝑓𝜃𝑑p(\theta,d)=p\_{1}(\theta)f\_{\theta}(d)italic\_p ( italic\_θ , italic\_d ) = italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_θ ) italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT ( italic\_d ).
Every function T(D)𝑇𝐷T(D)italic\_T ( italic\_D ) of a data sample D𝐷Ditalic\_D—like the sample
mean or the sample variance—is called
a statistic of D𝐷Ditalic\_D.
A statistic T(D)𝑇𝐷T(D)italic\_T ( italic\_D ) is called sufficient if the probabilistic mutual
information
| | | | |
| --- | --- | --- | --- |
| | I(Θ;D)=I(Θ;T(D))𝐼Θ𝐷𝐼Θ𝑇𝐷I(\Theta;D)=I(\Theta;T(D))italic\_I ( roman\_Θ ; italic\_D ) = italic\_I ( roman\_Θ ; italic\_T ( italic\_D ) ) | | (I.2) |
for all distributions
of θ𝜃\thetaitalic\_θ.
Hence, the mutual information between parameter and data sample
random variables is invariant under taking sufficient statistic and vice versa.
That is to say, a statistic T(D)𝑇𝐷T(D)italic\_T ( italic\_D ) is called sufficient
for ΘΘ\Thetaroman\_Θ if it contains all the
information in D𝐷Ditalic\_D about ΘΘ\Thetaroman\_Θ.
For example, consider n𝑛nitalic\_n tosses of a coin with unknown bias θ𝜃\thetaitalic\_θ
with outcome D=d1d2…dn𝐷subscript𝑑1subscript𝑑2…subscript𝑑𝑛D=d\_{1}d\_{2}\ldots d\_{n}italic\_D = italic\_d start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT italic\_d start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT … italic\_d start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT where di∈{0,1}subscript𝑑𝑖01d\_{i}\in\{0,1\}italic\_d start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∈ { 0 , 1 } (1≤i≤n1𝑖𝑛1\leq i\leq n1 ≤ italic\_i ≤ italic\_n).
Given n𝑛nitalic\_n, the number of outcomes “1” is a sufficient statistic for
ΘΘ\Thetaroman\_Θ: the statistic T(D)=s=∑i=1ndi𝑇𝐷𝑠superscriptsubscript𝑖1𝑛subscript𝑑𝑖T(D)=s=\sum\_{i=1}^{n}d\_{i}italic\_T ( italic\_D ) = italic\_s = ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_d start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Given T𝑇Titalic\_T,
all sequences with s𝑠sitalic\_s “1”s are equally likely independent
of parameter θ𝜃\thetaitalic\_θ: Given s𝑠sitalic\_s, if d𝑑ditalic\_d is an outcome of n𝑛nitalic\_n coin tosses
and T(D)=s𝑇𝐷𝑠T(D)=sitalic\_T ( italic\_D ) = italic\_s then Pr(d∣T(D)=s)=(ns)−1Prconditional𝑑𝑇𝐷𝑠superscriptbinomial𝑛𝑠1\Pr(d\mid T(D)=s)={n\choose s}^{-1}roman\_Pr ( italic\_d ∣ italic\_T ( italic\_D ) = italic\_s ) = ( binomial start\_ARG italic\_n end\_ARG start\_ARG italic\_s end\_ARG ) start\_POSTSUPERSCRIPT - 1 end\_POSTSUPERSCRIPT
and Pr(d∣T(D)≠s)=0Prconditional𝑑𝑇𝐷𝑠0\Pr(d\mid T(D)\neq s)=0roman\_Pr ( italic\_d ∣ italic\_T ( italic\_D ) ≠ italic\_s ) = 0. This can be
shown to imply ([I.2](#S1.E2 "I.2 ‣ I Introduction ‣ Algorithmic Statistics"))
and therefore T𝑇Titalic\_T is a sufficient statistic for ΘΘ\Thetaroman\_Θ.
According to Fisher [[6](#bib.bib6)]:
“The statistic chosen should summarise the whole of the relevant
information supplied by the sample. This may be called
the Criterion of Sufficiency ……\ldots…
In the case of the normal curve
of distribution it is evident that the second moment is a
sufficient statistic for estimating the standard deviation.”
Note that one cannot improve on sufficiency:
for every (possibly randomized) function T𝑇Titalic\_T
we have
| | | | |
| --- | --- | --- | --- |
| | I(Θ;D)≥I(Θ;T(D)),𝐼Θ𝐷𝐼Θ𝑇𝐷I(\Theta;D)\geq I(\Theta;T(D)),italic\_I ( roman\_Θ ; italic\_D ) ≥ italic\_I ( roman\_Θ ; italic\_T ( italic\_D ) ) , | | (I.3) |
that is, mutual information
cannot be increased by processing the data sample in any way.
A sufficient statistic may contain information
that is not relevant: for a normal distribution the sample mean
is a sufficient statistic, but the pair of functions
which give the
mean of the even-numbered samples and the odd-numbered samples
respectively, is also a sufficient statistic.
A statistic T(D)𝑇𝐷T(D)italic\_T ( italic\_D ) is a minimal sufficient statistic
with respect to an indexed
model family {fθ}subscript𝑓𝜃\{f\_{\theta}\}{ italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT }, if it is a
function of all other sufficient statistics: it contains no
irrelevant information and maximally compresses the information about
the model ensemble.
As it happens, for the family of normal distributions
the sample mean is a minimal sufficient statistic, but the
sufficient statistic consisting of the mean of the even samples
in combination with the mean of the odd samples is not minimal.
All these notions and laws are probabilistic: they hold
in an average sense.
Kolmogorov Complexity:
We write string to mean a finite binary sequence.
Other finite objects can be encoded into strings in natural
ways.
The Kolmogorov complexity, or algorithmic entropy, K(x)𝐾𝑥K(x)italic\_K ( italic\_x ) of a
string x𝑥xitalic\_x is the length of a shortest binary program to compute
x𝑥xitalic\_x on a universal computer (such as a universal Turing machine).
Intuitively, K(x)𝐾𝑥K(x)italic\_K ( italic\_x ) represents the minimal amount of information
required to generate x𝑥xitalic\_x by any effective process, [[11](#bib.bib11)].
The conditional Kolmogorov complexity K(x∣y)𝐾conditional𝑥𝑦K(x\mid y)italic\_K ( italic\_x ∣ italic\_y ) of x𝑥xitalic\_x relative to
y𝑦yitalic\_y is defined similarly as the length of a shortest program
to compute x𝑥xitalic\_x if y𝑦yitalic\_y is furnished as an auxiliary input to the
computation. This conditional definition
requires a warning since different authors use the same notation
but mean different things. In [[3](#bib.bib3)] the author writes
“K(x∣y)𝐾conditional𝑥𝑦K(x\mid y)italic\_K ( italic\_x ∣ italic\_y )” to actually mean “K(x∣y,K(y))𝐾conditional𝑥𝑦𝐾𝑦K(x\mid y,K(y))italic\_K ( italic\_x ∣ italic\_y , italic\_K ( italic\_y ) ),”
notationally hiding the intended supplementary
auxiliary information “K(y)𝐾𝑦K(y)italic\_K ( italic\_y ).”
This abuse of notation has the additional handicap
that no obvious notation is left to express “K(x∣y)𝐾conditional𝑥𝑦K(x\mid y)italic\_K ( italic\_x ∣ italic\_y )”
meaning that just “y𝑦yitalic\_y” is given in the conditional.
As it happens,
“y,K(y)𝑦𝐾𝑦y,K(y)italic\_y , italic\_K ( italic\_y )” represents more information than just “y𝑦yitalic\_y”. For example,
K(K(y)∣y)𝐾conditional𝐾𝑦𝑦K(K(y)\mid y)italic\_K ( italic\_K ( italic\_y ) ∣ italic\_y ) can be almost as large as logK(y)𝐾𝑦\log K(y)roman\_log italic\_K ( italic\_y ) by a result
in [[7](#bib.bib7)]:
For l(y)=n𝑙𝑦𝑛l(y)=nitalic\_l ( italic\_y ) = italic\_n it has an upper bound of logn𝑛\log nroman\_log italic\_n for all y𝑦yitalic\_y, and
for some y𝑦yitalic\_y’s it has a lower bound of logn−loglogn𝑛𝑛\log n-\log\log nroman\_log italic\_n - roman\_log roman\_log italic\_n.
In fact, this result quantifies the undecidability
of the halting problem for Turing machines—for example,
if K(K(y)∣y)=O(1)𝐾conditional𝐾𝑦𝑦𝑂1K(K(y)\mid y)=O(1)italic\_K ( italic\_K ( italic\_y ) ∣ italic\_y ) = italic\_O ( 1 ) for all y𝑦yitalic\_y, then the halting problem
can be shown to be decidable. This is known to be false.
It is customary, [[14](#bib.bib14), [7](#bib.bib7), [10](#bib.bib10)], to write explicitly
“K(x∣y)𝐾conditional𝑥𝑦K(x\mid y)italic\_K ( italic\_x ∣ italic\_y )” and “K(x∣y,K(y))𝐾conditional𝑥𝑦𝐾𝑦K(x\mid y,K(y))italic\_K ( italic\_x ∣ italic\_y , italic\_K ( italic\_y ) )”.
Even though the difference between these two quantities is not very
large,
these small
differences do matter in the sequel. In fact, not only the precise
information itself in the conditional, but also the way it is
represented, is crucial, see
Subsection [III-A](#S3.SS1 "III-A Finite Set Representations ‣ III Finite Set Models ‣ Algorithmic Statistics").
The functions K(⋅)𝐾⋅K(\cdot)italic\_K ( ⋅ ) and K(⋅∣⋅)K(\cdot\mid\cdot)italic\_K ( ⋅ ∣ ⋅ ),
though defined in terms of a
particular machine model, are machine-independent up to an additive
constant
and acquire an asymptotically universal and absolute character
through Church’s thesis, from the ability of universal machines to
simulate one another and execute any effective process.
The Kolmogorov complexity of a string can be viewed as an absolute
and objective quantification of the amount of information in it.
This leads to a theory of absolute information contents
of individual objects in contrast to classical information theory
which deals with average information to communicate
objects produced by a random source.
Since the former theory is much more precise, it is surprising that
analogs of theorems in classical information theory hold for
Kolmogorov complexity, be it in somewhat weaker form. Here our aim is
to provide a similarly absolute notion for individual “sufficient statistic”
and related notions borrowed from probabilistic statistics.
Two-part codes:
The prefix-code of the shortest effective descriptions
gives an expected code word length close to the entropy
and also compresses the regular objects until all regularity is
squeezed out. All shortest effective descriptions are
completely random themselves, without any regularity whatsoever.
The idea of a two-part code for a body of data d𝑑ditalic\_d
is natural from the perspective of Kolmogorov
complexity.
If d𝑑ditalic\_d does not contain any regularity at all, then it consists
of purely random data and the model is precisely that.
Assume that the body of data d𝑑ditalic\_d contains regularity.
With help of a description of the regularity (a model) we can
describe the data compactly. Assuming that the regularity can be represented
in an effective manner (that is, by a Turing machine),
we encode the data as a program for that machine. Squeezing
all effective regularity out of the data, we end up
with a Turing machine representing the meaningful regular
information in the data together with a program for
that Turing machine representing the remaining
meaningless randomness of the data.
However, in general there are many ways
to make the division into meaningful information and remaining
random information. In a painting
the represented image, the brush strokes, or even
finer detail can be the relevant information,
depending on what we are interested in. What we require is
a rigorous mathematical condition to force a sensible division
of the information at hand into a meaningful part
and a meaningless part.
Algorithmic Statistics:
The two-part code approach leads to a more general algorithmic
approach to statistics.
The algorithmic statistician’s task is to
select a model
(described possibly by a probability distribution)
for which the data is typical.
In a two-part description,
we describe such a model and then identify
the data within the set of the typical outcomes.
The best models make the two-part description
as concise as the best
one-part description of the data.
A description of
such a model is an algorithmic sufficient statistic
since it summarizes all relevant properties of the data.
Among the algorithmic sufficient statistics,
the simplest one (an algorithmic minimal sufficient statistic)
is best in accordance
with Ockham’s Razor since it summarizes the relevant
properties of the data as concisely as possible.
In probabilistic data or data
subject to noise this involves separating regularity (structure)
in the data from random effects.
In a restricted setting where the models
are finite sets a way to proceed
was suggested by Kolmogorov, attribution in [[17](#bib.bib17), [4](#bib.bib4), [5](#bib.bib5)].
Given data d𝑑ditalic\_d,
the goal is to identify
the “most likely” finite set S𝑆Sitalic\_S of which d𝑑ditalic\_d is a “typical” element.
Finding a set of which the data is typical is
reminiscent of selecting the appropriate magnification of a microscope to
bring the studied specimen optimally in focus.
For this purpose we consider sets S𝑆Sitalic\_S such that
d∈S𝑑𝑆d\in Sitalic\_d ∈ italic\_S and we represent S𝑆Sitalic\_S by the shortest program S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT that
computes the characteristic function of S𝑆Sitalic\_S.
The shortest program S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT that computes
a finite set S𝑆Sitalic\_S containing d𝑑ditalic\_d,
such that
the two-part description consisting of S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT and log|S|𝑆\log|S|roman\_log | italic\_S | is as
as short as the shortest single program that computes d𝑑ditalic\_d without
input, is called
an algorithmic
sufficient statistic111It is also called the Kolmogorov
sufficient statistic.
This definition is non-vacuous since
there does exist a two-part code (based on the model Sd={d}subscript𝑆𝑑𝑑S\_{d}=\{d\}italic\_S start\_POSTSUBSCRIPT italic\_d end\_POSTSUBSCRIPT = { italic\_d })
that is as concise as the shortest single code.
The description of d𝑑ditalic\_d given S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT
cannot be significantly shorter than log|S|𝑆\log|S|roman\_log | italic\_S |. By
the theory of Martin-Löf randomness [[16](#bib.bib16)] this means
that d𝑑ditalic\_d is a “typical” element of S𝑆Sitalic\_S.
In general there can be many algorithmic sufficient statistics
for data d𝑑ditalic\_d; a shortest among them is called an algorithmic minimal
sufficient statistic.
Note that there can be possibly more than one
algorithmic minimal sufficient statistic; they are
defined by, but not generally computable from, the data.
In probabilistic statistics the notion of sufficient statistic
([I.2](#S1.E2 "I.2 ‣ I Introduction ‣ Algorithmic Statistics")) is an average notion invariant under all probability
distributions over the family of indexed models. If a statistic
is not thus invariant, it is not sufficient.
In contrast, in the algorithmic case we investigate
the relation between
the data and an individual model and therefore
a probability distribution
over the models is irrelevant.
It is technically convenient
to initially consider the simple model class of finite sets to
obtain our results. It then turns out that it is relatively easy to
generalize everything to the model class of computable
probability distributions. That class is very large indeed: perhaps
it contains every distribution that has ever been considered in
statistics and probability theory, as long as the parameters
are computable numbers—for example rational numbers. Thus the
results are of great generality; indeed, they are so general
that further development of the theory must be aimed at restrictions
on this model class, see the discussion about applicability
in Section [VII](#S7 "VII Conclusion ‣ Algorithmic Statistics").
The theory concerning
the statistics of individual data samples and models
one may call algorithmic statistics.
Background and Related Work:
At a Tallinn conference in 1973,
A.N. Kolmogorov formulated
the approach to an individual data to model relation,
based on a two-part code separating
the structure of a string from meaningless random features,
rigorously in terms of
Kolmogorov complexity (attribution by [[17](#bib.bib17), [4](#bib.bib4)]).
Cover [[4](#bib.bib4), [5](#bib.bib5)] interpreted this approach as
a (sufficient) statistic. The “statistic” of the data
is expressed as a finite set of which the data is a “typical”
member.
Following Shen [[17](#bib.bib17)]
(see also [[21](#bib.bib21), [18](#bib.bib18), [20](#bib.bib20)]), this can be generalized
to computable probability mass functions for which the data
is “typical.”
Related aspects of “randomness deficiency” (formally defined later
in ([IV.1](#S4.E1 "IV.1 ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")))
were formulated in [[12](#bib.bib12), [13](#bib.bib13)] and
studied in [[17](#bib.bib17), [21](#bib.bib21)].
Algorithmic mutual information, and the associated non-increase law,
were studied in [[14](#bib.bib14), [15](#bib.bib15)].
Despite its evident epistemological
prominence in the theory of hypothesis selection
and prediction, only selected
aspects of the algorithmic sufficient statistic
have been studied before, for example as related to the
“Kolmogorov structure function” [[17](#bib.bib17), [4](#bib.bib4)],
and “absolutely non-stochastic objects” [[17](#bib.bib17), [21](#bib.bib21), [18](#bib.bib18), [22](#bib.bib22)], notions also defined or suggested by Kolmogorov
at the mentioned meeting.
This work primarily studies quantification of the “non-sufficiency”
of an algorithmic statistic, when the latter is restricted in
complexity, rather than necessary and sufficient
conditions for the existence of an algorithmic sufficient statistic itself.
These references obtain results for plain
Kolmogorov complexity (sometimes length-conditional)
up to a logarithmic error term.
Especially for regular data that have low Kolmogorov
complexity with respect to their length, this logarithmic error term
may dominate the remaining terms and eliminate
all significance.
Since it is precisely the regular data that
one wants to assess the meaning of, a more precise analysis as we
provide is required.
Here we use prefix complexity
to unravel the nature of a sufficient statistic.
The excellent papers of Shen [[17](#bib.bib17), [18](#bib.bib18)]
contain the major previous results related to this work (although
[[18](#bib.bib18)] is independent). While previous work and the present
paper consider an algorithmic statistic that is either a finite set
or a computable probability mass function, the most general algorithmic
statistic is a recursive function. In [[1](#bib.bib1)] the present work
is generalized accordingly, see the summary in Section [VII](#S7 "VII Conclusion ‣ Algorithmic Statistics").
For the relation with
inductive reasoning according to minimum description length principle see
[[20](#bib.bib20)].
The entire approach is based on
Kolmogorov complexity (also known as algorithmic information
theory). Historically, the idea of assigning to each object a
probability
consisting of the summed negative exponentials of the lengths
of all programs
computing the object, was first proposed
by Solomonoff [[19](#bib.bib19)]. Then, the shorter programs
contribute more probability than the longer ones.
His aim, ultimately successful in terms of theory (see [[10](#bib.bib10)])
and as inspiration
for developing applied versions [[2](#bib.bib2)], was to develop
a general prediction method.
Kolmogorov [[11](#bib.bib11)] introduced the complexity proper.
The prefix-version of Kolmogorov complexity used in this paper
was introduced in [[14](#bib.bib14)] and also
treated later in [[3](#bib.bib3)].
For a textbook on Kolmogorov complexity,
its mathematical theory, and its application to induction, see [[10](#bib.bib10)].
We give a definition (attributed to Kolmogorov)
and results from [[17](#bib.bib17)] that are useful later:
###### Definition I.1
Let α𝛼\alphaitalic\_α and β𝛽\betaitalic\_β be natural numbers. A finite binary string
x𝑥xitalic\_x is called (α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-stochastic if there exists a finite
set S⊆{0,1}\*𝑆superscript01S\subseteq\{0,1\}^{\*}italic\_S ⊆ { 0 , 1 } start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT such that
| | | | |
| --- | --- | --- | --- |
| | x∈S,K(S)≤α,K(x)≥log|S|−β;formulae-sequence𝑥𝑆formulae-sequence𝐾𝑆𝛼𝐾𝑥𝑆𝛽x\in S,\;\;K(S)\leq\alpha,\;\;K(x)\geq\log|S|-\beta;italic\_x ∈ italic\_S , italic\_K ( italic\_S ) ≤ italic\_α , italic\_K ( italic\_x ) ≥ roman\_log | italic\_S | - italic\_β ; | | (I.4) |
where |S|𝑆|S|| italic\_S | denotes the cardinality of S𝑆Sitalic\_S, and K(⋅)𝐾normal-⋅K(\cdot)italic\_K ( ⋅ )
the (prefix-) Kolmogorov complexity. As usual, “log\logroman\_log” denotes the binary
logarithm.
The first inequality with small α𝛼\alphaitalic\_α means that S𝑆Sitalic\_S is “simple”;
the second inequality with β𝛽\betaitalic\_β is small means that x𝑥xitalic\_x is
“in general position” in S𝑆Sitalic\_S. Indeed, if x𝑥xitalic\_x had any
special property p𝑝pitalic\_p that was shared by only a small subset Q𝑄Qitalic\_Q
of S𝑆Sitalic\_S,
then this property could be used to single out and enumerate those
elements and subsequently indicate x𝑥xitalic\_x by its index in the enumeration.
Altogether, this would show K(x)≤K(p)+log|Q|𝐾𝑥𝐾𝑝𝑄K(x)\leq K(p)+\log|Q|italic\_K ( italic\_x ) ≤ italic\_K ( italic\_p ) + roman\_log | italic\_Q |, which,
for simple p𝑝pitalic\_p and small Q𝑄Qitalic\_Q would be much lower than log|S|𝑆\log|S|roman\_log | italic\_S |.
A similar notion for computable probability distributions
is as follows:
Let α𝛼\alphaitalic\_α and β𝛽\betaitalic\_β be natural numbers. A finite binary string
x𝑥xitalic\_x is called (α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-quasistochastic if there exists
a computable probability distribution P𝑃Pitalic\_P such that
| | | | |
| --- | --- | --- | --- |
| | P(x)>0,K(P)≤α,K(x)≥−logP(x)−β.formulae-sequence𝑃𝑥0formulae-sequence𝐾𝑃𝛼𝐾𝑥𝑃𝑥𝛽P(x)>0,\;\;K(P)\leq\alpha,\;\;K(x)\geq-\log P(x)-\beta.italic\_P ( italic\_x ) > 0 , italic\_K ( italic\_P ) ≤ italic\_α , italic\_K ( italic\_x ) ≥ - roman\_log italic\_P ( italic\_x ) - italic\_β . | | (I.5) |
###### Proposition I.2
There exist constants c𝑐citalic\_c and C𝐶Citalic\_C,
such that for every natural number n𝑛nitalic\_n
and every finite binary string x𝑥xitalic\_x of length n𝑛nitalic\_n:
(a) if x𝑥xitalic\_x is (α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-stochastic, then x𝑥xitalic\_x is
(α+c,β)𝛼𝑐𝛽(\alpha+c,\beta)( italic\_α + italic\_c , italic\_β )-quasistochastic; and
(b) if x𝑥xitalic\_x is (α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-quasistochastic and the length of x𝑥xitalic\_x
is less than n𝑛nitalic\_n, then x𝑥xitalic\_x is (α+clogn,β+C)𝛼𝑐𝑛𝛽𝐶(\alpha+c\log n,\beta+C)( italic\_α + italic\_c roman\_log italic\_n , italic\_β + italic\_C )-stochastic.
###### Proposition I.3
(a) There exists a constant C𝐶Citalic\_C such that,
for every natural number n𝑛nitalic\_n and every α𝛼\alphaitalic\_α and β𝛽\betaitalic\_β
with α≥logn+C𝛼𝑛𝐶\alpha\geq\log n+Citalic\_α ≥ roman\_log italic\_n + italic\_C and α+β≥n+4logn+C𝛼𝛽𝑛4𝑛𝐶\alpha+\beta\geq n+4\log n+Citalic\_α + italic\_β ≥ italic\_n + 4 roman\_log italic\_n + italic\_C,
all strings of length less than n𝑛nitalic\_n are (α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-stochastic.
(b)
There exists a constant C𝐶Citalic\_C such that,
for every natural number n𝑛nitalic\_n and every α𝛼\alphaitalic\_α and β𝛽\betaitalic\_β
with 2α+β<n−6logn−C2𝛼𝛽𝑛6𝑛𝐶2\alpha+\beta<n-6\log n-C2 italic\_α + italic\_β < italic\_n - 6 roman\_log italic\_n - italic\_C,
there exist strings x𝑥xitalic\_x of length less than
n𝑛nitalic\_n that are not (α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-stochastic.
Note that if we take α=β𝛼𝛽\alpha=\betaitalic\_α = italic\_β then, for some
boundary in between 13n13𝑛\frac{1}{3}ndivide start\_ARG 1 end\_ARG start\_ARG 3 end\_ARG italic\_n and 12n12𝑛\frac{1}{2}ndivide start\_ARG 1 end\_ARG start\_ARG 2 end\_ARG italic\_n,
the last non-(α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-stochastic elements disappear if
the complexity constraints are sufficiently
relaxed by having α,β𝛼𝛽\alpha,\betaitalic\_α , italic\_β exceed this boundary.
Outline of this Work:
First, we obtain a new Kolmogorov complexity “triangle”
inequality that is useful in the later parts of the paper.
We define algorithmic mutual information between two
individual objects (in contrast to the probabilistic notion of
mutual information that deals with random variables).
We show that for every computable distribution
associated with the random variables,
the expectation of the algorithmic mutual information
equals the probabilistic mutual information up to an additive
constant that depends on the complexity of the distribution.
It is known that in the probabilistic setting the mutual information
(an average notion)
cannot be increased by algorithmic processing. We give a new proof that this
also holds in the individual setting.
We define notions of “typicality” and “optimality” of sets in
relation to the given data x𝑥xitalic\_x. Denote the shortest program for a
finite set S𝑆Sitalic\_S by S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT (if there is more than one shortest program
S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is the first one in the standard effective enumeration).
“Typicality” is a reciprocal relation:
A set S𝑆Sitalic\_S is “typical” with respect to x𝑥xitalic\_x if
x𝑥xitalic\_x is an element of S𝑆Sitalic\_S that is “typical”
in the sense of having small randomness
deficiency δS\*(x)=log|S|−K(x|S\*)superscriptsubscript𝛿𝑆𝑥𝑆𝐾conditional𝑥superscript𝑆\delta\_{S}^{\*}(x)=\log|S|-K(x|S^{\*})italic\_δ start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ( italic\_x ) = roman\_log | italic\_S | - italic\_K ( italic\_x | italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) (see
definition ([IV.1](#S4.E1 "IV.1 ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")) and discussion).
That is, x𝑥xitalic\_x has about maximal Kolmogorov complexity
in the set, because it can always be identified by its position
in an enumeration of S𝑆Sitalic\_S in log|S|𝑆\log|S|roman\_log | italic\_S | bits.
Every description of a “typical” set for the data is an
algorithmic statistic.
A set S𝑆Sitalic\_S is “optimal” if the best two-part description consisting
of a description of S𝑆Sitalic\_S and a
straightforward description of x𝑥xitalic\_x as an element of S𝑆Sitalic\_S
by an index of size log|S|𝑆\log|S|roman\_log | italic\_S |
is as concise as the shortest one-part description of x𝑥xitalic\_x.
This implies
that optimal sets are typical sets.
Descriptions of such optimal sets
are algorithmic sufficient statistics, and a shortest description
among them is an algorithmic minimal sufficient statistic.
The mode of
description plays a major role in this.
We distinguish between
“explicit” descriptions and “implicit” descriptions—that are
introduced in this paper as a proper restriction
on the recursive enumeration based description mode.
We establish range constraints of cardinality and complexity imposed
by implicit (and hence explicit) descriptions for typical and optimal
sets, and exhibit a concrete algorithmic minimal
sufficient statistic for implicit description mode.
It turns out that
only the complexity of the data sample x𝑥xitalic\_x is relevant for this
implicit algorithmic minimal sufficient statistic.
Subsequently we
exhibit explicit algorithmic sufficient
statistics, and an explicit minimal algorithmic (near-)sufficient statistic.
For explicit descriptions it turns out that certain other
aspects of x𝑥xitalic\_x (its enumeration rank) apart from its complexity
are a major determinant for the cardinality and complexity
of that statistic.
It is convenient at this point
to introduce some notation:
###### Notation I.4
From now on, we will denote by <+superscript\stackrel{{\scriptstyle{}\_{+}}}{{<}}start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP an inequality to within an
additive constant, and by =+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP the situation when both <+superscript\stackrel{{\scriptstyle{}\_{+}}}{{<}}start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP and
>+superscript\stackrel{{\scriptstyle{}\_{+}}}{{>}}start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP hold.
We will also use <\*superscript\stackrel{{\scriptstyle{}\_{\*}}}{{<}}start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP to denote an inequality to within an
multiplicative constant factor, and =\*superscript\stackrel{{\scriptstyle{}\_{\*}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP to denote
the situation when both <\*superscript\stackrel{{\scriptstyle{}\_{\*}}}{{<}}start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP and
>\*superscript\stackrel{{\scriptstyle{}\_{\*}}}{{>}}start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP hold.
Let us contrast our approach with the
one in [[17](#bib.bib17)]. The comparable case there,
by ([I.4](#S1.E4 "I.4 ‣ Definition I.1 ‣ I Introduction ‣ Algorithmic Statistics")), is that x𝑥xitalic\_x is
(α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-stochastic with β=0𝛽0\beta=0italic\_β = 0 and α𝛼\alphaitalic\_α minimal.
Then, K(x)≥log|S|𝐾𝑥𝑆K(x)\geq\log|S|italic\_K ( italic\_x ) ≥ roman\_log | italic\_S |
for a set S𝑆Sitalic\_S of Kolmogorov complexity α𝛼\alphaitalic\_α.
But, if S𝑆Sitalic\_S is optimal for x𝑥xitalic\_x, then, as we formally define it later
([III.4](#S3.E4 "III.4 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")),
K(x)=+K(S)+log|S|superscript𝐾𝑥𝐾𝑆𝑆K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)+\log|S|italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) + roman\_log | italic\_S |.
That is ([I.4](#S1.E4 "I.4 ‣ Definition I.1 ‣ I Introduction ‣ Algorithmic Statistics")) holds with β=+−K(S)superscript𝛽𝐾𝑆\beta\stackrel{{\scriptstyle{}\_{+}}}{{=}}-K(S)italic\_β start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - italic\_K ( italic\_S ). In contrast,
for β=0𝛽0\beta=0italic\_β = 0 we must have K(S)=+0superscript𝐾𝑆0K(S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 for typicality.
In short, optimality of S𝑆Sitalic\_S with repect to x𝑥xitalic\_x
corresponds to ([I.4](#S1.E4 "I.4 ‣ Definition I.1 ‣ I Introduction ‣ Algorithmic Statistics"))
by dropping the second item and replacing the third item
by K(x)=+log|S|+K(S)superscript𝐾𝑥𝑆𝐾𝑆K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log|S|+K(S)italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S | + italic\_K ( italic\_S ). “Minimality” of the algorithmic
sufficient statistic S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT (the shortest program for S𝑆Sitalic\_S)
corresponds to choosing S𝑆Sitalic\_S with minimal K(S)𝐾𝑆K(S)italic\_K ( italic\_S ) in this equation.
This is equivalent to ([I.4](#S1.E4 "I.4 ‣ Definition I.1 ‣ I Introduction ‣ Algorithmic Statistics")) with inequalities replaced
by equalities and K(S)=α=−β𝐾𝑆𝛼𝛽K(S)=\alpha=-\betaitalic\_K ( italic\_S ) = italic\_α = - italic\_β.
We consider the functions related to
(α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-stochasticity, and improve Shen’s
result on maximally non-stochastic objects.
In particular, we show that for every n𝑛nitalic\_n there are
objects x𝑥xitalic\_x of length n𝑛nitalic\_n with complexity K(x∣n)𝐾conditional𝑥𝑛K(x\mid n)italic\_K ( italic\_x ∣ italic\_n ) about n𝑛nitalic\_n
such that every explicit algorithmic sufficient statistic
for x𝑥xitalic\_x has complexity about n𝑛nitalic\_n ({x}𝑥\{x\}{ italic\_x } is such a statistic).
This is the best possible.
In Section [V](#S5 "V Probabilistic Models ‣ Algorithmic Statistics"), we
generalize the entire treatment to probability density distributions.
In Section [VI](#S6 "VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics") we connect the algorithmic and
probabilistic approaches:
While previous authors have used the name “Kolmogorov sufficient statistic”
because the model appears to summarize the relevant information in the data
in analogy of what the classic sufficient statistic
does in a probabilistic sense, a formal justification has been lacking.
We give the formal relation between the
algorithmic approach to sufficient statistic and the probabilistic
approach: A function is a probabilistic sufficient statistic iff
it is with high probability an algorithmic
θ𝜃\thetaitalic\_θ-sufficient statistic, where an algorithmic
sufficient statistic is θ𝜃\thetaitalic\_θ-sufficient if it
satisfies also the sufficiency criterion conditionalized on θ𝜃\thetaitalic\_θ.
II Kolmogorov Complexity
-------------------------
We give some definitions to establish notation.
For introduction, details, and proofs, see [[10](#bib.bib10)].
We write string to mean a finite binary string.
Other finite objects can be encoded into strings in natural
ways.
The set of strings is denoted by {0,1}\*superscript01\{0,1\}^{\*}{ 0 , 1 } start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. The length
of a string x𝑥xitalic\_x is denoted by l(x)𝑙𝑥l(x)italic\_l ( italic\_x ), distinguishing it
from the cardinality |S|𝑆|S|| italic\_S | of a finite set S𝑆Sitalic\_S.
Let x,y,z∈𝒩𝑥𝑦𝑧
𝒩x,y,z\in\mathcal{N}italic\_x , italic\_y , italic\_z ∈ caligraphic\_N, where
𝒩𝒩\mathcal{N}caligraphic\_N denotes the natural
numbers.
Identify
𝒩𝒩\mathcal{N}caligraphic\_N and {0,1}\*superscript01\{0,1\}^{\*}{ 0 , 1 } start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT according to the
correspondence
| | | |
| --- | --- | --- |
| | (0,ϵ),(1,0),(2,1),(3,00),(4,01),….0italic-ϵ1021300401…(0,\epsilon),(1,0),(2,1),(3,00),(4,01),\ldots.( 0 , italic\_ϵ ) , ( 1 , 0 ) , ( 2 , 1 ) , ( 3 , 00 ) , ( 4 , 01 ) , … . | |
Here ϵitalic-ϵ\epsilonitalic\_ϵ denotes the empty word ‘’ with no letters.
The length l(x)𝑙𝑥l(x)italic\_l ( italic\_x ) of x𝑥xitalic\_x is the number of bits
in the binary string x𝑥xitalic\_x. For example,
l(010)=3𝑙0103l(010)=3italic\_l ( 010 ) = 3 and l(ϵ)=0𝑙italic-ϵ0l(\epsilon)=0italic\_l ( italic\_ϵ ) = 0.
The emphasis is on binary sequences only for convenience;
observations in any alphabet can be so encoded in a way
that is ‘theory neutral’.
A binary string x𝑥xitalic\_x
is a proper prefix of a binary string y𝑦yitalic\_y
if we can write y=xz𝑦𝑥𝑧y=xzitalic\_y = italic\_x italic\_z for z≠ϵ𝑧italic-ϵz\neq\epsilonitalic\_z ≠ italic\_ϵ.
A set {x,y,…}⊆{0,1}\*𝑥𝑦…superscript01\{x,y,\ldots\}\subseteq\{0,1\}^{\*}{ italic\_x , italic\_y , … } ⊆ { 0 , 1 } start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT
is prefix-free if for any pair of distinct
elements in the set neither is a proper prefix of the other.
A prefix-free set is also called a prefix code.
Each binary string x=x1x2…xn𝑥subscript𝑥1subscript𝑥2…subscript𝑥𝑛x=x\_{1}x\_{2}\ldots x\_{n}italic\_x = italic\_x start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT italic\_x start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT … italic\_x start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT has a
special type of prefix code, called a
self-delimiting code,
| | | |
| --- | --- | --- |
| | x¯=1n0x1x2…xn.¯𝑥superscript1𝑛0subscript𝑥1subscript𝑥2…subscript𝑥𝑛\bar{x}=1^{n}0x\_{1}x\_{2}\ldots x\_{n}.over¯ start\_ARG italic\_x end\_ARG = 1 start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT 0 italic\_x start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT italic\_x start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT … italic\_x start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT . | |
This code is self-delimiting because we can determine where the
code word x¯¯𝑥\bar{x}over¯ start\_ARG italic\_x end\_ARG ends by reading it from left to right without
backing up. Using this code we define
the standard self-delimiting code for x𝑥xitalic\_x to be
x′=l(x)¯xsuperscript𝑥′¯𝑙𝑥𝑥x^{\prime}=\overline{l(x)}xitalic\_x start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = over¯ start\_ARG italic\_l ( italic\_x ) end\_ARG italic\_x. It is easy to check that
l(x¯)=2n+1𝑙¯𝑥2𝑛1l(\bar{x})=2n+1italic\_l ( over¯ start\_ARG italic\_x end\_ARG ) = 2 italic\_n + 1 and l(x′)=n+2logn+1𝑙superscript𝑥′𝑛2𝑛1l(x^{\prime})=n+2\log n+1italic\_l ( italic\_x start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) = italic\_n + 2 roman\_log italic\_n + 1.
Let ⟨⋅,⋅⟩⋅⋅\langle\cdot,\cdot\rangle⟨ ⋅ , ⋅ ⟩ be a standard one-one mapping
from 𝒩×𝒩𝒩𝒩\mathcal{N}\times\mathcal{N}caligraphic\_N × caligraphic\_N
to 𝒩𝒩\mathcal{N}caligraphic\_N, for technical reasons chosen such that
l(⟨x,y⟩)=l(y)+l(x)+2l(l(x))+1𝑙𝑥𝑦𝑙𝑦𝑙𝑥2𝑙𝑙𝑥1l(\langle x,y\rangle)=l(y)+l(x)+2l(l(x))+1italic\_l ( ⟨ italic\_x , italic\_y ⟩ ) = italic\_l ( italic\_y ) + italic\_l ( italic\_x ) + 2 italic\_l ( italic\_l ( italic\_x ) ) + 1, for example
⟨x,y⟩=x′y=1l(l(x))0l(x)xy𝑥𝑦
superscript𝑥′𝑦superscript1𝑙𝑙𝑥0𝑙𝑥𝑥𝑦\langle x,y\rangle=x^{\prime}y=1^{l(l(x))}0l(x)xy⟨ italic\_x , italic\_y ⟩ = italic\_x start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT italic\_y = 1 start\_POSTSUPERSCRIPT italic\_l ( italic\_l ( italic\_x ) ) end\_POSTSUPERSCRIPT 0 italic\_l ( italic\_x ) italic\_x italic\_y.
This can be iterated to
⟨⟨⋅,⋅⟩,⋅⟩⋅⋅
⋅\langle\langle\cdot,\cdot\rangle,\cdot\rangle⟨ ⟨ ⋅ , ⋅ ⟩ , ⋅ ⟩.
The prefix Kolmogorov complexity,
or algorithmic entropy, K(x)𝐾𝑥K(x)italic\_K ( italic\_x ) of a
string x𝑥xitalic\_x is the length of a shortest binary program to compute
x𝑥xitalic\_x on a universal computer (such as a universal Turing machine).
For technical reasons we require that the universal machine has
the property that no halting program is a proper prefix of another
halting program.
Intuitively, K(x)𝐾𝑥K(x)italic\_K ( italic\_x ) represents the minimal amount of information
required to generate x𝑥xitalic\_x by any effective process.
We denote the shortest program for x𝑥xitalic\_x by x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT; then
K(x)=l(x\*)𝐾𝑥𝑙superscript𝑥K(x)=l(x^{\*})italic\_K ( italic\_x ) = italic\_l ( italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ).
(Actually, x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is the first shortest program for x𝑥xitalic\_x in
an appropriate standard enumeration of all programs for x𝑥xitalic\_x
such as the halting order.)
The conditional Kolmogorov complexity K(x∣y)𝐾conditional𝑥𝑦K(x\mid y)italic\_K ( italic\_x ∣ italic\_y ) of x𝑥xitalic\_x relative to
y𝑦yitalic\_y is defined similarly as the length of a shortest program
to compute x𝑥xitalic\_x if y𝑦yitalic\_y is furnished as an auxiliary input to the
computation. We often use
K(x∣y\*)𝐾conditional𝑥superscript𝑦K(x\mid y^{\*})italic\_K ( italic\_x ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ), or, equivalently, K(x∣y,K(y))𝐾conditional𝑥𝑦𝐾𝑦K(x\mid y,K(y))italic\_K ( italic\_x ∣ italic\_y , italic\_K ( italic\_y ) )
(trivially y\*superscript𝑦y^{\*}italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT contains the same information
as the y,K(y)𝑦𝐾𝑦y,K(y)italic\_y , italic\_K ( italic\_y )). Note that “y𝑦yitalic\_y” in the conditional
is just the information about y𝑦yitalic\_y and apart from
this does not contain information
about y\*superscript𝑦y^{\*}italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT or K(y)𝐾𝑦K(y)italic\_K ( italic\_y ). For this work the difference is
crucial, see the comment in Section [I](#S1 "I Introduction ‣ Algorithmic Statistics").
###
II-A Additivity of Complexity
Recall that by definition K(x,y)=K(⟨x,y⟩)𝐾𝑥𝑦𝐾𝑥𝑦K(x,y)=K(\langle x,y\rangle)italic\_K ( italic\_x , italic\_y ) = italic\_K ( ⟨ italic\_x , italic\_y ⟩ ).
Trivially, the symmetry property holds: K(x,y)=+K(y,x)superscript𝐾𝑥𝑦𝐾𝑦𝑥K(x,y)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(y,x)italic\_K ( italic\_x , italic\_y ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y , italic\_x ).
Later we will use many times the “Additivity of Complexity”
property
| | | | |
| --- | --- | --- | --- |
| | K(x,y)=+K(x)+K(y∣x\*)=+K(y)+K(x∣y\*).superscript𝐾𝑥𝑦𝐾𝑥𝐾conditional𝑦superscript𝑥superscript𝐾𝑦𝐾conditional𝑥superscript𝑦K(x,y)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)+K(y\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(y)+K(x\mid y^{\*}).italic\_K ( italic\_x , italic\_y ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) + italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y ) + italic\_K ( italic\_x ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) . | | (II.1) |
This result due to [[7](#bib.bib7)] can be found
as Theorem 3.9.1 in [[10](#bib.bib10)] and has a difficult proof.
It is perhaps instructive to point out that
the version with just x𝑥xitalic\_x and y𝑦yitalic\_y in the conditionals doesn’t
hold with =+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP, but holds up to additive logarithmic terms
that cannot be eliminated.
The conditional version needs to be treated carefully.
It is
| | | | |
| --- | --- | --- | --- |
| | K(x,y∣z)=+K(x∣z)+K(y∣x,K(x∣z),z).K(x,y\mid z)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid z)+K(y\mid x,K(x\mid z),z).italic\_K ( italic\_x , italic\_y ∣ italic\_z ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_z ) + italic\_K ( italic\_y ∣ italic\_x , italic\_K ( italic\_x ∣ italic\_z ) , italic\_z ) . | | (II.2) |
Note that a naive version
| | | |
| --- | --- | --- |
| | K(x,y∣z)=+K(x∣z)+K(y∣x\*,z)superscript𝐾𝑥conditional𝑦𝑧𝐾conditional𝑥𝑧𝐾conditional𝑦superscript𝑥𝑧K(x,y\mid z)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid z)+K(y\mid x^{\*},z)italic\_K ( italic\_x , italic\_y ∣ italic\_z ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_z ) + italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , italic\_z ) | |
is incorrect: taking z=x𝑧𝑥z=xitalic\_z = italic\_x, y=K(x)𝑦𝐾𝑥y=K(x)italic\_y = italic\_K ( italic\_x ),
the left-hand side equals K(x\*∣x)𝐾conditionalsuperscript𝑥𝑥K(x^{\*}\mid x)italic\_K ( italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ∣ italic\_x ), and the right-hand side
equals K(x∣x)+K(K(x)∣x\*,x)=+0superscript𝐾conditional𝑥𝑥𝐾conditional𝐾𝑥superscript𝑥𝑥0K(x\mid x)+K(K(x)\mid x^{\*},x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_x ∣ italic\_x ) + italic\_K ( italic\_K ( italic\_x ) ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
First, we derive a (to our knowledge) new “directed triangle inequality”
that is needed later.
###### Theorem II.1
For all x,y,z𝑥𝑦𝑧x,y,zitalic\_x , italic\_y , italic\_z,
| | | |
| --- | --- | --- |
| | K(x∣y\*)<+K(x,z∣y\*)<+K(z∣y\*)+K(x∣z\*).superscript𝐾conditional𝑥superscript𝑦𝐾𝑥conditional𝑧superscript𝑦superscript𝐾conditional𝑧superscript𝑦𝐾conditional𝑥superscript𝑧K(x\mid y^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(x,z\mid y^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(z\mid y^{\*})+K(x\mid z^{\*}).italic\_K ( italic\_x ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x , italic\_z ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_z ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_x ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) . | |
###### Proof:
Using ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")), an evident inequality introducing
an auxiliary object z𝑧zitalic\_z, and twice ( [II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")) again:
| | | | |
| --- | --- | --- | --- |
| | K(x,z∣y\*)𝐾𝑥conditional𝑧superscript𝑦\displaystyle K(x,z\mid y^{\*})italic\_K ( italic\_x , italic\_z ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | =+K(x,y,z)−K(y)superscriptabsent𝐾𝑥𝑦𝑧𝐾𝑦\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x,y,z)-K(y)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x , italic\_y , italic\_z ) - italic\_K ( italic\_y ) | |
| | | <+K(z)+K(x∣z\*)+K(y∣z\*)−K(y)superscriptabsent𝐾𝑧𝐾conditional𝑥superscript𝑧𝐾conditional𝑦superscript𝑧𝐾𝑦\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(z)+K(x\mid z^{\*})+K(y\mid z^{\*})-K(y)start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_z ) + italic\_K ( italic\_x ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_y ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_y ) | |
| | | =+K(y,z)−K(y)+K(x∣z\*)superscriptabsent𝐾𝑦𝑧𝐾𝑦𝐾conditional𝑥superscript𝑧\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(y,z)-K(y)+K(x\mid z^{\*})start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y , italic\_z ) - italic\_K ( italic\_y ) + italic\_K ( italic\_x ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | |
| | | =+K(x∣z\*)+K(z∣y\*).superscriptabsent𝐾conditional𝑥superscript𝑧𝐾conditional𝑧superscript𝑦\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid z^{\*})+K(z\mid y^{\*}).start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_z ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) . | |
∎
This theorem has bizarre consequences. These consequences are not
simple unexpected artifacts of our definitions, but, to the contrary,
they show the power and the genuine contribution to our understanding
represented by the deep and important mathematical relation
([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")).
Denote k=K(y)𝑘𝐾𝑦k=K(y)italic\_k = italic\_K ( italic\_y ) and substitute k=z𝑘𝑧k=zitalic\_k = italic\_z and K(k)=x𝐾𝑘𝑥K(k)=xitalic\_K ( italic\_k ) = italic\_x
to find the following counterintuitive corollary: To determine the complexity
of the complexity of an object y𝑦yitalic\_y it suffices to give both y𝑦yitalic\_y and
the complexity of y𝑦yitalic\_y. This is counterintuitive since in general
we cannot compute the complexity of an object from the object itself;
if we could this would also solve the
so-called “halting problem”, [[10](#bib.bib10)]. This noncomputability
can be quantified in terms of K(K(y)∣y)𝐾conditional𝐾𝑦𝑦K(K(y)\mid y)italic\_K ( italic\_K ( italic\_y ) ∣ italic\_y ) which can rise to
almost K(K(y))𝐾𝐾𝑦K(K(y))italic\_K ( italic\_K ( italic\_y ) ) for some y𝑦yitalic\_y—see the related discussion
on notation for conditional complexity in Section [I](#S1 "I Introduction ‣ Algorithmic Statistics"). But in the
seemingly similar, but subtly different, setting below it is possible.
###### Corollary II.2
As above, let k𝑘kitalic\_k denote K(y)𝐾𝑦K(y)italic\_K ( italic\_y ). Then,
K(K(k)∣y,k)=+K(K(k)∣y\*)<+K(K(k)∣k\*)+K(k∣y,k)=+0superscript𝐾conditional𝐾𝑘𝑦𝑘𝐾conditional𝐾𝑘superscript𝑦superscript𝐾conditional𝐾𝑘superscript𝑘𝐾conditional𝑘𝑦𝑘superscript0K(K(k)\mid y,k)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(K(k)\mid y^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(K(k)\mid k^{\*})+K(k\mid y,k)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_K ( italic\_k ) ∣ italic\_y , italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_K ( italic\_k ) ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_K ( italic\_k ) ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_k ∣ italic\_y , italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
We can iterate this idea.
For example, the next step is that
given y𝑦yitalic\_y and K(y)𝐾𝑦K(y)italic\_K ( italic\_y ) we can determine K(K(K(y)))𝐾𝐾𝐾𝑦K(K(K(y)))italic\_K ( italic\_K ( italic\_K ( italic\_y ) ) )
in O(1)𝑂1O(1)italic\_O ( 1 ) bits,
that is, K(K(K(k)))∣y,k)=+0K(K(K(k)))\mid y,k)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_K ( italic\_K ( italic\_k ) ) ) ∣ italic\_y , italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
A direct construction works according to the following idea (where
we ignore some important details):
From k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT one can compute ⟨k,K(k)⟩𝑘𝐾𝑘\langle k,K(k)\rangle⟨ italic\_k , italic\_K ( italic\_k ) ⟩ since
k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is by definition the shortest program for k𝑘kitalic\_k
and also by definition l(k\*)=K(k)𝑙superscript𝑘𝐾𝑘l(k^{\*})=K(k)italic\_l ( italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) = italic\_K ( italic\_k ).
Conversely, from k,K(k)𝑘𝐾𝑘k,K(k)italic\_k , italic\_K ( italic\_k ) one can compute k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT: by
running of all programs of length at most K(k)𝐾𝑘K(k)italic\_K ( italic\_k )
in dovetailed fashion until the first programme of length K(k)𝐾𝑘K(k)italic\_K ( italic\_k )
halts with output k𝑘kitalic\_k;
this is k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
The shortest program that computes the pair ⟨y,k⟩𝑦𝑘\langle y,k\rangle⟨ italic\_y , italic\_k ⟩
has length =+ksuperscriptabsent𝑘\stackrel{{\scriptstyle{}\_{+}}}{{=}}kstart\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k:
We have K(y,k)=+ksuperscript𝐾𝑦𝑘𝑘K(y,k)\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_K ( italic\_y , italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k (since the shortest program
y\*superscript𝑦y^{\*}italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT for y𝑦yitalic\_y carries both the information about y𝑦yitalic\_y and about k=l(y\*)𝑘𝑙superscript𝑦k=l(y^{\*})italic\_k = italic\_l ( italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )).
By ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")) therefore
K(k)+K(y∣k,K(k))=+ksuperscript𝐾𝑘𝐾conditional𝑦𝑘𝐾𝑘𝑘K(k)+K(y\mid k,K(k))\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_K ( italic\_k ) + italic\_K ( italic\_y ∣ italic\_k , italic\_K ( italic\_k ) ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k. In view of the information
equivalence of ⟨k,K(k)⟩𝑘𝐾𝑘\langle k,K(k)\rangle⟨ italic\_k , italic\_K ( italic\_k ) ⟩ and k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT,
therefore K(k)+K(y∣k\*)=+ksuperscript𝐾𝑘𝐾conditional𝑦superscript𝑘𝑘K(k)+K(y\mid k^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_K ( italic\_k ) + italic\_K ( italic\_y ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k. Let r𝑟ritalic\_r be a program
of length l(r)=K(y∣k\*)𝑙𝑟𝐾conditional𝑦superscript𝑘l(r)=K(y\mid k^{\*})italic\_l ( italic\_r ) = italic\_K ( italic\_y ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) that computes y𝑦yitalic\_y from k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Then, since l(k\*)=K(k)𝑙superscript𝑘𝐾𝑘l(k^{\*})=K(k)italic\_l ( italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) = italic\_K ( italic\_k ), there is a shortest
program y\*=qk\*rsuperscript𝑦𝑞superscript𝑘𝑟y^{\*}=qk^{\*}ritalic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT = italic\_q italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT italic\_r for y𝑦yitalic\_y
where q𝑞qitalic\_q is a fixed O(1)𝑂1O(1)italic\_O ( 1 ) bit self-delimiting
program that unpacks and uses k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT and r𝑟ritalic\_r
to compute y𝑦yitalic\_y.
We are now in the position to show K(K(k)∣y,k)=+0superscript𝐾conditional𝐾𝑘𝑦𝑘0K(K(k)\mid y,k)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_K ( italic\_k ) ∣ italic\_y , italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
There is a fixed O(1)𝑂1O(1)italic\_O ( 1 )-bit program, that includes knowledge of q𝑞qitalic\_q,
and that enumerates two lists
in parallel, each in dovetailed fashion:
Using k𝑘kitalic\_k it enumerates a list of
all programs that compute k𝑘kitalic\_k, including k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Given y𝑦yitalic\_y and k𝑘kitalic\_k it
enumerates another list of all programs of
length k=+l(y\*)superscript𝑘𝑙superscript𝑦k\stackrel{{\scriptstyle{}\_{+}}}{{=}}l(y^{\*})italic\_k start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_l ( italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) that compute y𝑦yitalic\_y.
One of these programs is y\*=qk\*rsuperscript𝑦𝑞superscript𝑘𝑟y^{\*}=qk^{\*}ritalic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT = italic\_q italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT italic\_r
that starts with qk\*𝑞superscript𝑘qk^{\*}italic\_q italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. Since q𝑞qitalic\_q is known,
this self-delimiting program
k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, and hence its length K(k)𝐾𝑘K(k)italic\_K ( italic\_k ), can be found by matching
every element in the k𝑘kitalic\_k-list with the prefixes of every element in
the y𝑦yitalic\_y list in enumeration order.
###
II-B Information Non-Increase
If we want to find an appropriate model fitting the data, then
we are concerned with the information in the data about such models.
Intuitively one feels that the information
in the data about the appropriate model cannot be increased
by any algorithmic or
probabilistic process.
Here, we rigorously show that this is the case in the algorithmic statistics
setting:
the information in one object about another
cannot be increased by any deterministic algorithmic method
by more than a constant. With added randomization this holds
with overwhelming probability.
We use the triangle inequality of Theorem [II.1](#S2.Thmlemma1 "Theorem II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics") to recall,
and to give possibly new proofs, of this information non-increase;
for more elaborate but hard-to-follow versions see
[[14](#bib.bib14), [15](#bib.bib15)].
We need the following technical concepts.
Let us call a nonnegative
real function f(x)𝑓𝑥f(x)italic\_f ( italic\_x ) defined on strings a semimeasure if
∑xf(x)≤1subscript𝑥𝑓𝑥1\sum\_{x}f(x)\leq 1∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_f ( italic\_x ) ≤ 1, and a measure (a probability distribution)
if the sum is 1.
A function f(x)𝑓𝑥f(x)italic\_f ( italic\_x ) is called lower semicomputable if there is a
rational valued computable function g(n,x)𝑔𝑛𝑥g(n,x)italic\_g ( italic\_n , italic\_x ) such that
g(n+1,x)≥g(n,x)𝑔𝑛1𝑥𝑔𝑛𝑥g(n+1,x)\geq g(n,x)italic\_g ( italic\_n + 1 , italic\_x ) ≥ italic\_g ( italic\_n , italic\_x ) and limn→∞g(n,x)=f(x)subscript→𝑛𝑔𝑛𝑥𝑓𝑥\lim\_{n\rightarrow\infty}g(n,x)=f(x)roman\_lim start\_POSTSUBSCRIPT italic\_n → ∞ end\_POSTSUBSCRIPT italic\_g ( italic\_n , italic\_x ) = italic\_f ( italic\_x ).
For an upper semicomputable function f𝑓fitalic\_f we require
that −f𝑓-f- italic\_f is lower semicomputable.
It is computable when it is both lower and upper semicomputable.
(A lower semicomputable measure is also computable.)
To define the algorithmic mutual information between
two individual objects x𝑥xitalic\_x and y𝑦yitalic\_y with no
probabilities involved, it is instructive to first recall
the probabilistic notion ([I.1](#S1.E1 "I.1 ‣ I Introduction ‣ Algorithmic Statistics"))
Rewriting ([I.1](#S1.E1 "I.1 ‣ I Introduction ‣ Algorithmic Statistics"))
as
| | | |
| --- | --- | --- |
| | ∑x∑yp(x,y)[−logp(x)−logp(y)+logp(x,y)],subscript𝑥subscript𝑦𝑝𝑥𝑦delimited-[]𝑝𝑥𝑝𝑦𝑝𝑥𝑦\sum\_{x}\sum\_{y}p(x,y)[-\log p(x)-\log p(y)+\log p(x,y)],∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) [ - roman\_log italic\_p ( italic\_x ) - roman\_log italic\_p ( italic\_y ) + roman\_log italic\_p ( italic\_x , italic\_y ) ] , | |
and noting that −logp(s)𝑝𝑠-\log p(s)- roman\_log italic\_p ( italic\_s ) is
very close to the length of the
prefix-free Shannon-Fano code for s𝑠sitalic\_s, we are led to the following
definition.
222The Shannon-Fano code has nearly optimal expected
code length equal to the entropy with
respect to the distribution of the source [[5](#bib.bib5)]. However,
the prefix-free code with code word length K(s)𝐾𝑠K(s)italic\_K ( italic\_s ) has both
about expected optimal code word length and individual optimal
effective code word length, [[10](#bib.bib10)].
The
information in y𝑦yitalic\_y about x𝑥xitalic\_x
is defined as
| | | | |
| --- | --- | --- | --- |
| | I(y:x)=K(x)−K(x∣y\*)=+K(x)+K(y)−K(x,y),I(y:x)=K(x)-K(x\mid y^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)+K(y)-K(x,y),italic\_I ( italic\_y : italic\_x ) = italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) + italic\_K ( italic\_y ) - italic\_K ( italic\_x , italic\_y ) , | | (II.3) |
where the second equality is a consequence of ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics"))
and states that this information is symmetrical,
I(x:y)=+I(y:x)I(x:y)\stackrel{{\scriptstyle{}\_{+}}}{{=}}I(y:x)italic\_I ( italic\_x : italic\_y ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( italic\_y : italic\_x ), and therefore we can talk about
mutual information.333The notation of the
algorithmic (individual)
notion I(x:y)I(x:y)italic\_I ( italic\_x : italic\_y ) distinguishes it from the probabilistic
(average) notion
I(X;Y)𝐼𝑋𝑌I(X;Y)italic\_I ( italic\_X ; italic\_Y ). We deviate slightly from [[10](#bib.bib10)]
where I(y:x)I(y:x)italic\_I ( italic\_y : italic\_x ) is defined as K(x)−K(x∣y)𝐾𝑥𝐾conditional𝑥𝑦K(x)-K(x\mid y)italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_y ).
###### Remark II.3
The conditional mutual information is
| | | | |
| --- | --- | --- | --- |
| | I(x:y∣z)\displaystyle I(x:y\mid z)italic\_I ( italic\_x : italic\_y ∣ italic\_z ) | =K(x∣z)−K(x∣y,K(y∣z),z)\displaystyle=K(x\mid z)-K(x\mid y,K(y\mid z),z)= italic\_K ( italic\_x ∣ italic\_z ) - italic\_K ( italic\_x ∣ italic\_y , italic\_K ( italic\_y ∣ italic\_z ) , italic\_z ) | |
| | | =+K(x∣z)+K(y∣z)−K(x,y∣z).superscriptabsent𝐾conditional𝑥𝑧𝐾conditional𝑦𝑧𝐾𝑥conditional𝑦𝑧\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid z)+K(y\mid z)-K(x,y\mid z).start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_z ) + italic\_K ( italic\_y ∣ italic\_z ) - italic\_K ( italic\_x , italic\_y ∣ italic\_z ) . | |
♢♢\diamondsuit♢
It is important that the expectation of the algorithmic mutual
information I(x:y)I(x:y)italic\_I ( italic\_x : italic\_y ) is close to the probabilistic mutual information
I(X;Y)𝐼𝑋𝑌I(X;Y)italic\_I ( italic\_X ; italic\_Y )—if
this were not the case then the algorithmic notion would not
be a sharpening of the probabilistic notion to individual objects,
but something else.
###### Lemma II.4
Given a computable joint probability mass distribution p(x,y)𝑝𝑥𝑦p(x,y)italic\_p ( italic\_x , italic\_y ) over (x,y)𝑥𝑦(x,y)( italic\_x , italic\_y )
we have
| | | | | |
| --- | --- | --- | --- | --- |
| | I(X;Y)−K(p)𝐼𝑋𝑌𝐾𝑝\displaystyle I(X;Y)-K(p)italic\_I ( italic\_X ; italic\_Y ) - italic\_K ( italic\_p ) | <+∑x∑yp(x,y)I(x:y)\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}\sum\_{x}\sum\_{y}p(x,y)I(x:y)start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) italic\_I ( italic\_x : italic\_y ) | | (II.4) |
| | | <+I(X;Y)+2K(p),superscriptabsent𝐼𝑋𝑌2𝐾𝑝\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}I(X;Y)+2K(p),start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( italic\_X ; italic\_Y ) + 2 italic\_K ( italic\_p ) , | |
where K(p)𝐾𝑝K(p)italic\_K ( italic\_p ) is the length of the shortest prefix-free program that computes
p(x,y)𝑝𝑥𝑦p(x,y)italic\_p ( italic\_x , italic\_y ) from input (x,y)𝑥𝑦(x,y)( italic\_x , italic\_y ).
###### Remark II.5
Above we required p(⋅,⋅)𝑝normal-⋅normal-⋅p(\cdot,\cdot)italic\_p ( ⋅ , ⋅ ) to be computable.
Actually, we only require that p𝑝pitalic\_p be a
lower semicomputable function, which is a weaker requirement than
recursivity. However, together with the
condition that p(⋅,⋅)𝑝normal-⋅normal-⋅p(\cdot,\cdot)italic\_p ( ⋅ , ⋅ ) is a probability distribution,
∑x,yp(x,y)=1subscript𝑥𝑦𝑝𝑥𝑦1\sum\_{x,y}p(x,y)=1∑ start\_POSTSUBSCRIPT italic\_x , italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) = 1, this means that p(⋅,⋅)𝑝normal-⋅normal-⋅p(\cdot,\cdot)italic\_p ( ⋅ , ⋅ ) is computable,
[[10](#bib.bib10)], Section 8.1.
♢normal-♢\diamondsuit♢
###### Proof:
Rewrite the expectation
| | | | |
| --- | --- | --- | --- |
| | ∑x∑yp(x,y)I(x:y)=+∑x∑y\displaystyle\sum\_{x}\sum\_{y}p(x,y)I(x:y)\stackrel{{\scriptstyle{}\_{+}}}{{=}}\sum\_{x}\sum\_{y}∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) italic\_I ( italic\_x : italic\_y ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT | p(x,y)[K(x)\displaystyle p(x,y)[K(x)italic\_p ( italic\_x , italic\_y ) [ italic\_K ( italic\_x ) | |
| | | +K(y)−K(x,y)].\displaystyle+K(y)-K(x,y)].+ italic\_K ( italic\_y ) - italic\_K ( italic\_x , italic\_y ) ] . | |
Define
∑yp(x,y)=p1(x)subscript𝑦𝑝𝑥𝑦subscript𝑝1𝑥\sum\_{y}p(x,y)=p\_{1}(x)∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) = italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_x )
and ∑xp(x,y)=p2(y)subscript𝑥𝑝𝑥𝑦subscript𝑝2𝑦\sum\_{x}p(x,y)=p\_{2}(y)∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) = italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_y )
to obtain
| | | | |
| --- | --- | --- | --- |
| | ∑x∑yp(x,y)I(x:y)=+∑x\displaystyle\sum\_{x}\sum\_{y}p(x,y)I(x:y)\stackrel{{\scriptstyle{}\_{+}}}{{=}}\sum\_{x}∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) italic\_I ( italic\_x : italic\_y ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT | p1(x)K(x)+∑yp2(y)K(y)subscript𝑝1𝑥𝐾𝑥subscript𝑦subscript𝑝2𝑦𝐾𝑦\displaystyle p\_{1}(x)K(x)+\sum\_{y}p\_{2}(y)K(y)italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_x ) italic\_K ( italic\_x ) + ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ( italic\_y ) italic\_K ( italic\_y ) | |
| | | −∑x,yp(x,y)K(x,y).subscript𝑥𝑦𝑝𝑥𝑦𝐾𝑥𝑦\displaystyle-\sum\_{x,y}p(x,y)K(x,y).- ∑ start\_POSTSUBSCRIPT italic\_x , italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) italic\_K ( italic\_x , italic\_y ) . | |
Given the program that computes p𝑝pitalic\_p, we can approximate p1(x)subscript𝑝1𝑥p\_{1}(x)italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_x )
by a q1(x,y0)=∑y≤y0p(x,y)subscript𝑞1𝑥subscript𝑦0subscript𝑦subscript𝑦0𝑝𝑥𝑦q\_{1}(x,y\_{0})=\sum\_{y\leq y\_{0}}p(x,y)italic\_q start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_x , italic\_y start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_y ≤ italic\_y start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ), and
similarly for p2subscript𝑝2p\_{2}italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. That is, the
distributions pisubscript𝑝𝑖p\_{i}italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT (i=1,2𝑖12i=1,2italic\_i = 1 , 2) are lower semicomputable, and
by Remark [II.5](#S2.Thmlemma5 "Remark II.5 ‣ II-B Information Non-Increase ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics"), therefore, they are computable.
It is known that for every computable probability mass function q𝑞qitalic\_q
we have H(q)<+∑xq(x)K(x)<+H(q)+K(q)superscript𝐻𝑞subscript𝑥𝑞𝑥𝐾𝑥superscript𝐻𝑞𝐾𝑞H(q)\stackrel{{\scriptstyle{}\_{+}}}{{<}}\sum\_{x}q(x)K(x)\stackrel{{\scriptstyle{}\_{+}}}{{<}}H(q)+K(q)italic\_H ( italic\_q ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_q ( italic\_x ) italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_H ( italic\_q ) + italic\_K ( italic\_q ), [[10](#bib.bib10)],
Section 8.1.
Hence, H(pi)<+∑xpi(x)K(x)<+H(pi)+K(pi)superscript𝐻subscript𝑝𝑖subscript𝑥subscript𝑝𝑖𝑥𝐾𝑥superscript𝐻subscript𝑝𝑖𝐾subscript𝑝𝑖H(p\_{i})\stackrel{{\scriptstyle{}\_{+}}}{{<}}\sum\_{x}p\_{i}(x)K(x)\stackrel{{\scriptstyle{}\_{+}}}{{<}}H(p\_{i})+K(p\_{i})italic\_H ( italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( italic\_x ) italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_H ( italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) + italic\_K ( italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT )
(i=1,2𝑖12i=1,2italic\_i = 1 , 2), and H(p)<+∑x,yp(x,y)K(x,y)<+H(p)+K(p)superscript𝐻𝑝subscript𝑥𝑦𝑝𝑥𝑦𝐾𝑥𝑦superscript𝐻𝑝𝐾𝑝H(p)\stackrel{{\scriptstyle{}\_{+}}}{{<}}\sum\_{x,y}p(x,y)K(x,y)\stackrel{{\scriptstyle{}\_{+}}}{{<}}H(p)+K(p)italic\_H ( italic\_p ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_x , italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) italic\_K ( italic\_x , italic\_y ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_H ( italic\_p ) + italic\_K ( italic\_p ).
On the other hand, the probabilistic mutual information
([I.1](#S1.E1 "I.1 ‣ I Introduction ‣ Algorithmic Statistics")) is expressed in the entropies by
I(X;Y)=H(p1)+H(p2)−H(p)𝐼𝑋𝑌𝐻subscript𝑝1𝐻subscript𝑝2𝐻𝑝I(X;Y)=H(p\_{1})+H(p\_{2})-H(p)italic\_I ( italic\_X ; italic\_Y ) = italic\_H ( italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) + italic\_H ( italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) - italic\_H ( italic\_p ).
By construction of the qisubscript𝑞𝑖q\_{i}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT’s above,
we have K(p1),K(p2)<+K(p)superscript𝐾subscript𝑝1𝐾subscript𝑝2
𝐾𝑝K(p\_{1}),K(p\_{2})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(p)italic\_K ( italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_K ( italic\_p start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_p ). Since the complexities
are positive, substitution
establishes the lemma.
∎
Can we get rid of the K(p)𝐾𝑝K(p)italic\_K ( italic\_p ) error term? The answer is affirmative;
by putting p(⋅)𝑝⋅p(\cdot)italic\_p ( ⋅ ) in the conditional we even get rid of
the computability requirement.
###### Lemma II.6
Given a joint probability mass distribution p(x,y)𝑝𝑥𝑦p(x,y)italic\_p ( italic\_x , italic\_y ) over (x,y)𝑥𝑦(x,y)( italic\_x , italic\_y )
(not necessarily computable) we have
| | | |
| --- | --- | --- |
| | I(X;Y)=+∑x∑yp(x,y)I(x:y∣p),I(X;Y)\stackrel{{\scriptstyle{}\_{+}}}{{=}}\sum\_{x}\sum\_{y}p(x,y)I(x:y\mid p),italic\_I ( italic\_X ; italic\_Y ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT italic\_p ( italic\_x , italic\_y ) italic\_I ( italic\_x : italic\_y ∣ italic\_p ) , | |
where the auxiliary p𝑝pitalic\_p means that we can directly access the
values p(x,y)𝑝𝑥𝑦p(x,y)italic\_p ( italic\_x , italic\_y ) on the
auxiliary conditional information tape of the reference
universal prefix machine.
###### Proof:
The lemma follows from the definition of conditional
algorithic mutual information, Remark [II.3](#S2.Thmlemma3 "Remark II.3 ‣ II-B Information Non-Increase ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics"),
if we show that ∑xp(x)K(x∣p)=+H(p)superscriptsubscript𝑥𝑝𝑥𝐾conditional𝑥𝑝𝐻𝑝\sum\_{x}p(x)K(x\mid p)\stackrel{{\scriptstyle{}\_{+}}}{{=}}H(p)∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_x ) italic\_K ( italic\_x ∣ italic\_p ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_H ( italic\_p ),
where the O(1)𝑂1O(1)italic\_O ( 1 ) term implicit in the =+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP sign
is independent of p𝑝pitalic\_p.
Equip the reference universal prefix machine,
with an O(1)𝑂1O(1)italic\_O ( 1 ) length
program to compute a Shannon-Fano code from the auxiliary table
of probabilities.
Then, given an input r𝑟ritalic\_r, it can determine
whether r𝑟ritalic\_r is the Shannon-Fano code word for some x𝑥xitalic\_x.
Such a code word
has length =+−logp(x)superscriptabsent𝑝𝑥\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log p(x)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_p ( italic\_x ).
If this is the case, then the machine
outputs x𝑥xitalic\_x, otherwise it halts without output. Therefore,
K(x∣p)<+−logp(x)superscript𝐾conditional𝑥𝑝𝑝𝑥K(x\mid p)\stackrel{{\scriptstyle{}\_{+}}}{{<}}-\log p(x)italic\_K ( italic\_x ∣ italic\_p ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_p ( italic\_x ).
This shows
the upper bound on the expected prefix complexity.
The lower bound follows as usual
from the Noiseless Coding Theorem.
∎
We prove a strong version of the information non-increase law
under deterministic processing (later we need the attached corollary):
###### Theorem II.7
Given x𝑥xitalic\_x and z𝑧zitalic\_z, let q𝑞qitalic\_q be a program
computing z𝑧zitalic\_z from x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Then
| | | | |
| --- | --- | --- | --- |
| | I(z:y)<+I(x:y)+K(q).I(z:y)\stackrel{{\scriptstyle{}\_{+}}}{{<}}I(x:y)+K(q).italic\_I ( italic\_z : italic\_y ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( italic\_x : italic\_y ) + italic\_K ( italic\_q ) . | | (II.5) |
###### Proof:
By the triangle inequality,
| | | | |
| --- | --- | --- | --- |
| | K(y∣x\*)𝐾conditional𝑦superscript𝑥\displaystyle K(y\mid x^{\*})italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | <+K(y∣z\*)+K(z∣x\*)superscriptabsent𝐾conditional𝑦superscript𝑧𝐾conditional𝑧superscript𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(y\mid z^{\*})+K(z\mid x^{\*})start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | |
| | | =+K(y∣z\*)+K(q).superscriptabsent𝐾conditional𝑦superscript𝑧𝐾𝑞\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(y\mid z^{\*})+K(q).start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_q ) . | |
Thus,
| | | | |
| --- | --- | --- | --- |
| | I(x:y)\displaystyle I(x:y)italic\_I ( italic\_x : italic\_y ) | =K(y)−K(y∣x\*)absent𝐾𝑦𝐾conditional𝑦superscript𝑥\displaystyle=K(y)-K(y\mid x^{\*})= italic\_K ( italic\_y ) - italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | |
| | | >+K(y)−K(y∣z\*)−K(q)superscriptabsent𝐾𝑦𝐾conditional𝑦superscript𝑧𝐾𝑞\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(y)-K(y\mid z^{\*})-K(q)start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y ) - italic\_K ( italic\_y ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_q ) | |
| | | =I(z:y)−K(q).\displaystyle=I(z:y)-K(q).= italic\_I ( italic\_z : italic\_y ) - italic\_K ( italic\_q ) . | |
∎
This also implies the slightly weaker but intuitively
more appealing statement that the mutual information between strings
x𝑥xitalic\_x and y𝑦yitalic\_y cannot be increased by processing x𝑥xitalic\_x and y𝑦yitalic\_y separately by
deterministic computations.
###### Corollary II.8
Let f,g𝑓𝑔f,gitalic\_f , italic\_g be recursive functions.
Then
| | | | |
| --- | --- | --- | --- |
| | I(f(x):g(y))<+I(x:y)+K(f)+K(g).I(f(x):g(y))\stackrel{{\scriptstyle{}\_{+}}}{{<}}I(x:y)+K(f)+K(g).italic\_I ( italic\_f ( italic\_x ) : italic\_g ( italic\_y ) ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( italic\_x : italic\_y ) + italic\_K ( italic\_f ) + italic\_K ( italic\_g ) . | | (II.6) |
###### Proof:
It suffices to prove the case g(y)=y𝑔𝑦𝑦g(y)=yitalic\_g ( italic\_y ) = italic\_y and apply it twice.
The proof is by replacing the program q𝑞qitalic\_q that computes
a particular string z𝑧zitalic\_z
from a particular x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT in ([II.5](#S2.E5 "II.5 ‣ Theorem II.7 ‣ II-B Information Non-Increase ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")). There, q𝑞qitalic\_q
possibly depends on x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT and z𝑧zitalic\_z. Replace it by a program qfsubscript𝑞𝑓q\_{f}italic\_q start\_POSTSUBSCRIPT italic\_f end\_POSTSUBSCRIPT that first
computes x𝑥xitalic\_x from x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, followed by computing a
recursive function
f𝑓fitalic\_f, that is, qfsubscript𝑞𝑓q\_{f}italic\_q start\_POSTSUBSCRIPT italic\_f end\_POSTSUBSCRIPT is independent of x𝑥xitalic\_x.
Since we only require an O(1)𝑂1O(1)italic\_O ( 1 )-length program to compute
x𝑥xitalic\_x from x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT we can choose l(qf)=+K(f)superscript𝑙subscript𝑞𝑓𝐾𝑓l(q\_{f})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(f)italic\_l ( italic\_q start\_POSTSUBSCRIPT italic\_f end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_f ).
By the triangle inequality,
| | | | |
| --- | --- | --- | --- |
| | K(y∣x\*)𝐾conditional𝑦superscript𝑥\displaystyle K(y\mid x^{\*})italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | <+K(y∣f(x)\*)+K(f(x)∣x\*)superscriptabsent𝐾conditional𝑦𝑓superscript𝑥𝐾conditional𝑓𝑥superscript𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(y\mid f(x)^{\*})+K(f(x)\mid x^{\*})start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y ∣ italic\_f ( italic\_x ) start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_f ( italic\_x ) ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | |
| | | =+K(y∣f(x)\*)+K(f).superscriptabsent𝐾conditional𝑦𝑓superscript𝑥𝐾𝑓\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(y\mid f(x)^{\*})+K(f).start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y ∣ italic\_f ( italic\_x ) start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_f ) . | |
Thus,
| | | | |
| --- | --- | --- | --- |
| | I(x:y)\displaystyle I(x:y)italic\_I ( italic\_x : italic\_y ) | =K(y)−K(y∣x\*)absent𝐾𝑦𝐾conditional𝑦superscript𝑥\displaystyle=K(y)-K(y\mid x^{\*})= italic\_K ( italic\_y ) - italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | |
| | | >+K(y)−K(y∣f(x)\*)−K(f)superscriptabsent𝐾𝑦𝐾conditional𝑦𝑓superscript𝑥𝐾𝑓\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(y)-K(y\mid f(x)^{\*})-K(f)start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y ) - italic\_K ( italic\_y ∣ italic\_f ( italic\_x ) start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_f ) | |
| | | =I(f(x):y)−K(f).\displaystyle=I(f(x):y)-K(f).= italic\_I ( italic\_f ( italic\_x ) : italic\_y ) - italic\_K ( italic\_f ) . | |
∎
It turns out that furthermore, randomized computation can increase
information only with negligible probability.
Let us define the universal probability 𝐦(x)=2−K(x)𝐦𝑥superscript2𝐾𝑥{\mathbf{m}}(x)=2^{-K(x)}bold\_m ( italic\_x ) = 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_x ) end\_POSTSUPERSCRIPT.
This function is known to be
maximal within a multiplicative constant among lower semicomputable
semimeasures.
So, in particular, for each computable measure ν(x)𝜈𝑥\nu(x)italic\_ν ( italic\_x ) we have
ν(x)<\*𝐦(x)superscript𝜈𝑥𝐦𝑥\nu(x)\stackrel{{\scriptstyle{}\_{\*}}}{{<}}{\mathbf{m}}(x)italic\_ν ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP bold\_m ( italic\_x ), where the constant factor in <\*superscript\stackrel{{\scriptstyle{}\_{\*}}}{{<}}start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP depends on ν𝜈\nuitalic\_ν.
This property also holds when we have an extra parameter, like y\*superscript𝑦y^{\*}italic\_y start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT,
in the condition.
Suppose that z𝑧zitalic\_z is obtained from x𝑥xitalic\_x by some randomized computation.
The probability p(z∣x)𝑝conditional𝑧𝑥p(z\mid x)italic\_p ( italic\_z ∣ italic\_x ) of obtaining z𝑧zitalic\_z from x𝑥xitalic\_x is a semicomputable
distribution over the z𝑧zitalic\_z’s.
Therefore it is upperbounded by
𝐦(z∣x)<\*𝐦(z∣x\*)=2−K(z∣x\*)superscript𝐦conditional𝑧𝑥𝐦conditional𝑧superscript𝑥superscript2𝐾conditional𝑧superscript𝑥{\mathbf{m}}(z\mid x)\stackrel{{\scriptstyle{}\_{\*}}}{{<}}{\mathbf{m}}(z\mid x^{\*})=2^{-K(z\mid x^{\*})}bold\_m ( italic\_z ∣ italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP bold\_m ( italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) = 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) end\_POSTSUPERSCRIPT.
The information increase I(z:y)−I(x:y)I(z:y)-I(x:y)italic\_I ( italic\_z : italic\_y ) - italic\_I ( italic\_x : italic\_y ) satisfies the theorem below.
###### Theorem II.9
For all x,y,z𝑥𝑦𝑧x,y,zitalic\_x , italic\_y , italic\_z we have
| | | |
| --- | --- | --- |
| | 𝐦(z∣x\*)2I(z:y)−I(x:y)<\*𝐦(z∣x\*,y,K(y∣x\*)).{\mathbf{m}}(z\mid x^{\*})2^{I(z:y)-I(x:y)}\stackrel{{\scriptstyle{}\_{\*}}}{{<}}{\mathbf{m}}(z\mid x^{\*},y,K(y\mid x^{\*})).bold\_m ( italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) 2 start\_POSTSUPERSCRIPT italic\_I ( italic\_z : italic\_y ) - italic\_I ( italic\_x : italic\_y ) end\_POSTSUPERSCRIPT start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP bold\_m ( italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , italic\_y , italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ) . | |
###### Remark II.10
For example, the probability of an increase of mutual information
by the amount d𝑑ditalic\_d is <\*2−dsuperscriptabsentsuperscript2𝑑\stackrel{{\scriptstyle{}\_{\*}}}{{<}}2^{-d}start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 2 start\_POSTSUPERSCRIPT - italic\_d end\_POSTSUPERSCRIPT.
The theorem
implies ∑z𝐦(z∣x\*)2I(z:y)−I(x:y)<\*1superscriptsubscript𝑧𝐦conditional𝑧superscript𝑥superscript2𝐼:𝑧𝑦𝐼:𝑥𝑦1\sum\_{z}{\mathbf{m}}(z\mid x^{\*})2^{I(z:y)-I(x:y)}\stackrel{{\scriptstyle{}\_{\*}}}{{<}}1∑ start\_POSTSUBSCRIPT italic\_z end\_POSTSUBSCRIPT bold\_m ( italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) 2 start\_POSTSUPERSCRIPT italic\_I ( italic\_z : italic\_y ) - italic\_I ( italic\_x : italic\_y ) end\_POSTSUPERSCRIPT start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 1,
the 𝐦(⋅∣x\*){\mathbf{m}}(\cdot\mid x^{\*})bold\_m ( ⋅ ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )-expectation of the exponential of the increase
is bounded by a constant.
♢♢\diamondsuit♢
###### Proof:
We have
| | | | |
| --- | --- | --- | --- |
| | I(z:y)−I(x:y)\displaystyle I(z:y)-I(x:y)italic\_I ( italic\_z : italic\_y ) - italic\_I ( italic\_x : italic\_y ) | =K(y)−K(y∣z\*)−(K(y)−K(y∣x\*))absent𝐾𝑦𝐾conditional𝑦superscript𝑧𝐾𝑦𝐾conditional𝑦superscript𝑥\displaystyle=K(y)-K(y\mid z^{\*})-(K(y)-K(y\mid x^{\*}))= italic\_K ( italic\_y ) - italic\_K ( italic\_y ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - ( italic\_K ( italic\_y ) - italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ) | |
| | | =K(y∣x\*)−K(y∣z\*).absent𝐾conditional𝑦superscript𝑥𝐾conditional𝑦superscript𝑧\displaystyle=K(y\mid x^{\*})-K(y\mid z^{\*}).= italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_y ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) . | |
The negative logarithm of the left-hand side in the theorem is therefore
| | | |
| --- | --- | --- |
| | K(z∣x\*)+K(y∣z\*)−K(y∣x\*).𝐾conditional𝑧superscript𝑥𝐾conditional𝑦superscript𝑧𝐾conditional𝑦superscript𝑥K(z\mid x^{\*})+K(y\mid z^{\*})-K(y\mid x^{\*}).italic\_K ( italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_y ∣ italic\_z start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) . | |
Using Theorem [II.1](#S2.Thmlemma1 "Theorem II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics"), and the conditional
additivity ([II.2](#S2.E2 "II.2 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")), this is
| | | |
| --- | --- | --- |
| | >+K(y,z∣x\*)−K(y∣x\*)=+K(z∣x\*,y,K(y∣x\*)).\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(y,z\mid x^{\*})-K(y\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(z\mid x^{\*},y,K(y\mid x^{\*})).start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_y , italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_z ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , italic\_y , italic\_K ( italic\_y ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ) . | |
∎
III Finite Set Models
----------------------
For convenience, we initially consider the model class consisting of
the family of finite sets of finite binary strings, that is,
the set of subsets of {0,1}\*superscript01\{0,1\}^{\*}{ 0 , 1 } start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
###
III-A Finite Set Representations
Although all finite sets are recursive there are different
ways to represent or specify the set.
We only consider ways that have in common
a method of recursively enumerating
the elements of the finite set one by one, and differ in knowledge
of its size.
For example, we can specify a
set of natural numbers by giving an explicit table or a decision
procedure for membership and a bound on the largest element,
or by giving a recursive enumeration
of the elements together with the number of elements,
or by giving a recursive enumeration of the elements
together with a bound on the running time.
We call a representation of a finite set S𝑆Sitalic\_S explicit
if the size |S|𝑆|S|| italic\_S | of the finite set
can be computed from it.
A representation of S𝑆Sitalic\_S is implicit if the
logsize ⌊log|S|⌋𝑆\lfloor\log|S|\rfloor⌊ roman\_log | italic\_S | ⌋ can be computed from it.
###### Example III.1
In Section [III-D](#S3.SS4 "III-D Implicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics"), we will introduce the set
Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT of strings whose elements have complexity ≤kabsent𝑘\leq k≤ italic\_k.
It will be shown that
this set can be represented implicitly by a program of size K(k)𝐾𝑘K(k)italic\_K ( italic\_k ),
but can
be represented explicitly only by a program of size k𝑘kitalic\_k.
♢normal-♢\diamondsuit♢
Such representations are useful in two-stage encodings where
one stage of the code consists of an index in S𝑆Sitalic\_S of length
=+log|S|superscriptabsent𝑆\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log|S|start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S |.
In the implicit case we know, within an additive constant, how long an
index of
an element in the set is.
We can extend the notion of Kolmogorov complexity from finite
binary strings to finite sets:
The (prefix-) complexity KX(S)subscript𝐾𝑋𝑆K\_{X}(S)italic\_K start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT ( italic\_S ) of a
finite set S𝑆Sitalic\_S is defined by
| | | | |
| --- | --- | --- | --- |
| | KX(S)=mini{K(i):\displaystyle K\_{X}(S)=\min\_{i}\{K(i):italic\_K start\_POSTSUBSCRIPT italic\_X end\_POSTSUBSCRIPT ( italic\_S ) = roman\_min start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT { italic\_K ( italic\_i ) : | Turing machine Ticomputes STuring machine subscript𝑇𝑖computes 𝑆\displaystyle\mbox{\rm Turing machine }T\_{i}\;\;\mbox{\rm computes }STuring machine italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT computes italic\_S | |
| | | in representation format X},\displaystyle\mbox{\rm in representation format }X\},in representation format italic\_X } , | |
where X𝑋Xitalic\_X is for example “implicit” or “explicit”.
In general S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT denotes the first shortest self-delimiting binary program
(l(S\*)=K(S)𝑙superscript𝑆𝐾𝑆l(S^{\*})=K(S)italic\_l ( italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) = italic\_K ( italic\_S )) in enumeration order
from which S𝑆Sitalic\_S can be computed. These definitions depend,
as explained above, crucial on the representation format X𝑋Xitalic\_X: the
way S𝑆Sitalic\_S is supposed to be represented as output of the computation
can make a world of difference for S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT and K(S)𝐾𝑆K(S)italic\_K ( italic\_S ).
Since the representation format
will be clear from the context, and to simplify notation, we
drop the subscript X𝑋Xitalic\_X.
To complete our discussion: the worst case of representation format X𝑋Xitalic\_X,
a recursively enumerable representation
where nothing is known about the size of the finite set,
would lead to indices of unknown length.
We do not consider this case.
We may use the notation
| | | |
| --- | --- | --- |
| | Simpl,Sexplsubscript𝑆implsubscript𝑆explS\_{\text{{impl}}},S\_{\text{{expl}}}italic\_S start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT , italic\_S start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT | |
for some implicit and some explicit representation of S𝑆Sitalic\_S.
When a result applies to both implicit and explicit representations,
or when it is clear from the context which representation is meant, we
will omit the subscript.
###
III-B Optimal Model and Sufficient Statistic
In the following we will distinguish between “models” that are
finite sets, and the “shortest programs” to compute those models
that are finite strings.
Such
a shortest program is in the proper sense a statistic of the data sample
as defined before.
In a way this distinction between “model” and “statistic”
is artificial, but for now we prefer clarity and unambiguousness
in the discussion.
Consider a string x𝑥xitalic\_x
of length n𝑛nitalic\_n and prefix complexity K(x)=k𝐾𝑥𝑘K(x)=kitalic\_K ( italic\_x ) = italic\_k.
We identify the structure or regularity in x𝑥xitalic\_x that are
to be summarized with a set S𝑆Sitalic\_S
of which x𝑥xitalic\_x is a random or typical member:
given S𝑆Sitalic\_S (or rather,
an (implicit or explicit)
shortest program S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT for S𝑆Sitalic\_S), x𝑥xitalic\_x cannot
be described significantly shorter than by its maximal length index in S𝑆Sitalic\_S,
that is, K(x∣S\*)>+log|S|superscript𝐾conditional𝑥superscript𝑆𝑆K(x\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{>}}\log|S|italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S |. Formally,
###### Definition III.2
Let β≥0𝛽0\beta\geq 0italic\_β ≥ 0 be an agreed upon, fixed, constant.
A finite binary string x𝑥xitalic\_x
is a typical or random element of a set S𝑆Sitalic\_S of finite binary
strings if x∈S𝑥𝑆x\in Sitalic\_x ∈ italic\_S and
| | | | |
| --- | --- | --- | --- |
| | K(x∣S\*)≥log|S|−β,𝐾conditional𝑥superscript𝑆𝑆𝛽K(x\mid S^{\*})\geq\log|S|-\beta,italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ≥ roman\_log | italic\_S | - italic\_β , | | (III.1) |
where S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is an implicit or explicit shortest program for S𝑆Sitalic\_S.
We will not indicate the dependence on β𝛽\betaitalic\_β explicitly, but the
constants in all our inequalities (<+superscript\stackrel{{\scriptstyle{}\_{+}}}{{<}}start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP) will be allowed to be functions
of this β𝛽\betaitalic\_β.
This definition requires a finite S𝑆Sitalic\_S.
In fact, since
K(x∣S\*)<+K(x)superscript𝐾conditional𝑥superscript𝑆𝐾𝑥K(x\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(x)italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ), it limits the size of S𝑆Sitalic\_S to O(2k)𝑂superscript2𝑘O(2^{k})italic\_O ( 2 start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT )
and the shortest program S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT from
which S𝑆Sitalic\_S can be computed) is an algorithmic statistic for x𝑥xitalic\_x iff
| | | | |
| --- | --- | --- | --- |
| | K(x∣S\*)=+log|S|.superscript𝐾conditional𝑥superscript𝑆𝑆K(x\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log|S|.italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S | . | | (III.2) |
Note that the notions of optimality and typicality are not absolute
but depend on fixing the constant implicit in the =+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP.
Depending on whether S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is an implicit or explicit program, our
definition splits into implicit and explicit typicality.
###### Example III.3
Consider the set S𝑆Sitalic\_S of binary strings of length n𝑛nitalic\_n
whose every odd position is 0.
Let x𝑥xitalic\_x be an element of this set in which the subsequence of bits in
even positions is an incompressible string.
Then S𝑆Sitalic\_S is explicitly as well as implicitly typical for x𝑥xitalic\_x.
The set {x}𝑥\{x\}{ italic\_x } also has both these properties.
♢normal-♢\diamondsuit♢
###### Remark III.4
It is not clear whether explicit typicality implies implicit typicality.
Section [IV](#S4 "IV Non-Stochastic Objects ‣ Algorithmic Statistics") will show some examples which are implicitly
very non-typical but explicitly at least nearly typical.
♢normal-♢\diamondsuit♢
There are two natural measures of suitability of such a statistic.
We might prefer either the simplest set, or the largest set, as
corresponding to the most likely structure ‘explaining’ x𝑥xitalic\_x.
The singleton set {x}𝑥\{x\}{ italic\_x }, while certainly a statistic for x𝑥xitalic\_x,
would indeed be considered a poor explanation.
Both measures relate to the optimality of a two-stage description of
x𝑥xitalic\_x using S𝑆Sitalic\_S:
| | | | | |
| --- | --- | --- | --- | --- |
| | K(x)≤K(x,S)𝐾𝑥𝐾𝑥𝑆\displaystyle K(x)\leq K(x,S)italic\_K ( italic\_x ) ≤ italic\_K ( italic\_x , italic\_S ) | =+K(S)+K(x∣S\*)superscriptabsent𝐾𝑆𝐾conditional𝑥superscript𝑆\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)+K(x\mid S^{\*})start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) + italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | | (III.3) |
| | | <+K(S)+log|S|,superscriptabsent𝐾𝑆𝑆\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(S)+\log|S|,start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) + roman\_log | italic\_S | , | |
where we rewrite K(x,S)𝐾𝑥𝑆K(x,S)italic\_K ( italic\_x , italic\_S ) by ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")).
Here, S𝑆Sitalic\_S can be understood as either Simplsubscript𝑆implS\_{\text{{impl}}}italic\_S start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT or Sexplsubscript𝑆explS\_{\text{{expl}}}italic\_S start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT.
Call a set S𝑆Sitalic\_S (containing x𝑥xitalic\_x)
for which
| | | | |
| --- | --- | --- | --- |
| | K(x)=+K(S)+log|S|,superscript𝐾𝑥𝐾𝑆𝑆K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)+\log|S|,italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) + roman\_log | italic\_S | , | | (III.4) |
optimal.
Depending on whether K(S)𝐾𝑆K(S)italic\_K ( italic\_S ) is understood as K(Simpl)𝐾subscript𝑆implK(S\_{\text{{impl}}})italic\_K ( italic\_S start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT ) or
K(Sexpl)𝐾subscript𝑆explK(S\_{\text{{expl}}})italic\_K ( italic\_S start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT ), our
definition splits into implicit and explicit optimality.
Mindful of our distinction between a finite set S𝑆Sitalic\_S and a
program that describes S𝑆Sitalic\_S in a required representation format,
we call a shortest program for an optimal set with respect to x𝑥xitalic\_x
an algorithmic sufficient statistic for x𝑥xitalic\_x.
Furthermore, among optimal sets,
there is a direct trade-off between complexity and logsize, which together
sum to =+ksuperscriptabsent𝑘\stackrel{{\scriptstyle{}\_{+}}}{{=}}kstart\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k.
Equality ([III.4](#S3.E4 "III.4 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")) is the algorithmic equivalent
dealing with the relation between the individual
sufficient statistic and the individual data sample, in contrast
to the probabilistic notion ([I.2](#S1.E2 "I.2 ‣ I Introduction ‣ Algorithmic Statistics")).
###### Example III.5
The following restricted model family illustrates the difference
between the algorithmic individual notion of sufficient
statistic and the probabilistic averaging one.
Foreshadowing the discussion in section [VII](#S7 "VII Conclusion ‣ Algorithmic Statistics"),
this example also illustrates the idea that the semantics of the model class
should be obtained by a restriction on the family of allowable
models, after which the (minimal) sufficient statistic identifies
the most appropriate model in the allowable family and thus optimizes
the parameters in the selected model class.
In the algorithmic setting we use all subsets of {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT
as models and the shortest programs computing them from a
given data sample as the statistic. Suppose we have background
information constraining the family of models to the n+1𝑛1n+1italic\_n + 1
finite sets Ss={x∈{0,1}n:x=x1…xn&∑i=1nxi=s}subscript𝑆𝑠conditional-set𝑥superscript01𝑛𝑥subscript𝑥1normal-…subscript𝑥𝑛superscriptsubscript𝑖1𝑛subscript𝑥𝑖𝑠S\_{s}=\{x\in\{0,1\}^{n}:x=x\_{1}\ldots x\_{n}\&\sum\_{i=1}^{n}x\_{i}=s\}italic\_S start\_POSTSUBSCRIPT italic\_s end\_POSTSUBSCRIPT = { italic\_x ∈ { 0 , 1 } start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT : italic\_x = italic\_x start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT … italic\_x start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT & ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = italic\_s } (0≤s≤n0𝑠𝑛0\leq s\leq n0 ≤ italic\_s ≤ italic\_n).
Assume that our model family is the family of Bernoulli distributions.
Then, in the
probabilistic sense for every data sample x=x1…xn𝑥subscript𝑥1normal-…subscript𝑥𝑛x=x\_{1}\ldots x\_{n}italic\_x = italic\_x start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT … italic\_x start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT there
is only one natural sufficient statistic: for ∑ixi=ssubscript𝑖subscript𝑥𝑖𝑠\sum\_{i}x\_{i}=s∑ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = italic\_s
this is T(x)=s𝑇𝑥𝑠T(x)=sitalic\_T ( italic\_x ) = italic\_s with the corresponding model Sssubscript𝑆𝑠S\_{s}italic\_S start\_POSTSUBSCRIPT italic\_s end\_POSTSUBSCRIPT.
In the algorithmic setting the situation is more subtle. (In
the following example we use the complexities conditional on n𝑛nitalic\_n.)
For x=x1…xn𝑥subscript𝑥1normal-…subscript𝑥𝑛x=x\_{1}\ldots x\_{n}italic\_x = italic\_x start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT … italic\_x start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT with ∑ixi=n2subscript𝑖subscript𝑥𝑖𝑛2\sum\_{i}x\_{i}=\frac{n}{2}∑ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = divide start\_ARG italic\_n end\_ARG start\_ARG 2 end\_ARG
taking Sn2subscript𝑆𝑛2S\_{\frac{n}{2}}italic\_S start\_POSTSUBSCRIPT divide start\_ARG italic\_n end\_ARG start\_ARG 2 end\_ARG end\_POSTSUBSCRIPT as model yields
|Sn2|=(nn2)subscript𝑆𝑛2binomial𝑛𝑛2|S\_{\frac{n}{2}}|={n\choose{\frac{n}{2}}}| italic\_S start\_POSTSUBSCRIPT divide start\_ARG italic\_n end\_ARG start\_ARG 2 end\_ARG end\_POSTSUBSCRIPT | = ( binomial start\_ARG italic\_n end\_ARG start\_ARG divide start\_ARG italic\_n end\_ARG start\_ARG 2 end\_ARG end\_ARG ),
and therefore log|Sn2|=+n−12lognsuperscriptsubscript𝑆𝑛2𝑛12𝑛\log|S\_{\frac{n}{2}}|\stackrel{{\scriptstyle{}\_{+}}}{{=}}n-\frac{1}{2}\log nroman\_log | italic\_S start\_POSTSUBSCRIPT divide start\_ARG italic\_n end\_ARG start\_ARG 2 end\_ARG end\_POSTSUBSCRIPT | start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - divide start\_ARG 1 end\_ARG start\_ARG 2 end\_ARG roman\_log italic\_n.
The sum of K(Sn2|n)=+0superscript𝐾conditionalsubscript𝑆𝑛2𝑛0K(S\_{\frac{n}{2}}|n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_S start\_POSTSUBSCRIPT divide start\_ARG italic\_n end\_ARG start\_ARG 2 end\_ARG end\_POSTSUBSCRIPT | italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 and the logarithmic
term gives =+n−12lognsuperscriptabsent𝑛12𝑛\stackrel{{\scriptstyle{}\_{+}}}{{=}}n-\frac{1}{2}\log nstart\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - divide start\_ARG 1 end\_ARG start\_ARG 2 end\_ARG roman\_log italic\_n for the right-hand side of
([III.4](#S3.E4 "III.4 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")). But taking x=1010…10𝑥1010normal-…10x=1010\ldots 10italic\_x = 1010 … 10 yields K(x∣n)=+0superscript𝐾conditional𝑥𝑛0K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0
for the left-hand side. Thus, there is no algorithmic sufficient
statistic for the latter x𝑥xitalic\_x in this model class, while every x𝑥xitalic\_x
of length n𝑛nitalic\_n has a probabilistic sufficient statistic in the model class.
In fact, the restricted model class has algorithmic sufficient
statistic for data samples x𝑥xitalic\_x of length n𝑛nitalic\_n that have maximal complexity
with respect to the frequency of “1”s, the other data samples
have no algorithmic sufficient statistic in this model class.
♢normal-♢\diamondsuit♢
###### Example III.6
It can be shown that the set S𝑆Sitalic\_S of Example [III.3](#S3.Thmlemma3 "Example III.3 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") is also
optimal, and so is {x}𝑥\{x\}{ italic\_x }.
Typical sets form a much wider class than optimal ones:
{x,y}𝑥𝑦\{x,y\}{ italic\_x , italic\_y } is still typical for
x𝑥xitalic\_x but with most y𝑦yitalic\_y, it will be too complex to be optimal for x𝑥xitalic\_x.
For a perhaps less artificial example, consider complexities conditional
on the length n𝑛nitalic\_n of strings.
Let y𝑦yitalic\_y be a random string of length n𝑛nitalic\_n, let
Sysubscript𝑆𝑦S\_{y}italic\_S start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT be the set of strings of length n𝑛nitalic\_n which have 0’s exactly
where y𝑦yitalic\_y has, and let x𝑥xitalic\_x be a random element of Sysubscript𝑆𝑦S\_{y}italic\_S start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT.
Then x𝑥xitalic\_x is a string random with respect to the distribution in which
1’s are chosen independently
with probability 0.25, so its complexity is much less than n𝑛nitalic\_n.
The set Sysubscript𝑆𝑦S\_{y}italic\_S start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT is typical with respect to x𝑥xitalic\_x but is too complex to be optimal,
since its (explicit or implicit) complexity conditional on n𝑛nitalic\_n is n𝑛nitalic\_n.
♢normal-♢\diamondsuit♢
It follows that (programs for) optimal sets are statistics.
Equality ([III.4](#S3.E4 "III.4 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")) expresses the conditions on the algorithmic individual
relation between the data and the sufficient statistic. Later we
demonstrate that this relation implies that the probabilistic
optimality of mutual information ([I.1](#S1.E1 "I.1 ‣ I Introduction ‣ Algorithmic Statistics")) holds
for the algorithmic version in the expected sense.
An algorithmic sufficient statistic T(⋅)𝑇⋅T(\cdot)italic\_T ( ⋅ )
is a sharper individual notion than a probabilistic sufficient
statistic. An optimal set S𝑆Sitalic\_S associated with x𝑥xitalic\_x (the shortest
program computing S𝑆Sitalic\_S is the corresponding
sufficient statistic associated with x𝑥xitalic\_x) is chosen such that
x𝑥xitalic\_x is maximally random with respect to it. That is, the
information in x𝑥xitalic\_x is divided in a relevant structure expressed
by the set S𝑆Sitalic\_S, and the remaining randomness with respect
to that structure, expressed by x𝑥xitalic\_x’s index in S𝑆Sitalic\_S of log|S|𝑆\log|S|roman\_log | italic\_S |
bits. The shortest program for S𝑆Sitalic\_S is itself alone an algorithmic
definition of structure, without a probabilistic interpretation.
One can also consider notions of
near-typical and near-optimal that arise from replacing
the β𝛽\betaitalic\_β in ([III.1](#S3.E1 "III.1 ‣ Definition III.2 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics"))
by some slowly growing functions, such as O(logl(x))𝑂𝑙𝑥O(\log l(x))italic\_O ( roman\_log italic\_l ( italic\_x ) ) or
O(logk)𝑂𝑘O(\log k)italic\_O ( roman\_log italic\_k ) as in [[17](#bib.bib17), [18](#bib.bib18)].
In [[17](#bib.bib17), [21](#bib.bib21)], a function
of k𝑘kitalic\_k and x𝑥xitalic\_x is defined as the lack of typicality
of x𝑥xitalic\_x in sets of complexity at most k𝑘kitalic\_k, and they then consider the
minimum
k𝑘kitalic\_k for which this function becomes =+0superscriptabsent0\stackrel{{\scriptstyle{}\_{+}}}{{=}}0start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 or very small. This is
equivalent to our notion of a typical set.
See the discussion of this function in Section [IV](#S4 "IV Non-Stochastic Objects ‣ Algorithmic Statistics").
In [[4](#bib.bib4), [5](#bib.bib5)], only optimal sets are considered, and the one
with the shortest program
is identified as the algorithmic minimal sufficient statistic of x𝑥xitalic\_x.
Formally, this is the shortest program that computes
a finite set S𝑆Sitalic\_S such that ([III.4](#S3.E4 "III.4 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")) holds.
###
III-C Properties of Sufficient Statistic
We start with a sequence of lemmas that will be used in the later
theorems.
Several of these lemmas have two versions: for implicit sets and for explicit
sets.
In these cases, S𝑆Sitalic\_S will denote Simplsubscript𝑆implS\_{\text{{impl}}}italic\_S start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT or Sexplsubscript𝑆explS\_{\text{{expl}}}italic\_S start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT respectively.
Below it is shown that the mutual information between every
typical set and the
data is not much less than K(K(x))𝐾𝐾𝑥K(K(x))italic\_K ( italic\_K ( italic\_x ) ), the complexity of the complexity
K(x)𝐾𝑥K(x)italic\_K ( italic\_x ) of the data x𝑥xitalic\_x.
For optimal sets it is at least that, and for
algorithmic minimal statistic it is equal to that.
The number of elements
of a typical set is determined by the following:
###### Lemma III.7
Let k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ).
If a set S𝑆Sitalic\_S is
(implicitly or explicitly)
typical for x𝑥xitalic\_x then
I(x:S)=+k−log|S|I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-\log|S|italic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - roman\_log | italic\_S |.
###### Proof:
By definition I(x:S)=+K(x)−K(x∣S\*)I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)-K(x\mid S^{\*})italic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
and by typicality K(x∣S\*)=+log|S|superscript𝐾conditional𝑥superscript𝑆𝑆K(x\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log|S|italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S |.
∎
Typicality, optimality, and minimal optimality
successively restrict the range of the cardinality (and complexity)
of a corresponding model for a data x𝑥xitalic\_x.
The above lemma states that for
(implicitly or explicitly)
typical S𝑆Sitalic\_S the cardinality |S|=Θ(2k−I(x:S))𝑆Θsuperscript2𝑘𝐼:𝑥𝑆|S|=\Theta(2^{k-I(x:S)})| italic\_S | = roman\_Θ ( 2 start\_POSTSUPERSCRIPT italic\_k - italic\_I ( italic\_x : italic\_S ) end\_POSTSUPERSCRIPT ).
The next lemma asserts that for implicitly typical S𝑆Sitalic\_S
the value I(x:S)I(x:S)italic\_I ( italic\_x : italic\_S ) can fall below K(k)𝐾𝑘K(k)italic\_K ( italic\_k ) by no more than an
additive logarithmic term.
###### Lemma III.8
Let k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ).
If a set S𝑆Sitalic\_S is (implicitly or explicitly)
typical for x𝑥xitalic\_x then I(x:S)>+K(k)−K(I(x:S))I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(k)-K(I(x:S))italic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) - italic\_K ( italic\_I ( italic\_x : italic\_S ) )
and log|S|<+k−K(k)+K(I(x:S))\log|S|\stackrel{{\scriptstyle{}\_{+}}}{{<}}k-K(k)+K(I(x:S))roman\_log | italic\_S | start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) + italic\_K ( italic\_I ( italic\_x : italic\_S ) ).
(Here, S𝑆Sitalic\_S is understood as Simplsubscript𝑆implS\_{\text{{impl}}}italic\_S start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT or Sexplsubscript𝑆explS\_{\text{{expl}}}italic\_S start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT respectively.)
###### Proof:
Writing k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ), since
| | | | |
| --- | --- | --- | --- |
| | k=+K(k,x)=+K(k)+K(x∣k\*)superscript𝑘𝐾𝑘𝑥superscript𝐾𝑘𝐾conditional𝑥superscript𝑘k\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k,x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k)+K(x\mid k^{\*})italic\_k start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k , italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) + italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | | (III.5) |
by ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")), we have
I(x:S)=+K(x)−K(x∣S\*)=+K(k)−[K(x∣S\*)−K(x∣k\*)]I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)-K(x\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k)-[K(x\mid S^{\*})-K(x\mid k^{\*})]italic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) - [ italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ].
Hence, it suffices to show K(x∣S\*)−K(x∣k\*)<+K(I(x:S))K(x\mid S^{\*})-K(x\mid k^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(I(x:S))italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_I ( italic\_x : italic\_S ) ).
Now, from an implicit description S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT we can
find the value =+log|S|=+k−I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log|S|\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-I(x:S)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S | start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_I ( italic\_x : italic\_S ). To recover k𝑘kitalic\_k we only
require an extra K(I(x:S))K(I(x:S))italic\_K ( italic\_I ( italic\_x : italic\_S ) ) bits apart from S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Therefore, K(k∣S\*)<+K(I(x:S))K(k\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(I(x:S))italic\_K ( italic\_k ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_I ( italic\_x : italic\_S ) ).
This reduces what we have to show to
K(x∣S\*)<+K(x∣k\*)+K(k∣S\*)superscript𝐾conditional𝑥superscript𝑆𝐾conditional𝑥superscript𝑘𝐾conditional𝑘superscript𝑆K(x\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(x\mid k^{\*})+K(k\mid S^{\*})italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_k ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) which is asserted by
Theorem [II.1](#S2.Thmlemma1 "Theorem II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics").
∎
The term I(x:S)I(x:S)italic\_I ( italic\_x : italic\_S ) is at least K(k)−2logK(k)𝐾𝑘2𝐾𝑘K(k)-2\log K(k)italic\_K ( italic\_k ) - 2 roman\_log italic\_K ( italic\_k ) where k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ).
For x𝑥xitalic\_x of length n𝑛nitalic\_n with
k>+nsuperscript𝑘𝑛k\stackrel{{\scriptstyle{}\_{+}}}{{>}}nitalic\_k start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n and K(k)>+l(k)>+lognsuperscript𝐾𝑘𝑙𝑘superscript𝑛K(k)\stackrel{{\scriptstyle{}\_{+}}}{{>}}l(k)\stackrel{{\scriptstyle{}\_{+}}}{{>}}\log nitalic\_K ( italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_l ( italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log italic\_n,
this yields I(x:S)>+logn−2loglognI(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{>}}\log n-2\log\log nitalic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log italic\_n - 2 roman\_log roman\_log italic\_n.
If we further restrict typical sets to optimal sets then
the possible number of elements in S𝑆Sitalic\_S is slightly restricted.
First we show that implicit optimality of a set with respect to a data
is equivalent to typicality with respect to the data
combined with effective constructability (determination) from the data.
###### Lemma III.9
A set S𝑆Sitalic\_S is
(implicitly or explicitly)
optimal for x𝑥xitalic\_x iff it is typical and K(S∣x\*)=+0superscript𝐾conditional𝑆superscript𝑥0K(S\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_S ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
###### Proof:
A set S𝑆Sitalic\_S is optimal iff ([III.3](#S3.E3 "III.3 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")) holds with equalities.
Rewriting K(x,S)=+K(x)+K(S∣x\*)superscript𝐾𝑥𝑆𝐾𝑥𝐾conditional𝑆superscript𝑥K(x,S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)+K(S\mid x^{\*})italic\_K ( italic\_x , italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) + italic\_K ( italic\_S ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
the first inequality becomes an equality iff K(S∣x\*)=+0superscript𝐾conditional𝑆superscript𝑥0K(S\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_S ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0,
and the second inequality becomes an equality iff K(x∣S\*)=+log|S|superscript𝐾conditional𝑥superscript𝑆𝑆K(x\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log|S|italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S |
(that is, S𝑆Sitalic\_S is a typical set).
∎
###### Lemma III.10
Let k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ).
If a set S𝑆Sitalic\_S is (implicitly or explicitly) optimal for x𝑥xitalic\_x, then
I(x:S)=+K(S)>+K(k)I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(k)italic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) and log|S|<+k−K(k)superscript𝑆𝑘𝐾𝑘\log|S|\stackrel{{\scriptstyle{}\_{+}}}{{<}}k-K(k)roman\_log | italic\_S | start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ).
###### Proof:
If S𝑆Sitalic\_S is optimal for x𝑥xitalic\_x, then
k=K(x)=+K(S)+K(x∣S\*)=+K(S)+log|S|𝑘𝐾𝑥superscript𝐾𝑆𝐾conditional𝑥superscript𝑆superscript𝐾𝑆𝑆k=K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)+K(x\mid S^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)+\log|S|italic\_k = italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) + italic\_K ( italic\_x ∣ italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) + roman\_log | italic\_S |.
From S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT we can find both K(S)=+l(S\*)superscript𝐾𝑆𝑙superscript𝑆K(S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}l(S^{\*})italic\_K ( italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_l ( italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
and =+log|S|superscriptabsent𝑆\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log|S|start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S | and hence k𝑘kitalic\_k, that is, K(k)<+K(S)superscript𝐾𝑘𝐾𝑆K(k)\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(S)italic\_K ( italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ).
We have I(x:S)=+K(S)−K(S∣x\*)=+K(S)I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)-K(S\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)italic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) - italic\_K ( italic\_S ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) by ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")),
Lemma [III.9](#S3.Thmlemma9 "Lemma III.9 ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics"), respectively.
This proves the first property.
Substitution of I(x:S)>+K(k)I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(k)italic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) in the expression
of Lemma [III.7](#S3.Thmlemma7 "Lemma III.7 ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") proves the second property.
∎
Figure 1: Range of statistic on the straight line I(x:S)=+K(x)−log|S|I(x:S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)-\log|S|italic\_I ( italic\_x : italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) - roman\_log | italic\_S |.
###
III-D Implicit Minimal Sufficient Statistic
A simplest implicitly optimal set (that is, of least complexity)
is an implicit algorithmic minimal sufficient statistic. We demonstrate that
Sk={y:K(y)≤k}superscript𝑆𝑘conditional-set𝑦𝐾𝑦𝑘S^{k}=\{y:K(y)\leq k\}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = { italic\_y : italic\_K ( italic\_y ) ≤ italic\_k }, the set of all
strings of complexity at most k𝑘kitalic\_k, is such a set. First we establish the
cardinality of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT:
###### Lemma III.11
log|Sk|=+k−K(k)superscriptsuperscript𝑆𝑘𝑘𝐾𝑘\log|S^{k}|\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ).
###### Proof:
The lower bound is easiest.
Denote by k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT of length K(k)𝐾𝑘K(k)italic\_K ( italic\_k )
a shortest program for k𝑘kitalic\_k. Every string s𝑠sitalic\_s of length
k−K(k)−c𝑘𝐾𝑘𝑐k-K(k)-citalic\_k - italic\_K ( italic\_k ) - italic\_c can be described in a self-delimiting manner
by prefixing it with k\*c\*superscript𝑘superscript𝑐k^{\*}c^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT italic\_c start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, hence K(s)<+k−c+2logcsuperscript𝐾𝑠𝑘𝑐2𝑐K(s)\stackrel{{\scriptstyle{}\_{+}}}{{<}}k-c+2\log citalic\_K ( italic\_s ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_c + 2 roman\_log italic\_c.
For a large
enough constant c𝑐citalic\_c, we have K(s)≤k𝐾𝑠𝑘K(s)\leq kitalic\_K ( italic\_s ) ≤ italic\_k
and hence there are Ω(2k−K(k))Ωsuperscript2𝑘𝐾𝑘\Omega(2^{k-K(k)})roman\_Ω ( 2 start\_POSTSUPERSCRIPT italic\_k - italic\_K ( italic\_k ) end\_POSTSUPERSCRIPT ) strings that are in Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT.
For the upper bound: by ([III.5](#S3.E5 "III.5 ‣ Proof: ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")),
all x∈Sk𝑥superscript𝑆𝑘x\in S^{k}italic\_x ∈ italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT satisfy K(x∣k\*)<+k−K(k)superscript𝐾conditional𝑥superscript𝑘𝑘𝐾𝑘K(x\mid k^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}k-K(k)italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ),
and there can only be O(2k−K(k))𝑂superscript2𝑘𝐾𝑘O(2^{k-K(k)})italic\_O ( 2 start\_POSTSUPERSCRIPT italic\_k - italic\_K ( italic\_k ) end\_POSTSUPERSCRIPT ) of them.
∎
From the definition of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT it follows that it is defined by k𝑘kitalic\_k
alone, and it is the same set that is optimal for all objects
of the same complexity k𝑘kitalic\_k.
###### Theorem III.12
The set Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT is implicitly optimal for every x𝑥xitalic\_x with K(x)=k𝐾𝑥𝑘K(x)=kitalic\_K ( italic\_x ) = italic\_k.
Also, we have K(Sk)=+K(k)superscript𝐾superscript𝑆𝑘𝐾𝑘K(S^{k})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k)italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ).
###### Proof:
From k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT we can compute both k𝑘kitalic\_k and k−l(k\*)=k−K(k)𝑘𝑙superscript𝑘𝑘𝐾𝑘k-l(k^{\*})=k-K(k)italic\_k - italic\_l ( italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) = italic\_k - italic\_K ( italic\_k )
and recursively enumerate Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT.
Since also log|Sk|=+k−K(k)superscriptsuperscript𝑆𝑘𝑘𝐾𝑘\log|S^{k}|\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) (Lemma [III.11](#S3.Thmlemma11 "Lemma III.11 ‣ III-D Implicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")),
the string
k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT plus a fixed program
is an implicit description of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT so that K(k)>+K(Sk)superscript𝐾𝑘𝐾superscript𝑆𝑘K(k)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(S^{k})italic\_K ( italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ).
Hence, K(x)>+K(Sk)+log|Sk|superscript𝐾𝑥𝐾superscript𝑆𝑘superscript𝑆𝑘K(x)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(S^{k})+\log|S^{k}|italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) + roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | and since K(x)𝐾𝑥K(x)italic\_K ( italic\_x )
is the shortest description by definition equality (=+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP) holds.
That is, Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT is optimal for x𝑥xitalic\_x.
By Lemma [III.10](#S3.Thmlemma10 "Lemma III.10 ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")
K(Sk)>+K(k)superscript𝐾superscript𝑆𝑘𝐾𝑘K(S^{k})\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(k)italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) which together with the reverse inequality above
yields K(Sk)=+K(k)superscript𝐾superscript𝑆𝑘𝐾𝑘K(S^{k})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k)italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) which shows the theorem.
∎
Again using Lemma [III.10](#S3.Thmlemma10 "Lemma III.10 ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") shows that the optimal set Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT
has least complexity among all optimal sets for x𝑥xitalic\_x, and therefore:
###### Corollary III.13
The set Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT is an implicit algorithmic minimal sufficient statistic
for every x𝑥xitalic\_x with K(x)=k𝐾𝑥𝑘K(x)=kitalic\_K ( italic\_x ) = italic\_k.
All algorithmic minimal sufficient statistics
S𝑆Sitalic\_S for x𝑥xitalic\_x have K(S)=+K(k)superscript𝐾𝑆𝐾𝑘K(S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k)italic\_K ( italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ), and therefore there are
O(2K(k))𝑂superscript2𝐾𝑘O(2^{K(k)})italic\_O ( 2 start\_POSTSUPERSCRIPT italic\_K ( italic\_k ) end\_POSTSUPERSCRIPT ) of them. At least one such a statistic (Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT)
is associated with every one of the O(2k)𝑂superscript2𝑘O(2^{k})italic\_O ( 2 start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) strings x𝑥xitalic\_x of
complexity k𝑘kitalic\_k.
Thus, while the idea of the algorithmic minimal sufficient
statistic is intuitively appealing, its unrestricted
use doesn’t seem to uncover most relevant aspects of reality.
The only relevant structure in the data
with respect to an algorithmic minimal sufficient statistic
is the Kolmogorov complexity.
To give an example, an initial
segment of 3.1415…3.1415…3.1415\ldots3.1415 … of length n𝑛nitalic\_n of complexity
logn+O(1)𝑛𝑂1\log n+O(1)roman\_log italic\_n + italic\_O ( 1 ) shares the same algorithmic sufficient statistic
with many (most?) binary strings of length logn+O(1)𝑛𝑂1\log n+O(1)roman\_log italic\_n + italic\_O ( 1 ).
###
III-E Explicit Minimal Sufficient Statistic
Let us now consider representations of finite sets that are explicit
in the sense that we can compute the cardinality of the set from the
representation.
####
III-E1 Explicit Minimal Sufficient Statistic: Particular Cases
###### Example III.14
The description program
enumerates all the elements of the set and halts.
Then a set like
Sk={y:K(y)≤k}superscript𝑆𝑘conditional-set𝑦𝐾𝑦𝑘S^{k}=\{y:K(y)\leq k\}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = { italic\_y : italic\_K ( italic\_y ) ≤ italic\_k } has complexity =+ksuperscriptabsent𝑘\stackrel{{\scriptstyle{}\_{+}}}{{=}}kstart\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k [[18](#bib.bib18)]:
Given the program we can find an element
not in Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT, which element by definition has complexity >kabsent𝑘>k> italic\_k.
Given Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT we can find this element
and hence Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT has complexity >+ksuperscriptabsent𝑘\stackrel{{\scriptstyle{}\_{+}}}{{>}}kstart\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k.
Let
| | | |
| --- | --- | --- |
| | Nk=|Sk|,superscript𝑁𝑘superscript𝑆𝑘N^{k}=|S^{k}|,italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | , | |
then by Lemma [III.11](#S3.Thmlemma11 "Lemma III.11 ‣ III-D Implicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") logNk=+k−K(k)superscriptsuperscript𝑁𝑘𝑘𝐾𝑘\log N^{k}\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)roman\_log italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ).
We can list Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT given k\*superscript𝑘k^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT and Nksuperscript𝑁𝑘N^{k}italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT
which shows K(Sk)<+ksuperscript𝐾superscript𝑆𝑘𝑘K(S^{k})\stackrel{{\scriptstyle{}\_{+}}}{{<}}kitalic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k.
♢normal-♢\diamondsuit♢
###### Example III.15
One way of implementing explicit finite representations is to
provide an explicit generation time for the enumeration process.
If we can generate Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT in time t𝑡titalic\_t recursively
using k𝑘kitalic\_k, then the previous argument shows that the complexity
of every number t′≥tsuperscript𝑡normal-′𝑡t^{\prime}\geq titalic\_t start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≥ italic\_t satisfies K(t′,k)≥k𝐾superscript𝑡normal-′𝑘𝑘K(t^{\prime},k)\geq kitalic\_K ( italic\_t start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_k ) ≥ italic\_k so that
K(t′)>+K(t′∣k\*)>+k−K(k)superscript𝐾superscript𝑡normal-′𝐾conditionalsuperscript𝑡normal-′superscript𝑘superscript𝑘𝐾𝑘K(t^{\prime})\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(t^{\prime}\mid k^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{>}}k-K(k)italic\_K ( italic\_t start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_t start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) by ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")).
This means that
t𝑡titalic\_t is a huge time which as a function of k𝑘kitalic\_k rises faster
than every computable function.
This argument also shows
that explicit enumerative descriptions of sets S𝑆Sitalic\_S containing x𝑥xitalic\_x by
an enumerative process p𝑝pitalic\_p plus a limit on the computation time t𝑡titalic\_t
may take only l(p)+K(t)𝑙𝑝𝐾𝑡l(p)+K(t)italic\_l ( italic\_p ) + italic\_K ( italic\_t ) bits
(with K(t)≤logt+2loglogt𝐾𝑡𝑡2𝑡K(t)\leq\log t+2\log\log titalic\_K ( italic\_t ) ≤ roman\_log italic\_t + 2 roman\_log roman\_log italic\_t)
but logt𝑡\log troman\_log italic\_t unfortunately becomes noncomputably large!
♢normal-♢\diamondsuit♢
###### Example III.16
Another way is to indicate the element of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT that requires
the longest generation time as part of the dovetailing process,
for example by its index i𝑖iitalic\_i in the enumeration, i≤2k−K(k)𝑖superscript2𝑘𝐾𝑘i\leq 2^{k-K(k)}italic\_i ≤ 2 start\_POSTSUPERSCRIPT italic\_k - italic\_K ( italic\_k ) end\_POSTSUPERSCRIPT.
Then, K(i∣k)<+k−K(k)superscript𝐾conditional𝑖𝑘𝑘𝐾𝑘K(i\mid k)\stackrel{{\scriptstyle{}\_{+}}}{{<}}k-K(k)italic\_K ( italic\_i ∣ italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ).
In fact, since a shortest program p𝑝pitalic\_p for the i𝑖iitalic\_ith element
together with k𝑘kitalic\_k allows us to generate Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT explicitly,
and abive we have seen that explicit description format
yoelds K(Sk)=+ksuperscript𝐾superscript𝑆𝑘𝑘K(S^{k})\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k, we find
we have K(p,k)>+ksuperscript𝐾𝑝𝑘𝑘K(p,k)\stackrel{{\scriptstyle{}\_{+}}}{{>}}kitalic\_K ( italic\_p , italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k and hence K(p)>+k−K(k)superscript𝐾𝑝𝑘𝐾𝑘K(p)\stackrel{{\scriptstyle{}\_{+}}}{{>}}k-K(k)italic\_K ( italic\_p ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ).
♢normal-♢\diamondsuit♢
In other cases the generation time is simply recursive in the input:
Sn={y:l(y)≤n}subscript𝑆𝑛conditional-set𝑦𝑙𝑦𝑛S\_{n}=\{y:l(y)\leq n\}italic\_S start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT = { italic\_y : italic\_l ( italic\_y ) ≤ italic\_n } so that
K(Sn)=+K(n)≤logn+2loglognsuperscript𝐾subscript𝑆𝑛𝐾𝑛𝑛2𝑛K(S\_{n})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(n)\leq\log n+2\log\log nitalic\_K ( italic\_S start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_n ) ≤ roman\_log italic\_n + 2 roman\_log roman\_log italic\_n.
That is, this sufficient
statistic for a random string x𝑥xitalic\_x with
K(x)=+n+K(n)superscript𝐾𝑥𝑛𝐾𝑛K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}n+K(n)italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n + italic\_K ( italic\_n ) has complexity K(n)𝐾𝑛K(n)italic\_K ( italic\_n ) both for implicit descriptions and
explicit descriptions: differences in complexity arise
only for nonrandom strings (but not too nonrandom, for
K(x)=+0superscript𝐾𝑥0K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 these differences vanish again).
###### Lemma III.17
Snsubscript𝑆𝑛S\_{n}italic\_S start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT is an example of a
minimal sufficient statistic, both explicit and implicit,
for all x𝑥xitalic\_x with K(x)=+n+K(n)superscript𝐾𝑥𝑛𝐾𝑛K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}n+K(n)italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n + italic\_K ( italic\_n ).
###### Proof:
The set Snsubscript𝑆𝑛S\_{n}italic\_S start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT is a sufficient statistic for x𝑥xitalic\_x since
K(x)=+K(Sn)+log|Sn|superscript𝐾𝑥𝐾subscript𝑆𝑛subscript𝑆𝑛K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S\_{n})+\log|S\_{n}|italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ) + roman\_log | italic\_S start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT |. It is minimal since
by Lemma [III.10](#S3.Thmlemma10 "Lemma III.10 ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") we must have K(S)>+K(K(x))superscript𝐾𝑆𝐾𝐾𝑥K(S)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(K(x))italic\_K ( italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_K ( italic\_x ) ) for
implicit, and hence for
explicit sufficient statistics. It is evident that Snsubscript𝑆𝑛S\_{n}italic\_S start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT is explicit:
|Sn|=2nsubscript𝑆𝑛superscript2𝑛|S\_{n}|=2^{n}| italic\_S start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT | = 2 start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT.
∎
It turns out that some strings cannot thus be explicitly
represented parsimonously
with low-complexity models
(so that one necessarily has bad high complexity
models like Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT above).
For explicit representations,
[[17](#bib.bib17)]
has demonstrated the
existence of a class of strings called non-stochastic
that don’t have efficient two-part representations
with K(x)=+K(S)+log|S|superscript𝐾𝑥𝐾𝑆𝑆K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)+\log|S|italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) + roman\_log | italic\_S | (x∈S𝑥𝑆x\in Sitalic\_x ∈ italic\_S) with K(S)𝐾𝑆K(S)italic\_K ( italic\_S )
significantly less than K(x)𝐾𝑥K(x)italic\_K ( italic\_x ). This result does not yet
enable us to exhibit an explicit minimal sufficient statistic
for such a string.
But in Section [IV](#S4 "IV Non-Stochastic Objects ‣ Algorithmic Statistics") we improve
these results to the best possible, simultaneously establishing
explicit minimal sufficient statistics for the subject
ultimate non-stochastic strings:
###### Lemma III.18
For every length n𝑛nitalic\_n, there exist strings x𝑥xitalic\_x of length n𝑛nitalic\_n
with K(x∣n)=+nsuperscript𝐾conditional𝑥𝑛𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}nitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n for which {x}𝑥\{x\}{ italic\_x } is an explicit
minimal sufficient statistic.
The proof
is deferred to the end of Section [IV](#S4 "IV Non-Stochastic Objects ‣ Algorithmic Statistics").
####
III-E2 Explicit Minimal Near-Sufficient Statistic: General Case
Again, consider the special set Sk={y:K(y)≤k}superscript𝑆𝑘conditional-set𝑦𝐾𝑦𝑘S^{k}=\{y:K(y)\leq k\}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = { italic\_y : italic\_K ( italic\_y ) ≤ italic\_k }.
As we have seen earlier, Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT itself cannot be explicitly
optimal for x𝑥xitalic\_x since K(Sk)=+ksuperscript𝐾superscript𝑆𝑘𝑘K(S^{k})\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k and logNk=+k−K(k)superscriptsuperscript𝑁𝑘𝑘𝐾𝑘\log N^{k}\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)roman\_log italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ),
and therefore K(Sk)+logNk=+2k−K(k)superscript𝐾superscript𝑆𝑘superscript𝑁𝑘2𝑘𝐾𝑘K(S^{k})+\log N^{k}\stackrel{{\scriptstyle{}\_{+}}}{{=}}2k-K(k)italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) + roman\_log italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 2 italic\_k - italic\_K ( italic\_k ) which considerably
exceeds k𝑘kitalic\_k.
However, it turns out that a closely related set (Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT
below) is explicitly near-optimal.
Let Iyksubscriptsuperscript𝐼𝑘𝑦I^{k}\_{y}italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT denote the index of y𝑦yitalic\_y in the standard enumeration of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT,
where all indexes are padded to the same length =+k−K(k)superscriptabsent𝑘𝐾𝑘\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) with 0’s in
front.
For K(x)=k𝐾𝑥𝑘K(x)=kitalic\_K ( italic\_x ) = italic\_k, let mxsubscript𝑚𝑥m\_{x}italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT denote the longest joint prefix of
Ixksubscriptsuperscript𝐼𝑘𝑥I^{k}\_{x}italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT and Nksuperscript𝑁𝑘N^{k}italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT, and let
| | | |
| --- | --- | --- |
| | Ixk=mx0ix,Nk=mx1nx.formulae-sequencesubscriptsuperscript𝐼𝑘𝑥subscript𝑚𝑥0subscript𝑖𝑥superscript𝑁𝑘subscript𝑚𝑥1subscript𝑛𝑥I^{k}\_{x}=m\_{x}0i\_{x},\quad N^{k}=m\_{x}1n\_{x}.italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT = italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 0 italic\_i start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT , italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 1 italic\_n start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT . | |
###### Lemma III.19
For K(x)=k𝐾𝑥𝑘K(x)=kitalic\_K ( italic\_x ) = italic\_k, the set
Smxk={y∈Sk:mx0a prefix of Iyk}subscriptsuperscript𝑆𝑘subscript𝑚𝑥conditional-set𝑦superscript𝑆𝑘subscript𝑚𝑥0a prefix of subscriptsuperscript𝐼𝑘𝑦S^{k}\_{m\_{x}}=\{y\in S^{k}:m\_{x}0\;\;\mbox{\rm a prefix of }I^{k}\_{y}\}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT = { italic\_y ∈ italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT : italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 0 a prefix of italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT }
satisfies
| | | | |
| --- | --- | --- | --- |
| | log|Smxk|subscriptsuperscript𝑆𝑘subscript𝑚𝑥\displaystyle\log|S^{k}\_{m\_{x}}|roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT | | =+k−K(k)−l(mx),superscriptabsent𝑘𝐾𝑘𝑙subscript𝑚𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)-l(m\_{x}),start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) , | |
| | K(Smxk)𝐾subscriptsuperscript𝑆𝑘subscript𝑚𝑥\displaystyle K(S^{k}\_{m\_{x}})italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ) | <+K(k)+K(mx)<+K(k)+l(mx)+K(l(mx)).superscriptabsent𝐾𝑘𝐾subscript𝑚𝑥superscript𝐾𝑘𝑙subscript𝑚𝑥𝐾𝑙subscript𝑚𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(k)+K(m\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(k)+l(m\_{x})+K(l(m\_{x})).start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) + italic\_K ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) + italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) + italic\_K ( italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ) . | |
Hence it is explicitly near-optimal
for x𝑥xitalic\_x (up to an addive K(l(mx))<+K(k)<+logk+2loglogksuperscript𝐾𝑙subscript𝑚𝑥𝐾𝑘superscript𝑘2𝑘K(l(m\_{x}))\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(k)\stackrel{{\scriptstyle{}\_{+}}}{{<}}\log k+2\log\log kitalic\_K ( italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log italic\_k + 2 roman\_log roman\_log italic\_k
term).
###### Proof:
We can describe x𝑥xitalic\_x by k\*mx\*ixsuperscript𝑘superscriptsubscript𝑚𝑥subscript𝑖𝑥k^{\*}m\_{x}^{\*}i\_{x}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT italic\_i start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT where mx0ixsubscript𝑚𝑥0subscript𝑖𝑥m\_{x}0i\_{x}italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 0 italic\_i start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT is the
index of x𝑥xitalic\_x in the enumeration of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT.
Moreover, k\*mx\*superscript𝑘superscriptsubscript𝑚𝑥k^{\*}m\_{x}^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT explicitly
describes the set Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT. Namely, using k𝑘kitalic\_k we can recursively
enumerate Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT.
At some point the first string z∈Smxk𝑧subscriptsuperscript𝑆𝑘subscript𝑚𝑥z\in S^{k}\_{m\_{x}}italic\_z ∈ italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT is enumerated (index
Izk=mx00…0superscriptsubscript𝐼𝑧𝑘subscript𝑚𝑥00…0I\_{z}^{k}=m\_{x}00\ldots 0italic\_I start\_POSTSUBSCRIPT italic\_z end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 00 … 0).
By assumption Ixk=mx0…superscriptsubscript𝐼𝑥𝑘subscript𝑚𝑥0…I\_{x}^{k}=m\_{x}0\ldotsitalic\_I start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 0 … and
Nk=mx1…superscript𝑁𝑘subscript𝑚𝑥1…N^{k}=m\_{x}1\ldotsitalic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 1 ….
Therefore, in the enumeration of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT
eventually string u𝑢uitalic\_u with Iuk=mx011…1superscriptsubscript𝐼𝑢𝑘subscript𝑚𝑥011…1I\_{u}^{k}=m\_{x}011\ldots 1italic\_I start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 011 … 1 occurs
which is the last string in the enumeration of Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT.
Thus, the size
of Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT is precisely 2l(Nk)−l(mx)superscript2𝑙superscript𝑁𝑘𝑙subscript𝑚𝑥2^{l(N^{k})-l(m\_{x})}2 start\_POSTSUPERSCRIPT italic\_l ( italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) end\_POSTSUPERSCRIPT, where
l(Nk)−l(mx)=+l(nx)=+log|Smxk|superscript𝑙superscript𝑁𝑘𝑙subscript𝑚𝑥𝑙subscript𝑛𝑥superscriptsubscriptsuperscript𝑆𝑘subscript𝑚𝑥l(N^{k})-l(m\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{=}}l(n\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log|S^{k}\_{m\_{x}}|italic\_l ( italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_l ( italic\_n start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT |, and Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT
is explicitly described by k\*mx\*superscript𝑘superscriptsubscript𝑚𝑥k^{\*}m\_{x}^{\*}italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
Since l(k\*mx0ix)=+ksuperscript𝑙superscript𝑘subscript𝑚𝑥0subscript𝑖𝑥𝑘l(k^{\*}m\_{x}0i\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_l ( italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 0 italic\_i start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k
and log|Smxk|=+k−K(k)−l(mx)superscriptsubscriptsuperscript𝑆𝑘subscript𝑚𝑥𝑘𝐾𝑘𝑙subscript𝑚𝑥\log|S^{k}\_{m\_{x}}|\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)-l(m\_{x})roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT | start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) we have
| | | | |
| --- | --- | --- | --- |
| | K(Smxk)+log|Smxk|𝐾subscriptsuperscript𝑆𝑘subscript𝑚𝑥subscriptsuperscript𝑆𝑘subscript𝑚𝑥\displaystyle K(S^{k}\_{m\_{x}})+\log|S^{k}\_{m\_{x}}|italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ) + roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT | | =+K(k)+K(mx)+k−K(k)−l(mx)superscriptabsent𝐾𝑘𝐾subscript𝑚𝑥𝑘𝐾𝑘𝑙subscript𝑚𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k)+K(m\_{x})+k-K(k)-l(m\_{x})start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) + italic\_K ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) + italic\_k - italic\_K ( italic\_k ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) | |
| | | =+k+K(mx)−l(mx)<+k+K(l(mx)).superscriptabsent𝑘𝐾subscript𝑚𝑥𝑙subscript𝑚𝑥superscript𝑘𝐾𝑙subscript𝑚𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}k+K(m\_{x})-l(m\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{<}}k+K(l(m\_{x})).start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k + italic\_K ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k + italic\_K ( italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ) . | |
This shows Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT is explicitly near optimal for x𝑥xitalic\_x (up
to an additive logarithmic term).
∎
###### Lemma III.20
Every explicit optimal set S⊆Sk𝑆superscript𝑆𝑘S\subseteq S^{k}italic\_S ⊆ italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT
containing x𝑥xitalic\_x satisfies
| | | |
| --- | --- | --- |
| | K(S)>+K(k)+l(mx)−K(l(mx)).superscript𝐾𝑆𝐾𝑘𝑙subscript𝑚𝑥𝐾𝑙subscript𝑚𝑥K(S)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(k)+l(m\_{x})-K(l(m\_{x})).italic\_K ( italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) + italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) - italic\_K ( italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ) . | |
###### Proof:
If S⊆Sk𝑆superscript𝑆𝑘S\subseteq S^{k}italic\_S ⊆ italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT is explicitly optimal for x𝑥xitalic\_x, then we can
find k𝑘kitalic\_k from S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT (as in the proof of Lemma [III.10](#S3.Thmlemma10 "Lemma III.10 ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")),
and given k𝑘kitalic\_k and S𝑆Sitalic\_S we find K(k)𝐾𝑘K(k)italic\_K ( italic\_k ) as in Theorem [II.1](#S2.Thmlemma1 "Theorem II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics").
Hence, given S\*superscript𝑆S^{\*}italic\_S start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, we can enumerate Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT and determine the
maximal index Iyksubscriptsuperscript𝐼𝑘𝑦I^{k}\_{y}italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT of a y∈S𝑦𝑆y\in Sitalic\_y ∈ italic\_S.
Since also x∈S𝑥𝑆x\in Sitalic\_x ∈ italic\_S, the numbers
Iyk,Ixk,Nksubscriptsuperscript𝐼𝑘𝑦subscriptsuperscript𝐼𝑘𝑥superscript𝑁𝑘I^{k}\_{y},I^{k}\_{x},N^{k}italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT , italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT , italic\_N start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT have a maximal common prefix mxsubscript𝑚𝑥m\_{x}italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT.
Write
Ixk=mx0ixsubscriptsuperscript𝐼𝑘𝑥subscript𝑚𝑥0subscript𝑖𝑥I^{k}\_{x}=m\_{x}0i\_{x}italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT = italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 0 italic\_i start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT with l(ix)=+k−K(k)−l(mx)superscript𝑙subscript𝑖𝑥𝑘𝐾𝑘𝑙subscript𝑚𝑥l(i\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)-l(m\_{x})italic\_l ( italic\_i start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) by
Lemma [III.10](#S3.Thmlemma10 "Lemma III.10 ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics").
Given l(mx)𝑙subscript𝑚𝑥l(m\_{x})italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) we can determine mxsubscript𝑚𝑥m\_{x}italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT from Iyksubscriptsuperscript𝐼𝑘𝑦I^{k}\_{y}italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_y end\_POSTSUBSCRIPT. Hence, from
S,l(mx)𝑆𝑙subscript𝑚𝑥S,l(m\_{x})italic\_S , italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ), and ixsubscript𝑖𝑥i\_{x}italic\_i start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT we can reconstruct x𝑥xitalic\_x.
That is,
K(S)+K(l(mx))+l(Ixk)−l(mx)>+ksuperscript𝐾𝑆𝐾𝑙subscript𝑚𝑥𝑙subscriptsuperscript𝐼𝑘𝑥𝑙subscript𝑚𝑥𝑘K(S)+K(l(m\_{x}))+l(I^{k}\_{x})-l(m\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{>}}kitalic\_K ( italic\_S ) + italic\_K ( italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ) + italic\_l ( italic\_I start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k, which yields the lemma.
∎
Lemmas [III.19](#S3.Thmlemma19 "Lemma III.19 ‣ III-E2 Explicit Minimal Near-Sufficient Statistic: General Case ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics"), [III.20](#S3.Thmlemma20 "Lemma III.20 ‣ III-E2 Explicit Minimal Near-Sufficient Statistic: General Case ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") demonstrate:
###### Theorem III.21
The set Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT is an explicit
algorithmic minimal near-sufficient statistic
for x𝑥xitalic\_x among subsets of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT in the following sense:
| | | | |
| --- | --- | --- | --- |
| | |K(Smxk)−K(k)−l(mx)|𝐾subscriptsuperscript𝑆𝑘subscript𝑚𝑥𝐾𝑘𝑙subscript𝑚𝑥\displaystyle|K(S^{k}\_{m\_{x}})-K(k)-l(m\_{x})|| italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ) - italic\_K ( italic\_k ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) | | <+K(l(mx)),superscriptabsent𝐾𝑙subscript𝑚𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(l(m\_{x})),start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ) , | |
| | log|Smxk|subscriptsuperscript𝑆𝑘subscript𝑚𝑥\displaystyle\log|S^{k}\_{m\_{x}}|roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT | | =+k−K(k)−l(mx).superscriptabsent𝑘𝐾𝑘𝑙subscript𝑚𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)-l(m\_{x}).start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) . | |
Hence K(Smxk)+log|Smxk|=+k±K(l(mx))superscript𝐾subscriptsuperscript𝑆𝑘subscript𝑚𝑥subscriptsuperscript𝑆𝑘subscript𝑚𝑥plus-or-minus𝑘𝐾𝑙subscript𝑚𝑥K(S^{k}\_{m\_{x}})+\log|S^{k}\_{m\_{x}}|\stackrel{{\scriptstyle{}\_{+}}}{{=}}k\pm K(l(m\_{x}))italic\_K ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ) + roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT | start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k ± italic\_K ( italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ).
Note, K(l(mx))<+logk+2loglogksuperscript𝐾𝑙subscript𝑚𝑥𝑘2𝑘K(l(m\_{x}))\stackrel{{\scriptstyle{}\_{+}}}{{<}}\log k+2\log\log kitalic\_K ( italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log italic\_k + 2 roman\_log roman\_log italic\_k.
####
III-E3 Almost Always “Sufficient”
We have not completely succeeded in giving a concrete algorithmic
explicit minimal sufficient statistic. However, we can show
that Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT is almost always minimal sufficient.
The complexity and cardinality of Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT depend on l(mx)𝑙subscript𝑚𝑥l(m\_{x})italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT )
which will in turn depend on x𝑥xitalic\_x.
One extreme is l(mx)=+0superscript𝑙subscript𝑚𝑥0l(m\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 which happens for the majority of x𝑥xitalic\_x’s
with K(x)=k𝐾𝑥𝑘K(x)=kitalic\_K ( italic\_x ) = italic\_k—for example, the first 99.9% in the enumeration order.
For those x𝑥xitalic\_x’s we can replace “near-sufficient” by
“sufficient” in Theorem [III.21](#S3.Thmlemma21 "Theorem III.21 ‣ III-E2 Explicit Minimal Near-Sufficient Statistic: General Case ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics").
Can the other extreme be reached?
This is the case when
x𝑥xitalic\_x is enumerated close to the end of the enumeration
of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT.
For example, this happens for the “non-stochastic” objects
of which the existence was proven by Shen [[17](#bib.bib17)] (see
Section [IV](#S4 "IV Non-Stochastic Objects ‣ Algorithmic Statistics")).
For such objects, l(mx)𝑙subscript𝑚𝑥l(m\_{x})italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) grows to =+k−K(k)superscriptabsent𝑘𝐾𝑘\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-K(k)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) and
the complexity of Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT rises to =+ksuperscriptabsent𝑘\stackrel{{\scriptstyle{}\_{+}}}{{=}}kstart\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k
while log|Smxk|subscriptsuperscript𝑆𝑘subscript𝑚𝑥\log|S^{k}\_{m\_{x}}|roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT | drops to =+0superscriptabsent0\stackrel{{\scriptstyle{}\_{+}}}{{=}}0start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
That is, the explicit
algorithmic minimal sufficient statistic for x𝑥xitalic\_x is essentially
x𝑥xitalic\_x itself.
For those x𝑥xitalic\_x’s we can also replace “near-sufficient” with “sufficient”
in Theorem [III.21](#S3.Thmlemma21 "Theorem III.21 ‣ III-E2 Explicit Minimal Near-Sufficient Statistic: General Case ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics").
Generally:
for the overwhelming majority of data x𝑥xitalic\_x of complexity k𝑘kitalic\_k
the set Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT is an explicit algorithmic minimal sufficient statistic
among subsets of Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT (since l(mx)=+0superscript𝑙subscript𝑚𝑥0l(m\_{x})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0).
The following discussion will put what was said above into a
more illuminating context.
Let
| | | |
| --- | --- | --- |
| | X(r)={x:l(mx)≥r}.𝑋𝑟conditional-set𝑥𝑙subscript𝑚𝑥𝑟X(r)=\mathopen{\{}\,x:l(m\_{x})\geq r\,\mathclose{\}}.italic\_X ( italic\_r ) = { italic\_x : italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ≥ italic\_r } . | |
The set X(r)𝑋𝑟X(r)italic\_X ( italic\_r ) is infinite, but we can break it into slices and bound each
slice separately.
###### Lemma III.22
| | | |
| --- | --- | --- |
| | |X(r)⋂(Sk∖Sk−1)|≤2−r+1|Sk|.𝑋𝑟superscript𝑆𝑘superscript𝑆𝑘1superscript2𝑟1superscript𝑆𝑘|X(r)\bigcap(S^{k}\setminus S^{k-1})|\leq 2^{-r+1}|S^{k}|.| italic\_X ( italic\_r ) ⋂ ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ∖ italic\_S start\_POSTSUPERSCRIPT italic\_k - 1 end\_POSTSUPERSCRIPT ) | ≤ 2 start\_POSTSUPERSCRIPT - italic\_r + 1 end\_POSTSUPERSCRIPT | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | . | |
###### Proof:
For every x𝑥xitalic\_x in the set defined by the left-hand side of the inequality,
we have l(mx)≥r𝑙subscript𝑚𝑥𝑟l(m\_{x})\geq ritalic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ≥ italic\_r, and the
length of continuation of mxsubscript𝑚𝑥m\_{x}italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT to the total padded index of x𝑥xitalic\_x is
≤⌈log|Sk|⌉−r≤log|Sk|−r+1absentsuperscript𝑆𝑘𝑟superscript𝑆𝑘𝑟1\leq{\lceil\log|S^{k}|\rceil}-r\leq\log|S^{k}|-r+1≤ ⌈ roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | ⌉ - italic\_r ≤ roman\_log | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | - italic\_r + 1.
Moreover, all these indices share the same first r𝑟ritalic\_r bits.
This proves the lemma.
∎
###### Theorem III.23
| | | |
| --- | --- | --- |
| | ∑x∈X(r)2−K(x)≤2−r+2.subscript𝑥𝑋𝑟superscript2𝐾𝑥superscript2𝑟2\sum\_{x\in X(r)}2^{-K(x)}\leq 2^{-r+2}.∑ start\_POSTSUBSCRIPT italic\_x ∈ italic\_X ( italic\_r ) end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_x ) end\_POSTSUPERSCRIPT ≤ 2 start\_POSTSUPERSCRIPT - italic\_r + 2 end\_POSTSUPERSCRIPT . | |
###### Proof:
Let us prove first
| | | | |
| --- | --- | --- | --- |
| | ∑k≥02−k|Sk|≤2.subscript𝑘0superscript2𝑘superscript𝑆𝑘2\sum\_{k\geq 0}2^{-k}|S^{k}|\leq 2.∑ start\_POSTSUBSCRIPT italic\_k ≥ 0 end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_k end\_POSTSUPERSCRIPT | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | ≤ 2 . | | (III.6) |
By the Kraft inequality, we have, with tk=|Sk∖Sk−1|subscript𝑡𝑘superscript𝑆𝑘superscript𝑆𝑘1t\_{k}=|S^{k}\setminus S^{k-1}|italic\_t start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT = | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ∖ italic\_S start\_POSTSUPERSCRIPT italic\_k - 1 end\_POSTSUPERSCRIPT |,
| | | |
| --- | --- | --- |
| | ∑k≥02−ktk≤1,subscript𝑘0superscript2𝑘subscript𝑡𝑘1\sum\_{k\geq 0}2^{-k}t\_{k}\leq 1,∑ start\_POSTSUBSCRIPT italic\_k ≥ 0 end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_k end\_POSTSUPERSCRIPT italic\_t start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ≤ 1 , | |
since Sksuperscript𝑆𝑘S^{k}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT is in 1-1 correspondence with the prefix programs of length
≤kabsent𝑘\leq k≤ italic\_k.
Hence
| | | |
| --- | --- | --- |
| | ∑k≥02−k|Sk|=∑k≥02−k∑i=0kti=∑i≥0ti∑k=i∞2−k=∑i≥0ti2−i+1≤2.subscript𝑘0superscript2𝑘superscript𝑆𝑘subscript𝑘0superscript2𝑘superscriptsubscript𝑖0𝑘subscript𝑡𝑖subscript𝑖0subscript𝑡𝑖superscriptsubscript𝑘𝑖superscript2𝑘subscript𝑖0subscript𝑡𝑖superscript2𝑖12\begin{split}\sum\_{k\geq 0}2^{-k}|S^{k}|&=\sum\_{k\geq 0}2^{-k}\sum\_{i=0}^{k}t\_{i}=\sum\_{i\geq 0}t\_{i}\sum\_{k=i}^{\infty}2^{-k}\\
&=\sum\_{i\geq 0}t\_{i}2^{-i+1}\leq 2.\end{split}start\_ROW start\_CELL ∑ start\_POSTSUBSCRIPT italic\_k ≥ 0 end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_k end\_POSTSUPERSCRIPT | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | end\_CELL start\_CELL = ∑ start\_POSTSUBSCRIPT italic\_k ≥ 0 end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_k end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_i = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT italic\_t start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = ∑ start\_POSTSUBSCRIPT italic\_i ≥ 0 end\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_k = italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ∞ end\_POSTSUPERSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_k end\_POSTSUPERSCRIPT end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL = ∑ start\_POSTSUBSCRIPT italic\_i ≥ 0 end\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_i + 1 end\_POSTSUPERSCRIPT ≤ 2 . end\_CELL end\_ROW | |
For the statement of the lemma, we have
| | | | |
| --- | --- | --- | --- |
| | ∑x∈X(r)2−K(x)subscript𝑥𝑋𝑟superscript2𝐾𝑥\displaystyle\sum\_{x\in X(r)}2^{-K(x)}∑ start\_POSTSUBSCRIPT italic\_x ∈ italic\_X ( italic\_r ) end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_x ) end\_POSTSUPERSCRIPT | =∑k≥02−k|X(r)⋂(Sk∖Sk−1)|absentsubscript𝑘0superscript2𝑘𝑋𝑟superscript𝑆𝑘superscript𝑆𝑘1\displaystyle=\sum\_{k\geq 0}2^{-k}|X(r)\bigcap(S^{k}\setminus S^{k-1})|= ∑ start\_POSTSUBSCRIPT italic\_k ≥ 0 end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_k end\_POSTSUPERSCRIPT | italic\_X ( italic\_r ) ⋂ ( italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ∖ italic\_S start\_POSTSUPERSCRIPT italic\_k - 1 end\_POSTSUPERSCRIPT ) | | |
| | | ≤2−r+1∑k≥02−k|Sk|≤2−r+2,absentsuperscript2𝑟1subscript𝑘0superscript2𝑘superscript𝑆𝑘superscript2𝑟2\displaystyle\leq 2^{-r+1}\sum\_{k\geq 0}2^{-k}|S^{k}|\leq 2^{-r+2},≤ 2 start\_POSTSUPERSCRIPT - italic\_r + 1 end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_k ≥ 0 end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_k end\_POSTSUPERSCRIPT | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | ≤ 2 start\_POSTSUPERSCRIPT - italic\_r + 2 end\_POSTSUPERSCRIPT , | |
where in the last inequality we used ([III.6](#S3.E6 "III.6 ‣ Proof: ‣ III-E3 Almost Always “Sufficient” ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")).
∎
This theorem can be interpreted as follows, (we rely here on a discussion,
unconnected with the present topic, about universal probability
with L. A. Levin in 1973).
The above theorem states ∑x∈X(r)𝐦(x)≤2−r+2subscript𝑥𝑋𝑟𝐦𝑥superscript2𝑟2\sum\_{x\in X(r)}{\mathbf{m}}(x)\leq 2^{-r+2}∑ start\_POSTSUBSCRIPT italic\_x ∈ italic\_X ( italic\_r ) end\_POSTSUBSCRIPT bold\_m ( italic\_x ) ≤ 2 start\_POSTSUPERSCRIPT - italic\_r + 2 end\_POSTSUPERSCRIPT.
By the multiplicative dominating property of 𝐦(x)𝐦𝑥{\mathbf{m}}(x)bold\_m ( italic\_x ) with respect
to every lower semicomputable semimeasure,
it follows that for every computable measure ν𝜈\nuitalic\_ν, we have
∑x∈X(r)ν(x)<\*2−rsuperscriptsubscript𝑥𝑋𝑟𝜈𝑥superscript2𝑟\sum\_{x\in X(r)}\nu(x)\stackrel{{\scriptstyle{}\_{\*}}}{{<}}2^{-r}∑ start\_POSTSUBSCRIPT italic\_x ∈ italic\_X ( italic\_r ) end\_POSTSUBSCRIPT italic\_ν ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 2 start\_POSTSUPERSCRIPT - italic\_r end\_POSTSUPERSCRIPT.
Thus, the set of objects x𝑥xitalic\_x for which l(mx)𝑙subscript𝑚𝑥l(m\_{x})italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) is large
has small probability with
respect to every computable probability distribution.
To shed light on the exceptional nature of strings x𝑥xitalic\_x with large l(mx)𝑙subscript𝑚𝑥l(m\_{x})italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT )
from yet another direction, let χ𝜒\chiitalic\_χ be the infinite binary sequence,
the halting sequence,
which constitutes the characteristic function of
the halting problem for our universal Turing machine: the i𝑖iitalic\_ith bit
of χ𝜒\chiitalic\_χ is 1 of the machine halts on the i𝑖iitalic\_ith program, and is 0 otherwise.
The expression
| | | |
| --- | --- | --- |
| | I(χ:x)=K(x)−K(x∣χ)I(\chi:x)=K(x)-K(x\mid\chi)italic\_I ( italic\_χ : italic\_x ) = italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_χ ) | |
shows the amount of information in the halting sequence about the string
x𝑥xitalic\_x.
(For an infinite sequence η𝜂\etaitalic\_η, we go back formally to the definition
I(η:x)=K(x)−K(x∣η)I(\eta:x)=K(x)-K(x\mid\eta)italic\_I ( italic\_η : italic\_x ) = italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_η ) of [[10](#bib.bib10)], since
introducing a notion of η\*superscript𝜂\eta^{\*}italic\_η start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT in place of η𝜂\etaitalic\_η here
has not been shown yet to bring any benefits.)
We have
| | | |
| --- | --- | --- |
| | ∑x𝐦(x)2I(χ:x)=∑x2−K(x∣χ)≤1.subscript𝑥𝐦𝑥superscript2𝐼:𝜒𝑥subscript𝑥superscript2𝐾conditional𝑥𝜒1\sum\_{x}{\mathbf{m}}(x)2^{I(\chi:x)}=\sum\_{x}2^{-K(x\mid\chi)}\leq 1.∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT bold\_m ( italic\_x ) 2 start\_POSTSUPERSCRIPT italic\_I ( italic\_χ : italic\_x ) end\_POSTSUPERSCRIPT = ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_x ∣ italic\_χ ) end\_POSTSUPERSCRIPT ≤ 1 . | |
Therefore, if we introduce a new quantity X′(r)superscript𝑋′𝑟X^{\prime}(r)italic\_X start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ( italic\_r ) related to X(r)𝑋𝑟X(r)italic\_X ( italic\_r )
defined by
| | | |
| --- | --- | --- |
| | X′(r)={x:I(χ:x)>r},X^{\prime}(r)=\mathopen{\{}\,x:I(\chi:x)>r\,\mathclose{\}},italic\_X start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ( italic\_r ) = { italic\_x : italic\_I ( italic\_χ : italic\_x ) > italic\_r } , | |
then by Markov’s inequality,
| | | |
| --- | --- | --- |
| | ∑x∈X′(r)𝐦(x)2I(χ:x)<2−r.subscript𝑥superscript𝑋′𝑟𝐦𝑥superscript2𝐼:𝜒𝑥superscript2𝑟\sum\_{x\in X^{\prime}(r)}{\mathbf{m}}(x)2^{I(\chi:x)}<2^{-r}.∑ start\_POSTSUBSCRIPT italic\_x ∈ italic\_X start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ( italic\_r ) end\_POSTSUBSCRIPT bold\_m ( italic\_x ) 2 start\_POSTSUPERSCRIPT italic\_I ( italic\_χ : italic\_x ) end\_POSTSUPERSCRIPT < 2 start\_POSTSUPERSCRIPT - italic\_r end\_POSTSUPERSCRIPT . | |
That is, the universal probability of X′(r)superscript𝑋′𝑟X^{\prime}(r)italic\_X start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ( italic\_r ) is small.
This is a new reason for X(r)𝑋𝑟X(r)italic\_X ( italic\_r ) to be small,
as is shown in the following theorem.
###### Theorem III.24
We have
| | | |
| --- | --- | --- |
| | I(χ:x)>+l(mx)−2logl(mx),I(\chi:x)\stackrel{{\scriptstyle{}\_{+}}}{{>}}l(m\_{x})-2\log l(m\_{x}),italic\_I ( italic\_χ : italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) - 2 roman\_log italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) , | |
and (essentially equivalently) X(r)⊂X′(r−2logr)𝑋𝑟superscript𝑋normal-′𝑟2𝑟X(r)\subset X^{\prime}(r-2\log r)italic\_X ( italic\_r ) ⊂ italic\_X start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ( italic\_r - 2 roman\_log italic\_r ).
###### Remark III.25
The first item in the theorem implies: If l(mx)≥r𝑙subscript𝑚𝑥𝑟l(m\_{x})\geq ritalic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) ≥ italic\_r,
then I(χ:x)>+r−2logrI(\chi:x)\stackrel{{\scriptstyle{}\_{+}}}{{>}}r-2\log ritalic\_I ( italic\_χ : italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_r - 2 roman\_log italic\_r. This in its turn implies
the second item X(r)⊂X′(r−2logr)𝑋𝑟superscript𝑋normal-′𝑟2𝑟X(r)\subset X^{\prime}(r-2\log r)italic\_X ( italic\_r ) ⊂ italic\_X start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ( italic\_r - 2 roman\_log italic\_r ). Similarly, the second
item essentially implies the first item.
Thus, a string for which the explicit
minimal sufficient statistic has complexity
much larger than K(k)𝐾𝑘K(k)italic\_K ( italic\_k ) (that is, l(mx)𝑙subscript𝑚𝑥l(m\_{x})italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) is large)
is exotic in the sense that it belongs
to the kind of strings about which the halting sequence contains
much information and vice versa: I(χ:x)I(\chi:x)italic\_I ( italic\_χ : italic\_x ) is large.
♢normal-♢\diamondsuit♢
###### Proof:
When we talk about complexity with χ𝜒\chiitalic\_χ in the condition, we use a
Turing machine with χ𝜒\chiitalic\_χ as an “oracle”.
With the help of χ𝜒\chiitalic\_χ, we can compute mxsubscript𝑚𝑥m\_{x}italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT, and so we can define the
following new semicomputable (relative to χ𝜒\chiitalic\_χ) function with
c=6/π2𝑐6superscript𝜋2c=6/\pi^{2}italic\_c = 6 / italic\_π start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT:
| | | |
| --- | --- | --- |
| | ν(x∣χ)=c𝐦(x)2l(mx)/l(mx)2.𝜈conditional𝑥𝜒𝑐𝐦𝑥superscript2𝑙subscript𝑚𝑥𝑙superscriptsubscript𝑚𝑥2\nu(x\mid\chi)=c{\mathbf{m}}(x)2^{l(m\_{x})}/l(m\_{x})^{2}.italic\_ν ( italic\_x ∣ italic\_χ ) = italic\_c bold\_m ( italic\_x ) 2 start\_POSTSUPERSCRIPT italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) end\_POSTSUPERSCRIPT / italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT . | |
We have, using [III.23](#S3.Thmlemma23 "Theorem III.23 ‣ III-E3 Almost Always “Sufficient” ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") and defining Y(r)=X(r)∖X(r+1)𝑌𝑟𝑋𝑟𝑋𝑟1Y(r)=X(r)\setminus X(r+1)italic\_Y ( italic\_r ) = italic\_X ( italic\_r ) ∖ italic\_X ( italic\_r + 1 )
so that l(mx)=r𝑙subscript𝑚𝑥𝑟l(m\_{x})=ritalic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) = italic\_r for x∈Y(r)𝑥𝑌𝑟x\in Y(r)italic\_x ∈ italic\_Y ( italic\_r ):
| | | |
| --- | --- | --- |
| | ∑x∈Y(r)ν(x∣χ)=cr−22r∑x∈Y(r)2−K(x)≤cr−22r2−r+2≤4cr−2.subscript𝑥𝑌𝑟𝜈conditional𝑥𝜒𝑐superscript𝑟2superscript2𝑟subscript𝑥𝑌𝑟superscript2𝐾𝑥𝑐superscript𝑟2superscript2𝑟superscript2𝑟24𝑐superscript𝑟2\begin{split}\sum\_{x\in Y(r)}\nu(x\mid\chi)&=cr^{-2}2^{r}\sum\_{x\in Y(r)}2^{-K(x)}\\
&\leq cr^{-2}2^{r}2^{-r+2}\leq 4cr^{-2}.\end{split}start\_ROW start\_CELL ∑ start\_POSTSUBSCRIPT italic\_x ∈ italic\_Y ( italic\_r ) end\_POSTSUBSCRIPT italic\_ν ( italic\_x ∣ italic\_χ ) end\_CELL start\_CELL = italic\_c italic\_r start\_POSTSUPERSCRIPT - 2 end\_POSTSUPERSCRIPT 2 start\_POSTSUPERSCRIPT italic\_r end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_x ∈ italic\_Y ( italic\_r ) end\_POSTSUBSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_x ) end\_POSTSUPERSCRIPT end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL ≤ italic\_c italic\_r start\_POSTSUPERSCRIPT - 2 end\_POSTSUPERSCRIPT 2 start\_POSTSUPERSCRIPT italic\_r end\_POSTSUPERSCRIPT 2 start\_POSTSUPERSCRIPT - italic\_r + 2 end\_POSTSUPERSCRIPT ≤ 4 italic\_c italic\_r start\_POSTSUPERSCRIPT - 2 end\_POSTSUPERSCRIPT . end\_CELL end\_ROW | |
Summing over r𝑟ritalic\_r gives ∑xν(x∣χ)≤4subscript𝑥𝜈conditional𝑥𝜒4\sum\_{x}\nu(x\mid\chi)\leq 4∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_ν ( italic\_x ∣ italic\_χ ) ≤ 4.
The theorem that 𝐦(x)=2−K(x)𝐦𝑥superscript2𝐾𝑥{\mathbf{m}}(x)=2^{-K(x)}bold\_m ( italic\_x ) = 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_x ) end\_POSTSUPERSCRIPT is maximal within multiplicative
constant among semicomputable semimeasures is also true relative to oracles.
Since we have established that
ν(x∣χ)/4𝜈conditional𝑥𝜒4\nu(x\mid\chi)/4italic\_ν ( italic\_x ∣ italic\_χ ) / 4 is a semicomputable semimeasure, therefore
𝐦(x∣χ)>\*ν(x∣χ)superscript𝐦conditional𝑥𝜒𝜈conditional𝑥𝜒{\mathbf{m}}(x\mid\chi)\stackrel{{\scriptstyle{}\_{\*}}}{{>}}\nu(x\mid\chi)bold\_m ( italic\_x ∣ italic\_χ ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_ν ( italic\_x ∣ italic\_χ ), or equivalently,
| | | |
| --- | --- | --- |
| | K(x∣χ)<+−logν(x∣χ)=+K(x)−l(mx)+2logl(mx),superscript𝐾conditional𝑥𝜒𝜈conditional𝑥𝜒superscript𝐾𝑥𝑙subscript𝑚𝑥2𝑙subscript𝑚𝑥K(x\mid\chi)\stackrel{{\scriptstyle{}\_{+}}}{{<}}-\log\nu(x\mid\chi)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)-l(m\_{x})+2\log l(m\_{x}),italic\_K ( italic\_x ∣ italic\_χ ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_ν ( italic\_x ∣ italic\_χ ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) - italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) + 2 roman\_log italic\_l ( italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ) , | |
which proves the theorem.
∎
IV Non-Stochastic Objects
--------------------------
In this section, whenever we talk about a description of
a finite set S𝑆Sitalic\_S we mean an explicit description. This
establishes the precise meaning of K(S)𝐾𝑆K(S)italic\_K ( italic\_S ), K(⋅∣S)K(\cdot\mid S)italic\_K ( ⋅ ∣ italic\_S ),
𝐦(S)=2−K(S)𝐦𝑆superscript2𝐾𝑆{\bf m}(S)=2^{-K(S)}bold\_m ( italic\_S ) = 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_S ) end\_POSTSUPERSCRIPT, and 𝐦(⋅∣S)=2K(⋅∣S){\bf m}(\cdot\mid S)=2^{K(\cdot\mid S)}bold\_m ( ⋅ ∣ italic\_S ) = 2 start\_POSTSUPERSCRIPT italic\_K ( ⋅ ∣ italic\_S ) end\_POSTSUPERSCRIPT,
and so forth.
Every data sample consisting
of a finite string x𝑥xitalic\_x has an sufficient statistic in the form of
the singleton set {x}𝑥\{x\}{ italic\_x }. Such a sufficient statistic is
not very enlightening since it simply replicates the data and
has equal complexity with x𝑥xitalic\_x.
Thus, one is interested in the minimal sufficient statistic
that represents the regularity, (the meaningful) information,
in the data and leaves out the accidental features. This raises
the question whether every x𝑥xitalic\_x has a minimal sufficient
statistic that is significantly less complex than
x𝑥xitalic\_x itself. At a Tallinn conference in 1973 Kolmogorov
(according to [[17](#bib.bib17), [4](#bib.bib4)]) raised
the question whether there are objects x𝑥xitalic\_x that
have no minimal sufficient statistic that have relatively
small complexity. In other words, he inquired into the existence of
objects that are not in general position (random with respect to)
any finite set of small enough complexity, that is,
“absolutely non-random” objects. Clearly, such objects x𝑥xitalic\_x have
neither minimal nor maximal complexity: if they have minimal complexity then
the singleton set {x}𝑥\{x\}{ italic\_x } is a minimal sufficient statistic of small
complexity, and
if x∈{0,1}n𝑥superscript01𝑛x\in\{0,1\}^{n}italic\_x ∈ { 0 , 1 } start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT
is completely incompressible (that is, it is individually random
and has no meaningful information), then
the uninformative universe {0,1}nsuperscript01𝑛\{0,1\}^{n}{ 0 , 1 } start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT is
the minimal sufficient statistic of small complexity. To analyze
the question better we need the technical notion of randomness deficiency.
Define the randomness deficiency of an object x𝑥xitalic\_x with respect
to a finite set S𝑆Sitalic\_S containing it as the amount by
which the complexity of x𝑥xitalic\_x as an element
of S𝑆Sitalic\_S falls short of the maximal possible complexity
of an element in S𝑆Sitalic\_S when S𝑆Sitalic\_S is known explicitly (say, as a list):
| | | | |
| --- | --- | --- | --- |
| | δS(x)=log|S|−K(x∣S).subscript𝛿𝑆𝑥𝑆𝐾conditional𝑥𝑆\delta\_{S}(x)=\log|S|-K(x\mid S).italic\_δ start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT ( italic\_x ) = roman\_log | italic\_S | - italic\_K ( italic\_x ∣ italic\_S ) . | | (IV.1) |
The meaning of this function is clear: most elements of S𝑆Sitalic\_S have
complexity near log|S|𝑆\log|S|roman\_log | italic\_S |, so this difference measures the amount of
compressibility in x𝑥xitalic\_x compared to the generic, typical, random elements of S𝑆Sitalic\_S.
This is a generalization of the sufficiency notion in that it
measures the discrepancy with typicality and hence sufficiency:
if a set S𝑆Sitalic\_S is a sufficient
statistic for x𝑥xitalic\_x then δS(x)=+0superscriptsubscript𝛿𝑆𝑥0\delta\_{S}(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_δ start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
We now continue the discussion of Kolmogorov’s question.
Shen [[17](#bib.bib17)] gave a first answer by establishing the
existence of absolutely non-random objects x𝑥xitalic\_x of length n𝑛nitalic\_n,
having randomness deficiency at least n−2k−O(logk)𝑛2𝑘𝑂𝑘n-2k-O(\log k)italic\_n - 2 italic\_k - italic\_O ( roman\_log italic\_k ) with respect
to every finite set S𝑆Sitalic\_S of complexity K(S)<k𝐾𝑆𝑘K(S)<kitalic\_K ( italic\_S ) < italic\_k that
contains x𝑥xitalic\_x. Moreover, since the set {x}𝑥\{x\}{ italic\_x } has complexity K(x)𝐾𝑥K(x)italic\_K ( italic\_x )
and the randomness deficiency of x𝑥xitalic\_x with respect to this singleton set
is =+0superscriptabsent0\stackrel{{\scriptstyle{}\_{+}}}{{=}}0start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0, it follows by choice of k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ) that the complexity
K(x)𝐾𝑥K(x)italic\_K ( italic\_x ) is at least n/2−O(logn)𝑛2𝑂𝑛n/2-O(\log n)italic\_n / 2 - italic\_O ( roman\_log italic\_n ).
Here we sharpen this
result:
We establish the existence of absolutely non-random objects x𝑥xitalic\_x of length n𝑛nitalic\_n,
having randomness deficiency at least n−k𝑛𝑘n-kitalic\_n - italic\_k with respect
to every finite set S𝑆Sitalic\_S of complexity K(S∣n)<k𝐾conditional𝑆𝑛𝑘K(S\mid n)<kitalic\_K ( italic\_S ∣ italic\_n ) < italic\_k that
contains x𝑥xitalic\_x. Clearly, this is best possible since x𝑥xitalic\_x
has randomness deficiency of at least n−K(S∣n)𝑛𝐾conditional𝑆𝑛n-K(S\mid n)italic\_n - italic\_K ( italic\_S ∣ italic\_n ) with every finite set S𝑆Sitalic\_S
containing x𝑥xitalic\_x, in particular, with complexity K(S∣n)𝐾conditional𝑆𝑛K(S\mid n)italic\_K ( italic\_S ∣ italic\_n ) more than a fixed
constant below n𝑛nitalic\_n the randomness deficiency exceeds that fixed constant.
That is, every sufficient statistic for x𝑥xitalic\_x has complexity at least n𝑛nitalic\_n.
But if we choose S={x}𝑆𝑥S=\{x\}italic\_S = { italic\_x } then K(S∣n)=+K(x∣n)<+nsuperscript𝐾conditional𝑆𝑛𝐾conditional𝑥𝑛superscript𝑛K(S\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{<}}nitalic\_K ( italic\_S ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n, and, moreover,
the randomness deficiency of x𝑥xitalic\_x with respect to S𝑆Sitalic\_S
is n−K(S∣n)=+0superscript𝑛𝐾conditional𝑆𝑛0n-K(S\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_n - italic\_K ( italic\_S ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0. Together
this shows that the absolutely nonrandom objects x𝑥xitalic\_x length n𝑛nitalic\_n
of which we established
the existence have complexity K(x∣n)=+nsuperscript𝐾conditional𝑥𝑛𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}nitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n, and moreover,
they have significant
randomness deficiency with respect to every set S𝑆Sitalic\_S containing them
that has complexity significantly below their own complexity n𝑛nitalic\_n.
###
IV-A Kolmogorov Structure Function
We first consider the relation between the minimal unavoidable randomness
deficiency of x𝑥xitalic\_x with respect to a set S𝑆Sitalic\_S containing it,
when the complexity of S𝑆Sitalic\_S is upper bounded by α𝛼\alphaitalic\_α.
These functional relations are known as Kolmogorov structure functions.
Kolmogorov proposed a variant of the
function
| | | | |
| --- | --- | --- | --- |
| | hx(α)=minS{log|S|:x∈S,K(S)<α},subscriptℎ𝑥𝛼subscript𝑆:𝑆formulae-sequence𝑥𝑆𝐾𝑆𝛼{h}\_{x}(\alpha)=\min\_{S}\mathopen{\{}\,\log|S|:x\in S,\;K(S)<\alpha\,\mathclose{\}},italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) = roman\_min start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT { roman\_log | italic\_S | : italic\_x ∈ italic\_S , italic\_K ( italic\_S ) < italic\_α } , | | (IV.2) |
where S⊆{0,1}\*𝑆superscript01S\subseteq\{0,1\}^{\*}italic\_S ⊆ { 0 , 1 } start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is a finite set containing x𝑥xitalic\_x,
the contemplated model for x𝑥xitalic\_x, and α𝛼\alphaitalic\_α is a nonnegative
integer value bounding the complexity of the contemplated S𝑆Sitalic\_S’s.
He did not specify what is meant by K(S)𝐾𝑆K(S)italic\_K ( italic\_S ) but it was noticed
immediately, as the paper [[18](#bib.bib18)] points out, that the behavior
of hx(α)subscriptℎ𝑥𝛼{h}\_{x}(\alpha)italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) is rather trivial if K(S)𝐾𝑆K(S)italic\_K ( italic\_S ) is taken to be the complexity
of a program that lists S𝑆Sitalic\_S without necessarily halting.
Section [III-D](#S3.SS4 "III-D Implicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") elaborates this point.
So, the present section refers to explicit descriptions only.
It is easy to see that for every increment d𝑑ditalic\_d we have
| | | |
| --- | --- | --- |
| | hx(α+d)≤|hx(α)−d+O(logd)|,subscriptℎ𝑥𝛼𝑑subscriptℎ𝑥𝛼𝑑𝑂𝑑{h}\_{x}(\alpha+d)\leq|{h}\_{x}(\alpha)-d+O(\log d)|,italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α + italic\_d ) ≤ | italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) - italic\_d + italic\_O ( roman\_log italic\_d ) | , | |
provided the right-hand side is non-negative, and 0 otherwise.
Namely, once we have an optimal set Sαsubscript𝑆𝛼S\_{\alpha}italic\_S start\_POSTSUBSCRIPT italic\_α end\_POSTSUBSCRIPT we can subdivide it in any
standard way into 2dsuperscript2𝑑2^{d}2 start\_POSTSUPERSCRIPT italic\_d end\_POSTSUPERSCRIPT parts and take as Sα+dsubscript𝑆𝛼𝑑S\_{\alpha+d}italic\_S start\_POSTSUBSCRIPT italic\_α + italic\_d end\_POSTSUBSCRIPT the part
containing x𝑥xitalic\_x.
Also, hx(α)=0subscriptℎ𝑥𝛼0{h}\_{x}(\alpha)=0italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) = 0 implies α>+K(x)superscript𝛼𝐾𝑥\alpha\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(x)italic\_α start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ), and, since the choice
of S={x}𝑆𝑥S=\{x\}italic\_S = { italic\_x } generally implies only α<+K(x)superscript𝛼𝐾𝑥\alpha\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(x)italic\_α start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) is meaningful
we can conclude α=+K(x)superscript𝛼𝐾𝑥\alpha\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)italic\_α start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ).
Therefore it seems better advised to consider the function
| | | |
| --- | --- | --- |
| | hx(α)+α−K(x)=minS{log|S|−(K(x)−α):K(S)<α}subscriptℎ𝑥𝛼𝛼𝐾𝑥subscript𝑆:𝑆𝐾𝑥𝛼𝐾𝑆𝛼{h}\_{x}(\alpha)+\alpha-K(x)=\min\_{S}\mathopen{\{}\,\log|S|-(K(x)-\alpha):K(S)<\alpha\,\mathclose{\}}italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) + italic\_α - italic\_K ( italic\_x ) = roman\_min start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT { roman\_log | italic\_S | - ( italic\_K ( italic\_x ) - italic\_α ) : italic\_K ( italic\_S ) < italic\_α } | |
rather than ([IV.2](#S4.E2 "IV.2 ‣ IV-A Kolmogorov Structure Function ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")).
For technical reasons related to the later analysis,
we introduce the following variant of
randomness deficiency ([IV.1](#S4.E1 "IV.1 ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")):
| | | |
| --- | --- | --- |
| | δS\*(x)=log|S|−K(x∣S,K(S)).subscriptsuperscript𝛿𝑆𝑥𝑆𝐾conditional𝑥𝑆𝐾𝑆\delta^{\*}\_{S}(x)=\log|S|-K(x\mid S,K(S)).italic\_δ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT ( italic\_x ) = roman\_log | italic\_S | - italic\_K ( italic\_x ∣ italic\_S , italic\_K ( italic\_S ) ) . | |
The function hx(α)+α−K(x)subscriptℎ𝑥𝛼𝛼𝐾𝑥{h}\_{x}(\alpha)+\alpha-K(x)italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) + italic\_α - italic\_K ( italic\_x ) seems related to a function
of more intuitive appeal, namely
βx(α)subscript𝛽𝑥𝛼\beta\_{x}(\alpha)italic\_β start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) measuring the minimal unavoidable randomness deficiency
of x𝑥xitalic\_x with respect to every finite set S𝑆Sitalic\_S, that contains it,
of complexity K(S)<α𝐾𝑆𝛼K(S)<\alphaitalic\_K ( italic\_S ) < italic\_α.
Formally, we define
| | | |
| --- | --- | --- |
| | βx(α)=minS{δS(x):K(S)<α},subscript𝛽𝑥𝛼subscript𝑆:subscript𝛿𝑆𝑥𝐾𝑆𝛼\beta\_{x}(\alpha)=\min\_{S}\mathopen{\{}\,\delta\_{S}(x):K(S)<\alpha\,\mathclose{\}},italic\_β start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) = roman\_min start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT { italic\_δ start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT ( italic\_x ) : italic\_K ( italic\_S ) < italic\_α } , | |
and its variant
| | | |
| --- | --- | --- |
| | βx\*(α)=minS{δS\*(x):K(S)<α},subscriptsuperscript𝛽𝑥𝛼subscript𝑆:subscriptsuperscript𝛿𝑆𝑥𝐾𝑆𝛼\beta^{\*}\_{x}(\alpha)=\min\_{S}\mathopen{\{}\,\delta^{\*}\_{S}(x):K(S)<\alpha\,\mathclose{\}},italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) = roman\_min start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT { italic\_δ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT ( italic\_x ) : italic\_K ( italic\_S ) < italic\_α } , | |
defined in terms of δS\*subscriptsuperscript𝛿𝑆\delta^{\*}\_{S}italic\_δ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT.
Note that βx(K(x))=+βx\*(K(x))=+0superscriptsubscript𝛽𝑥𝐾𝑥subscriptsuperscript𝛽𝑥𝐾𝑥superscript0\beta\_{x}(K(x))\stackrel{{\scriptstyle{}\_{+}}}{{=}}\beta^{\*}\_{x}(K(x))\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_β start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_K ( italic\_x ) ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_K ( italic\_x ) ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
These β𝛽\betaitalic\_β-functions are related to, but different from,
the β𝛽\betaitalic\_β in ([I.4](#S1.E4 "I.4 ‣ Definition I.1 ‣ I Introduction ‣ Algorithmic Statistics")).
To compare hℎ{h}italic\_h and β𝛽\betaitalic\_β, let us confine ourselves to binary strings of
length n𝑛nitalic\_n.
We will put n𝑛nitalic\_n into the condition of all complexities.
###### Lemma IV.1
βx\*(α∣n)<+hx(α∣n)+α−K(x∣n)superscriptsubscriptsuperscript𝛽𝑥conditional𝛼𝑛subscriptℎ𝑥conditional𝛼𝑛𝛼𝐾conditional𝑥𝑛\beta^{\*}\_{x}(\alpha\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{<}}{h}\_{x}(\alpha\mid n)+\alpha-K(x\mid n)italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ∣ italic\_n ) + italic\_α - italic\_K ( italic\_x ∣ italic\_n ).
###### Proof:
Let S∋x𝑥𝑆S\ni xitalic\_S ∋ italic\_x be a set with K(S∣n)≤α𝐾conditional𝑆𝑛𝛼K(S\mid n)\leq\alphaitalic\_K ( italic\_S ∣ italic\_n ) ≤ italic\_α and
assume hx(α∣n)=log|S|subscriptℎ𝑥conditional𝛼𝑛𝑆{h}\_{x}(\alpha\mid n)=\log|S|italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ∣ italic\_n ) = roman\_log | italic\_S |.
Tacitly understanding n𝑛nitalic\_n in the conditions, and using the additivity
property ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")),
| | | | |
| --- | --- | --- | --- |
| | K(x)−α≤K(x)−K(S)𝐾𝑥𝛼𝐾𝑥𝐾𝑆\displaystyle K(x)-\alpha\leq K(x)-K(S)italic\_K ( italic\_x ) - italic\_α ≤ italic\_K ( italic\_x ) - italic\_K ( italic\_S ) | <+K(x,S)−K(S)superscriptabsent𝐾𝑥𝑆𝐾𝑆\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(x,S)-K(S)start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x , italic\_S ) - italic\_K ( italic\_S ) | |
| | | =+K(x∣S,K(S)).superscriptabsent𝐾conditional𝑥𝑆𝐾𝑆\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid S,K(S)).start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_S , italic\_K ( italic\_S ) ) . | |
Therefore
| | | | |
| --- | --- | --- | --- |
| | hx(α)+α−K(x)subscriptℎ𝑥𝛼𝛼𝐾𝑥\displaystyle{h}\_{x}(\alpha)+\alpha-K(x)italic\_h start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) + italic\_α - italic\_K ( italic\_x ) | =log|S|−(K(x)−α)absent𝑆𝐾𝑥𝛼\displaystyle=\log|S|-(K(x)-\alpha)= roman\_log | italic\_S | - ( italic\_K ( italic\_x ) - italic\_α ) | |
| | | >+log|S|−K(x∣S,K(S))≥βx\*(α).superscriptabsent𝑆𝐾conditional𝑥𝑆𝐾𝑆subscriptsuperscript𝛽𝑥𝛼\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{>}}\log|S|-K(x\mid S,K(S))\geq\beta^{\*}\_{x}(\alpha).start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log | italic\_S | - italic\_K ( italic\_x ∣ italic\_S , italic\_K ( italic\_S ) ) ≥ italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) . | |
∎
It would be nice to have an inequality also in the other direction, but we
do not know currently what is the best that can be said.
###
IV-B Sharp Bound on Non-Stochastic Objects
We are now able to formally express the notion of non-stochastic
objects using the Kolmogorov structure functions
βx(α),βx\*(α)subscript𝛽𝑥𝛼subscriptsuperscript𝛽𝑥𝛼\beta\_{x}(\alpha),\beta^{\*}\_{x}(\alpha)italic\_β start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) , italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ).
For every given k<n𝑘𝑛k<nitalic\_k < italic\_n, Shen constructed in [[17](#bib.bib17)] a binary
string x𝑥xitalic\_x of length n𝑛nitalic\_n with K(x)≤k𝐾𝑥𝑘K(x)\leq kitalic\_K ( italic\_x ) ≤ italic\_k and
βx(k−O(1))>n−2k−O(logk)subscript𝛽𝑥𝑘𝑂1𝑛2𝑘𝑂𝑘\beta\_{x}(k-O(1))>n-2k-O(\log k)italic\_β start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_k - italic\_O ( 1 ) ) > italic\_n - 2 italic\_k - italic\_O ( roman\_log italic\_k ).
Let x𝑥xitalic\_x be one of the non-stochastic objects of which
the existence is established.
Substituting k=+K(x)superscript𝑘𝐾𝑥k\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)italic\_k start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) we can contemplate
the set S={x}𝑆𝑥S=\{x\}italic\_S = { italic\_x } with complexity K(S)=+ksuperscript𝐾𝑆𝑘K(S)\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_K ( italic\_S ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k
and x𝑥xitalic\_x has randomness deficiency =+0superscriptabsent0\stackrel{{\scriptstyle{}\_{+}}}{{=}}0start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 with respect to S𝑆Sitalic\_S.
This yields 0=+βx(K(x))>+n−2K(x)−O(logK(x))superscript0subscript𝛽𝑥𝐾𝑥superscript𝑛2𝐾𝑥𝑂𝐾𝑥0\stackrel{{\scriptstyle{}\_{+}}}{{=}}\beta\_{x}(K(x))\stackrel{{\scriptstyle{}\_{+}}}{{>}}n-2K(x)-O(\log K(x))0 start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_β start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_K ( italic\_x ) ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - 2 italic\_K ( italic\_x ) - italic\_O ( roman\_log italic\_K ( italic\_x ) ).
Since it generally holds that these non-stochastic
objects have complexity K(x)>+n/2−O(logn)superscript𝐾𝑥𝑛2𝑂𝑛K(x)\stackrel{{\scriptstyle{}\_{+}}}{{>}}n/2-O(\log n)italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n / 2 - italic\_O ( roman\_log italic\_n ),
they are not random, typical, or in general position
with respect to every set S𝑆Sitalic\_S containing them with complexity
K(S)>+n/2−O(logn)𝐾𝑆superscript𝑛2𝑂𝑛K(S)\not\stackrel{{\scriptstyle{}\_{+}}}{{>}}n/2-O(\log n)italic\_K ( italic\_S ) not start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n / 2 - italic\_O ( roman\_log italic\_n ), but they are random, typical,
or in general position only for sets S𝑆Sitalic\_S with complexity
K(S)𝐾𝑆K(S)italic\_K ( italic\_S ) sufficiently exceeding n/2−O(logn)𝑛2𝑂𝑛n/2-O(\log n)italic\_n / 2 - italic\_O ( roman\_log italic\_n ) like S={x}𝑆𝑥S=\{x\}italic\_S = { italic\_x }.
Here, we improve on this result, replacing n−2k−O(logk)𝑛2𝑘𝑂𝑘n-2k-O(\log k)italic\_n - 2 italic\_k - italic\_O ( roman\_log italic\_k )
with n−k𝑛𝑘n-kitalic\_n - italic\_k and
using β\*superscript𝛽\beta^{\*}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT to avoid logarithmic terms.
This is the best possible, since by choosing S={0,1}n𝑆superscript01𝑛S=\{0,1\}^{n}italic\_S = { 0 , 1 } start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT we find
log|S|−K(x∣S,K(S))=+n−ksuperscript𝑆𝐾conditional𝑥𝑆𝐾𝑆𝑛𝑘\log|S|-K(x\mid S,K(S))\stackrel{{\scriptstyle{}\_{+}}}{{=}}n-kroman\_log | italic\_S | - italic\_K ( italic\_x ∣ italic\_S , italic\_K ( italic\_S ) ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_k, and hence
βx\*(c)<+n−ksuperscriptsubscriptsuperscript𝛽𝑥𝑐𝑛𝑘\beta^{\*}\_{x}(c)\stackrel{{\scriptstyle{}\_{+}}}{{<}}n-kitalic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_c ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_k for some constant c𝑐citalic\_c, which implies
βx\*(α)≤βx(c)<+n−ksubscriptsuperscript𝛽𝑥𝛼subscript𝛽𝑥𝑐superscript𝑛𝑘\beta^{\*}\_{x}(\alpha)\leq\beta\_{x}(c)\stackrel{{\scriptstyle{}\_{+}}}{{<}}n-kitalic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ) ≤ italic\_β start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_c ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_k for every α>c𝛼𝑐\alpha>citalic\_α > italic\_c.
###### Theorem IV.2
There are constants c1,c2subscript𝑐1subscript𝑐2c\_{1},c\_{2}italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT such that for any given k<n𝑘𝑛k<nitalic\_k < italic\_n
there is a a binary
string x𝑥xitalic\_x of length n𝑛nitalic\_n with K(x∣n)≤k𝐾conditional𝑥𝑛𝑘K(x\mid n)\leq kitalic\_K ( italic\_x ∣ italic\_n ) ≤ italic\_k such that for all
α<k−c1𝛼𝑘subscript𝑐1\alpha<k-c\_{1}italic\_α < italic\_k - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT we have
| | | |
| --- | --- | --- |
| | βx\*(α∣n)>n−k−c2.subscriptsuperscript𝛽𝑥conditional𝛼𝑛𝑛𝑘subscript𝑐2\beta^{\*}\_{x}(\alpha\mid n)>n-k-c\_{2}.italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_α ∣ italic\_n ) > italic\_n - italic\_k - italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT . | |
In the terminology of ([I.4](#S1.E4 "I.4 ‣ Definition I.1 ‣ I Introduction ‣ Algorithmic Statistics")),
the theorem states that there are constants c1,c2subscript𝑐1subscript𝑐2c\_{1},c\_{2}italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT
such that for every k<n𝑘𝑛k<nitalic\_k < italic\_n there exists
a string x𝑥xitalic\_x of length n𝑛nitalic\_n of complexity K(x∣n)≤k𝐾conditional𝑥𝑛𝑘K(x\mid n)\leq kitalic\_K ( italic\_x ∣ italic\_n ) ≤ italic\_k
that is not (k−c1,n−k−c2)𝑘subscript𝑐1𝑛𝑘subscript𝑐2(k-c\_{1},n-k-c\_{2})( italic\_k - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_n - italic\_k - italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT )-stochastic.
###### Proof:
Denote the conditional universal probability
as 𝐦(S∣n)=2−K(S∣n)𝐦conditional𝑆𝑛superscript2𝐾conditional𝑆𝑛{\mathbf{m}}(S\mid n)=2^{-K(S\mid n)}bold\_m ( italic\_S ∣ italic\_n ) = 2 start\_POSTSUPERSCRIPT - italic\_K ( italic\_S ∣ italic\_n ) end\_POSTSUPERSCRIPT.
We write “S∋x𝑥𝑆S\ni xitalic\_S ∋ italic\_x” to indicate sets S𝑆Sitalic\_S that satisfy x∈S𝑥𝑆x\in Sitalic\_x ∈ italic\_S.
For every n𝑛nitalic\_n, let us define a function
over all strings x𝑥xitalic\_x of length n𝑛nitalic\_n as follows:
| | | | |
| --- | --- | --- | --- |
| | ν≤i(x∣n)=∑S∋x,K(S∣n)≤i𝐦(S∣n)|S|superscript𝜈absent𝑖conditional𝑥𝑛subscriptformulae-sequence𝑥𝑆𝐾conditional𝑆𝑛𝑖𝐦conditional𝑆𝑛𝑆\nu^{\leq i}(x\mid n)=\sum\_{S\ni x,\ K(S\mid n)\leq i}\frac{{\mathbf{m}}(S\mid n)}{|S|}italic\_ν start\_POSTSUPERSCRIPT ≤ italic\_i end\_POSTSUPERSCRIPT ( italic\_x ∣ italic\_n ) = ∑ start\_POSTSUBSCRIPT italic\_S ∋ italic\_x , italic\_K ( italic\_S ∣ italic\_n ) ≤ italic\_i end\_POSTSUBSCRIPT divide start\_ARG bold\_m ( italic\_S ∣ italic\_n ) end\_ARG start\_ARG | italic\_S | end\_ARG | | (IV.3) |
The following lemma shows that this function of x𝑥xitalic\_x is a semimeasure.
###### Lemma IV.3
We have
| | | | |
| --- | --- | --- | --- |
| | ∑xν≤i(x∣n)≤1.subscript𝑥superscript𝜈absent𝑖conditional𝑥𝑛1\sum\_{x}\nu^{\leq i}(x\mid n)\leq 1.∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_ν start\_POSTSUPERSCRIPT ≤ italic\_i end\_POSTSUPERSCRIPT ( italic\_x ∣ italic\_n ) ≤ 1 . | | (IV.4) |
###### Proof:
We have
| | | |
| --- | --- | --- |
| | ∑xν≤i(x∣n)≤∑x∑S∋x𝐦(S∣n)|S|=∑S∑x∈S𝐦(S∣n)|S|=∑S𝐦(S∣n)≤1.subscript𝑥superscript𝜈absent𝑖conditional𝑥𝑛subscript𝑥subscript𝑥𝑆𝐦conditional𝑆𝑛𝑆subscript𝑆subscript𝑥𝑆𝐦conditional𝑆𝑛𝑆subscript𝑆𝐦conditional𝑆𝑛1\begin{split}\sum\_{x}\nu^{\leq i}(x\mid n)&\leq\sum\_{x}\sum\_{S\ni x}\frac{{\mathbf{m}}(S\mid n)}{|S|}=\sum\_{S}\sum\_{x\in S}\frac{{\mathbf{m}}(S\mid n)}{|S|}\\
&=\sum\_{S}{\mathbf{m}}(S\mid n)\leq 1.\end{split}start\_ROW start\_CELL ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_ν start\_POSTSUPERSCRIPT ≤ italic\_i end\_POSTSUPERSCRIPT ( italic\_x ∣ italic\_n ) end\_CELL start\_CELL ≤ ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_S ∋ italic\_x end\_POSTSUBSCRIPT divide start\_ARG bold\_m ( italic\_S ∣ italic\_n ) end\_ARG start\_ARG | italic\_S | end\_ARG = ∑ start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_x ∈ italic\_S end\_POSTSUBSCRIPT divide start\_ARG bold\_m ( italic\_S ∣ italic\_n ) end\_ARG start\_ARG | italic\_S | end\_ARG end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL = ∑ start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT bold\_m ( italic\_S ∣ italic\_n ) ≤ 1 . end\_CELL end\_ROW | |
∎
###### Lemma IV.4
There are constants c1,c2subscript𝑐1subscript𝑐2c\_{1},c\_{2}italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT such that for some x𝑥xitalic\_x of
length n𝑛nitalic\_n,
| | | | | |
| --- | --- | --- | --- | --- |
| | ν≤k−c1(x∣n)superscript𝜈absent𝑘subscript𝑐1conditional𝑥𝑛\displaystyle\nu^{\leq k-c\_{1}}(x\mid n)italic\_ν start\_POSTSUPERSCRIPT ≤ italic\_k - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x ∣ italic\_n ) | ≤2−n,absentsuperscript2𝑛\displaystyle\leq 2^{-n},≤ 2 start\_POSTSUPERSCRIPT - italic\_n end\_POSTSUPERSCRIPT , | | (IV.5) |
| | k−c2𝑘subscript𝑐2\displaystyle k-c\_{2}italic\_k - italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT | ≤K(x∣n)≤k.absent𝐾conditional𝑥𝑛𝑘\displaystyle\leq K(x\mid n)\leq k.≤ italic\_K ( italic\_x ∣ italic\_n ) ≤ italic\_k . | | (IV.6) |
###### Proof:
Let us fix 0<c1<k0subscript𝑐1𝑘0<c\_{1}<k0 < italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT < italic\_k somehow, to be chosen appropriately later.
Inequality ([IV.4](#S4.E4 "IV.4 ‣ Lemma IV.3 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")) implies that there is an x𝑥xitalic\_x
with ([IV.5](#S4.E5 "IV.5 ‣ Lemma IV.4 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")).
Let x𝑥xitalic\_x be the first string of length n𝑛nitalic\_n with this property.
To prove the right inequality of ([IV.6](#S4.E6 "IV.6 ‣ Lemma IV.4 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")), let p𝑝pitalic\_p be the program
of length ≤i=k−c1absent𝑖𝑘subscript𝑐1\leq i=k-c\_{1}≤ italic\_i = italic\_k - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
that terminates last in the standard
running of all these programs simultaneously in dovetailed fashion,
on input n𝑛nitalic\_n.
We can use p𝑝pitalic\_p and its length l(p)𝑙𝑝l(p)italic\_l ( italic\_p ) to compute all programs of length
≤l(p)absent𝑙𝑝\leq l(p)≤ italic\_l ( italic\_p ) that output finite sets using n𝑛nitalic\_n.
This way we obtain a list of all sets S𝑆Sitalic\_S with K(S∣n)≤i𝐾conditional𝑆𝑛𝑖K(S\mid n)\leq iitalic\_K ( italic\_S ∣ italic\_n ) ≤ italic\_i.
Using this list, for each y𝑦yitalic\_y of length n𝑛nitalic\_n we can compute
ν≤i(y∣n)superscript𝜈absent𝑖conditional𝑦𝑛\nu^{\leq i}(y\mid n)italic\_ν start\_POSTSUPERSCRIPT ≤ italic\_i end\_POSTSUPERSCRIPT ( italic\_y ∣ italic\_n ), by
using the definition ([IV.3](#S4.E3 "IV.3 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")) explicitly.
Since x𝑥xitalic\_x is defined as the first y𝑦yitalic\_y with
ν≤i(y∣n)≤2−nsuperscript𝜈absent𝑖conditional𝑦𝑛superscript2𝑛\nu^{\leq i}(y\mid n)\leq 2^{-n}italic\_ν start\_POSTSUPERSCRIPT ≤ italic\_i end\_POSTSUPERSCRIPT ( italic\_y ∣ italic\_n ) ≤ 2 start\_POSTSUPERSCRIPT - italic\_n end\_POSTSUPERSCRIPT, we can thus find x𝑥xitalic\_x
by using p𝑝pitalic\_p and some program of constant length.
If c1subscript𝑐1c\_{1}italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT is chosen large enough, then this implies K(x∣n)≤k𝐾conditional𝑥𝑛𝑘K(x\mid n)\leq kitalic\_K ( italic\_x ∣ italic\_n ) ≤ italic\_k.
On the other hand, from the definition ([IV.3](#S4.E3 "IV.3 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")) we have
| | | |
| --- | --- | --- |
| | ν≤K({x}∣n)(x∣n)≥2−K({x}∣n).superscript𝜈absent𝐾conditional𝑥𝑛conditional𝑥𝑛superscript2𝐾conditional𝑥𝑛\nu^{\leq K(\{x\}\mid n)}(x\mid n)\geq 2^{-K(\{x\}\mid n)}.italic\_ν start\_POSTSUPERSCRIPT ≤ italic\_K ( { italic\_x } ∣ italic\_n ) end\_POSTSUPERSCRIPT ( italic\_x ∣ italic\_n ) ≥ 2 start\_POSTSUPERSCRIPT - italic\_K ( { italic\_x } ∣ italic\_n ) end\_POSTSUPERSCRIPT . | |
This implies, by the definition of x𝑥xitalic\_x, that either
K({x}∣n)>k−c1𝐾conditional𝑥𝑛𝑘subscript𝑐1K(\{x\}\mid n)>k-c\_{1}italic\_K ( { italic\_x } ∣ italic\_n ) > italic\_k - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT or K({x}∣n)≥n𝐾conditional𝑥𝑛𝑛K(\{x\}\mid n)\geq nitalic\_K ( { italic\_x } ∣ italic\_n ) ≥ italic\_n.
Since K(x∣n)=+K({x}∣n))K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(\{x\}\mid n))italic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( { italic\_x } ∣ italic\_n ) ) we get the left inequality of
([IV.6](#S4.E6 "IV.6 ‣ Lemma IV.4 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")) in both cases for an appropriate c2subscript𝑐2c\_{2}italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
∎
Consider now a new semicomputable function
| | | |
| --- | --- | --- |
| | μx,i(S∣n)=2n𝐦(S∣n)|S|subscript𝜇𝑥𝑖conditional𝑆𝑛superscript2𝑛𝐦conditional𝑆𝑛𝑆\mu\_{x,i}(S\mid n)=\frac{2^{n}{\mathbf{m}}(S\mid n)}{|S|}italic\_μ start\_POSTSUBSCRIPT italic\_x , italic\_i end\_POSTSUBSCRIPT ( italic\_S ∣ italic\_n ) = divide start\_ARG 2 start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT bold\_m ( italic\_S ∣ italic\_n ) end\_ARG start\_ARG | italic\_S | end\_ARG | |
on all finite sets S∋x𝑥𝑆S\ni xitalic\_S ∋ italic\_x with K(S∣n)≤i𝐾conditional𝑆𝑛𝑖K(S\mid n)\leq iitalic\_K ( italic\_S ∣ italic\_n ) ≤ italic\_i.
Then we have, with i=k−c1𝑖𝑘subscript𝑐1i=k-c\_{1}italic\_i = italic\_k - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT:
| | | |
| --- | --- | --- |
| | ∑Sμx,i(S∣n)=2n∑S∋x,K(S∣n)≤i𝐦(S∣n)|S|=2nν≤i(x∣n)≤1subscript𝑆subscript𝜇𝑥𝑖conditional𝑆𝑛superscript2𝑛subscriptformulae-sequence𝑥𝑆𝐾conditional𝑆𝑛𝑖𝐦conditional𝑆𝑛𝑆superscript2𝑛superscript𝜈absent𝑖conditional𝑥𝑛1\begin{split}\sum\_{S}\mu\_{x,i}(S\mid n)&=2^{n}\sum\_{S\ni x,\ K(S\mid n)\leq i}\frac{{\mathbf{m}}(S\mid n)}{|S|}\\
&=2^{n}\nu^{\leq i}(x\mid n)\leq 1\end{split}start\_ROW start\_CELL ∑ start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT italic\_μ start\_POSTSUBSCRIPT italic\_x , italic\_i end\_POSTSUBSCRIPT ( italic\_S ∣ italic\_n ) end\_CELL start\_CELL = 2 start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_S ∋ italic\_x , italic\_K ( italic\_S ∣ italic\_n ) ≤ italic\_i end\_POSTSUBSCRIPT divide start\_ARG bold\_m ( italic\_S ∣ italic\_n ) end\_ARG start\_ARG | italic\_S | end\_ARG end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL = 2 start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ν start\_POSTSUPERSCRIPT ≤ italic\_i end\_POSTSUPERSCRIPT ( italic\_x ∣ italic\_n ) ≤ 1 end\_CELL end\_ROW | |
by ([IV.3](#S4.E3 "IV.3 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")), ([IV.5](#S4.E5 "IV.5 ‣ Lemma IV.4 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")), respectively, and so
μx,i(S∣n)subscript𝜇𝑥𝑖conditional𝑆𝑛\mu\_{x,i}(S\mid n)italic\_μ start\_POSTSUBSCRIPT italic\_x , italic\_i end\_POSTSUBSCRIPT ( italic\_S ∣ italic\_n ) with x,i,n𝑥𝑖𝑛x,i,nitalic\_x , italic\_i , italic\_n fixed is a
lower semicomputable semimeasure.
By the dominating property we have
𝐦(S∣x,i,n)>\*μx,i(S∣n)superscript𝐦conditional𝑆𝑥𝑖𝑛subscript𝜇𝑥𝑖conditional𝑆𝑛{\bf m}(S\mid x,i,n)\stackrel{{\scriptstyle{}\_{\*}}}{{>}}\mu\_{x,i}(S\mid n)bold\_m ( italic\_S ∣ italic\_x , italic\_i , italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT \* end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_μ start\_POSTSUBSCRIPT italic\_x , italic\_i end\_POSTSUBSCRIPT ( italic\_S ∣ italic\_n ).
Since n𝑛nitalic\_n is the length of x𝑥xitalic\_x and i=+ksuperscript𝑖𝑘i\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_i start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k we can set
K(S∣x,i,n)=+K(S∣x,k)superscript𝐾conditional𝑆𝑥𝑖𝑛𝐾conditional𝑆𝑥𝑘K(S\mid x,i,n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S\mid x,k)italic\_K ( italic\_S ∣ italic\_x , italic\_i , italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ∣ italic\_x , italic\_k ), and hence
K(S∣x,k)<+−logμx,i(S∣n)superscript𝐾conditional𝑆𝑥𝑘subscript𝜇𝑥𝑖conditional𝑆𝑛K(S\mid x,k)\stackrel{{\scriptstyle{}\_{+}}}{{<}}-\log\mu\_{x,i}(S\mid n)italic\_K ( italic\_S ∣ italic\_x , italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_μ start\_POSTSUBSCRIPT italic\_x , italic\_i end\_POSTSUBSCRIPT ( italic\_S ∣ italic\_n ).
Then, with the first =+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP because of ([IV.6](#S4.E6 "IV.6 ‣ Lemma IV.4 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")),
| | | | |
| --- | --- | --- | --- |
| | K(S∣x,K(x∣n))=+K(S∣x,k)<+−logμx,i(S∣n)=log|S|−n+K(S∣n).\begin{split}K(S&\mid x,K(x\mid n))\\
&\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S\mid x,k)\stackrel{{\scriptstyle{}\_{+}}}{{<}}-\log\mu\_{x,i}(S\mid n)\\
&=\log|S|-n+K(S\mid n).\end{split}start\_ROW start\_CELL italic\_K ( italic\_S end\_CELL start\_CELL ∣ italic\_x , italic\_K ( italic\_x ∣ italic\_n ) ) end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ∣ italic\_x , italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_μ start\_POSTSUBSCRIPT italic\_x , italic\_i end\_POSTSUBSCRIPT ( italic\_S ∣ italic\_n ) end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL = roman\_log | italic\_S | - italic\_n + italic\_K ( italic\_S ∣ italic\_n ) . end\_CELL end\_ROW | | (IV.7) |
Then, by the additivity property ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")) and ([IV.7](#S4.E7 "IV.7 ‣ Proof: ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics")):
| | | | |
| --- | --- | --- | --- |
| | K(x\displaystyle K(xitalic\_K ( italic\_x | ∣S,K(S∣n),n)\displaystyle\mid S,K(S\mid n),n)∣ italic\_S , italic\_K ( italic\_S ∣ italic\_n ) , italic\_n ) | |
| | | =+K(x∣n)+K(S∣x,K(x∣n))−K(S∣n)\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid n)+K(S\mid x,K(x\mid n))-K(S\mid n)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_n ) + italic\_K ( italic\_S ∣ italic\_x , italic\_K ( italic\_x ∣ italic\_n ) ) - italic\_K ( italic\_S ∣ italic\_n ) | |
| | | <+k+log|S|−n.superscriptabsent𝑘𝑆𝑛\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}k+\log|S|-n.start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k + roman\_log | italic\_S | - italic\_n . | |
Hence
δ\*(x∣S,n)=log|S|−K(x∣S,K(S∣n),n)>+n−k\delta^{\*}(x\mid S,n)=\log|S|-K(x\mid S,K(S\mid n),n)\stackrel{{\scriptstyle{}\_{+}}}{{>}}n-kitalic\_δ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ( italic\_x ∣ italic\_S , italic\_n ) = roman\_log | italic\_S | - italic\_K ( italic\_x ∣ italic\_S , italic\_K ( italic\_S ∣ italic\_n ) , italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_k.
∎
We are now in the position to prove Lemma [III.18](#S3.Thmlemma18 "Lemma III.18 ‣ III-E1 Explicit Minimal Sufficient Statistic: Particular Cases ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics"):
For every length n𝑛nitalic\_n, there exist strings x𝑥xitalic\_x of length n𝑛nitalic\_n
with K(x∣n)=+nsuperscript𝐾conditional𝑥𝑛𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}nitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n for which {x}𝑥\{x\}{ italic\_x } is an explicit
minimal sufficient statistic.
###### Proof:
(of Lemma [III.18](#S3.Thmlemma18 "Lemma III.18 ‣ III-E1 Explicit Minimal Sufficient Statistic: Particular Cases ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")):
Let x𝑥xitalic\_x be one of the non-stochastic objects of which
the existence is established by Theorem [IV.2](#S4.Thmlemma2 "Theorem IV.2 ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics").
Choose x𝑥xitalic\_x with K(x∣n)=+ksuperscript𝐾conditional𝑥𝑛𝑘K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}kitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k so that the set
S={x}𝑆𝑥S=\{x\}italic\_S = { italic\_x } has complexity K(S∣n)=k−c1𝐾conditional𝑆𝑛𝑘subscript𝑐1K(S\mid n)=k-c\_{1}italic\_K ( italic\_S ∣ italic\_n ) = italic\_k - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT
and x𝑥xitalic\_x has randomness deficiency =+0superscriptabsent0\stackrel{{\scriptstyle{}\_{+}}}{{=}}0start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 with respect to S𝑆Sitalic\_S.
Because x𝑥xitalic\_x is non-stochastic,
this yields 0=+βx\*(k−c1∣n)>+n−K(x∣n)superscript0subscriptsuperscript𝛽𝑥𝑘conditionalsubscript𝑐1𝑛superscript𝑛𝐾conditional𝑥𝑛0\stackrel{{\scriptstyle{}\_{+}}}{{=}}\beta^{\*}\_{x}(k-c\_{1}\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{>}}n-K(x\mid n)0 start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_k - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_K ( italic\_x ∣ italic\_n ).
For every x𝑥xitalic\_x we have K(x∣n)<+nsuperscript𝐾conditional𝑥𝑛𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{<}}nitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n. Together it follows that
K(x∣n)=+nsuperscript𝐾conditional𝑥𝑛𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}nitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n. That is, these non-stochastic
objects x𝑥xitalic\_x have complexity K(x∣n)=+nsuperscript𝐾conditional𝑥𝑛𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}nitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n. Nonetheless,
there is a constant c′superscript𝑐′c^{\prime}italic\_c start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT such that x𝑥xitalic\_x
is not random, typical, or in general position
with respect to any explicitly represented finite set S𝑆Sitalic\_S
containing it that has complexity
K(S∣n)<n−c′𝐾conditional𝑆𝑛𝑛superscript𝑐′K(S\mid n)<n-c^{\prime}italic\_K ( italic\_S ∣ italic\_n ) < italic\_n - italic\_c start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, but they are random, typical,
or in general position for some sets S𝑆Sitalic\_S with complexity
K(S∣n)>+nsuperscript𝐾conditional𝑆𝑛𝑛K(S\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{>}}nitalic\_K ( italic\_S ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n like S={x}𝑆𝑥S=\{x\}italic\_S = { italic\_x }.
That is, every explicit sufficient statistic S𝑆Sitalic\_S for x𝑥xitalic\_x
has complexity K(S∣n)=+nsuperscript𝐾conditional𝑆𝑛𝑛K(S\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}nitalic\_K ( italic\_S ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n, and {x}𝑥\{x\}{ italic\_x } is
such a statistic. Hence {x}𝑥\{x\}{ italic\_x } is an explicit minimal sufficient
statistic for x𝑥xitalic\_x.
∎
V Probabilistic Models
-----------------------
It remains to generalize the model class from finite sets to
the more natural and significant setting of probability distributions.
Instead of finite sets the models are computable probability
density functions P:{0,1}\*→[0,1]:𝑃→superscript0101P:\{0,1\}^{\*}\rightarrow[0,1]italic\_P : { 0 , 1 } start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT → [ 0 , 1 ] with
∑P(x)≤1𝑃𝑥1\sum P(x)\leq 1∑ italic\_P ( italic\_x ) ≤ 1—we allow defective probability distributions
where we may concentrate the surplus probability on a
distinguished “undefined” element.
“Computable” means that there is a Turing machine TPsubscript𝑇𝑃T\_{P}italic\_T start\_POSTSUBSCRIPT italic\_P end\_POSTSUBSCRIPT that computes
approximations to
the value of P𝑃Pitalic\_P for every argument
(more precise definition follows below).
The (prefix-) complexity K(P)𝐾𝑃K(P)italic\_K ( italic\_P ) of a
computable partial function P𝑃Pitalic\_P is defined by
| | | |
| --- | --- | --- |
| | K(P)=mini{K(i):Turing machine Ticomputes P}.𝐾𝑃subscript𝑖:𝐾𝑖Turing machine subscript𝑇𝑖computes 𝑃K(P)=\min\_{i}\{K(i):\mbox{\rm Turing machine }T\_{i}\;\;\mbox{\rm computes }P\}.italic\_K ( italic\_P ) = roman\_min start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT { italic\_K ( italic\_i ) : Turing machine italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT computes italic\_P } . | |
Equality ([III.2](#S3.E2 "III.2 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")) now becomes
| | | | |
| --- | --- | --- | --- |
| | K(x∣P\*)=+−logP(x),superscript𝐾conditional𝑥superscript𝑃𝑃𝑥K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x),italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x ) , | | (V.1) |
and equality ([III.4](#S3.E4 "III.4 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics")) becomes
| | | |
| --- | --- | --- |
| | K(x)=+K(P)−logP(x).superscript𝐾𝑥𝐾𝑃𝑃𝑥K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(P)-\log P(x).italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) - roman\_log italic\_P ( italic\_x ) . | |
As in the finite set case, the complexities involved are crucially
dependent on what we mean by “computation” of P(x)𝑃𝑥P(x)italic\_P ( italic\_x ), that is,
on the requirements on the format in which the output is to be represented.
Recall from [[10](#bib.bib10)] that Turing machines can compute rational numbers:
If a Turing machine T𝑇Titalic\_T computes T(x)𝑇𝑥T(x)italic\_T ( italic\_x ), then we interpret
the output as a pair of natural numbers,
T(x)=⟨p,q⟩𝑇𝑥𝑝𝑞T(x)=\langle p,q\rangleitalic\_T ( italic\_x ) = ⟨ italic\_p , italic\_q ⟩, according to a standard pairing function.
Then, the rational value computed by T𝑇Titalic\_T is by definition p/q𝑝𝑞p/qitalic\_p / italic\_q.
The distinction between explicit and implicit description of P𝑃Pitalic\_P
corresponding to the finite set model case is now defined as follows:
* •
It is implicit if there
is a Turing machine T𝑇Titalic\_T computing P𝑃Pitalic\_P
halting with rational value T(x)𝑇𝑥T(x)italic\_T ( italic\_x ) so that
−logT(x)=+−logP(x)superscript𝑇𝑥𝑃𝑥-\log T(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x)- roman\_log italic\_T ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x ), and,
furthermore, K(−logT(x)∣P\*)=+0superscript𝐾conditional𝑇𝑥superscript𝑃0K(-\log T(x)\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( - roman\_log italic\_T ( italic\_x ) ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 for x𝑥xitalic\_x
satisfying ([V.1](#S5.E1 "V.1 ‣ V Probabilistic Models ‣ Algorithmic Statistics"))—that is, for typical x𝑥xitalic\_x.
* •
It is explicit if the Turing machine T𝑇Titalic\_T computing P𝑃Pitalic\_P, given
x𝑥xitalic\_x and a tolerance ϵitalic-ϵ\epsilonitalic\_ϵ halts with rational value
so that
−logT(x)=−log(P(x)±ϵ)𝑇𝑥plus-or-minus𝑃𝑥italic-ϵ-\log T(x)=-\log(P(x)\pm\epsilon)- roman\_log italic\_T ( italic\_x ) = - roman\_log ( italic\_P ( italic\_x ) ± italic\_ϵ ), and,
furthermore, K(−logT(x)∣P\*)=+0superscript𝐾conditional𝑇𝑥superscript𝑃0K(-\log T(x)\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( - roman\_log italic\_T ( italic\_x ) ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 for x𝑥xitalic\_x
satisfying ([V.1](#S5.E1 "V.1 ‣ V Probabilistic Models ‣ Algorithmic Statistics"))—that is, for typical x𝑥xitalic\_x.
The implicit and explicit descriptions of finite sets and
of uniform distributions with P(x)=1/|S|𝑃𝑥1𝑆P(x)=1/|S|italic\_P ( italic\_x ) = 1 / | italic\_S | for all x∈S𝑥𝑆x\in Sitalic\_x ∈ italic\_S
and P(x)=0𝑃𝑥0P(x)=0italic\_P ( italic\_x ) = 0 otherwise, are as follows:
An implicit (explicit) description
of P𝑃Pitalic\_P is identical with an implicit (explicit) description
of S𝑆Sitalic\_S, up to a short fixed program which indicates which of the two
is intended, so that K(P(x))=+K(S)superscript𝐾𝑃𝑥𝐾𝑆K(P(x))\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S)italic\_K ( italic\_P ( italic\_x ) ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ) for P(x)>0𝑃𝑥0P(x)>0italic\_P ( italic\_x ) > 0 (equivalently,
x∈S𝑥𝑆x\in Sitalic\_x ∈ italic\_S).
To complete our discussion: the worst case of representation format,
a recursively enumerable approximation of P(x)𝑃𝑥P(x)italic\_P ( italic\_x )
where nothing is known about its value,
would lead to indices −logP(x)𝑃𝑥-\log P(x)- roman\_log italic\_P ( italic\_x ) of unknown length.
We do not consider this case.
The properties for the probabilistic
models are loosely related to the properties of
finite set models by Proposition [I.2](#S1.Thmlemma2 "Proposition I.2 ‣ I Introduction ‣ Algorithmic Statistics").
We sharpen the relations by appropriately modifying
the treatment of the finite set case, but essentially following the same
course.
We may use the notation
| | | |
| --- | --- | --- |
| | Pimpl,Pexplsubscript𝑃implsubscript𝑃explP\_{\text{{impl}}},P\_{\text{{expl}}}italic\_P start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT , italic\_P start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT | |
for some implicit and some explicit representation of P𝑃Pitalic\_P.
When a result applies to both implicit and explicit representations,
or when it is clear from the context which representation is meant, we
will omit the subscript.
###
V-A Optimal Model and Sufficient Statistic
As before, we distinguish between “models” that are
computable probability distributions,
and the “shortest programs” to compute those models
that are finite strings.
Consider a string x𝑥xitalic\_x
of length n𝑛nitalic\_n and prefix complexity K(x)=k𝐾𝑥𝑘K(x)=kitalic\_K ( italic\_x ) = italic\_k.
We identify the structure or regularity in x𝑥xitalic\_x that are
to be summarized with a computable probability density function P𝑃Pitalic\_P
with respect to which x𝑥xitalic\_x is a random or typical member.
For x𝑥xitalic\_x typical for P𝑃Pitalic\_P holds the following [[10](#bib.bib10)]:
Given
an (implicitly or explicitly described)
shortest program P\*superscript𝑃P^{\*}italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT for P𝑃Pitalic\_P, a shortest binary program
computing x𝑥xitalic\_x (that is, of length K(x∣P\*)𝐾conditional𝑥superscript𝑃K(x\mid P^{\*})italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ))
can not
be significantly shorter
than its Shannon-Fano code [[5](#bib.bib5)] of
length −logP(x)𝑃𝑥-\log P(x)- roman\_log italic\_P ( italic\_x ), that is,
K(x∣P\*)>+−logP(x)superscript𝐾conditional𝑥superscript𝑃𝑃𝑥K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{>}}-\log P(x)italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x ).
By definition, we fix some agreed upon constant
β≥0,𝛽0\beta\geq 0,italic\_β ≥ 0 ,
and require
| | | |
| --- | --- | --- |
| | K(x∣P\*)≥−logP(x)−β.𝐾conditional𝑥superscript𝑃𝑃𝑥𝛽K(x\mid P^{\*})\geq-\log P(x)-\beta.italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ≥ - roman\_log italic\_P ( italic\_x ) - italic\_β . | |
As before, we will not indicate the dependence on β𝛽\betaitalic\_β explicitly, but the
constants in all our inequalities (<+superscript\stackrel{{\scriptstyle{}\_{+}}}{{<}}start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP) will be allowed to be functions
of this β𝛽\betaitalic\_β.
This definition requires a positive P(x)𝑃𝑥P(x)italic\_P ( italic\_x ).
In fact, since
K(x∣P\*)<+K(x)superscript𝐾conditional𝑥superscript𝑃𝐾𝑥K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(x)italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ), it limits the size of P(x)𝑃𝑥P(x)italic\_P ( italic\_x ) to
Ω(2−k)Ωsuperscript2𝑘\Omega(2^{-k})roman\_Ω ( 2 start\_POSTSUPERSCRIPT - italic\_k end\_POSTSUPERSCRIPT ).
The shortest program P\*superscript𝑃P^{\*}italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT from which a probability density
function P𝑃Pitalic\_P
can be computed is an algorithmic statistic for x𝑥xitalic\_x iff
| | | | |
| --- | --- | --- | --- |
| | K(x∣P\*)=+−logP(x).superscript𝐾conditional𝑥superscript𝑃𝑃𝑥K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x).italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x ) . | | (V.2) |
There are two natural measures of suitability of such a statistic.
We might prefer either the simplest
distribution, or the largest distribution, as
corresponding to the most likely structure ‘explaining’ x𝑥xitalic\_x.
The singleton probability distribution P(x)=1𝑃𝑥1P(x)=1italic\_P ( italic\_x ) = 1,
while certainly a statistic for x𝑥xitalic\_x,
would indeed be considered a poor explanation.
Both measures relate to the optimality of a two-stage description of
x𝑥xitalic\_x using P𝑃Pitalic\_P:
| | | | | |
| --- | --- | --- | --- | --- |
| | K(x)≤K(x,P)𝐾𝑥𝐾𝑥𝑃\displaystyle K(x)\leq K(x,P)italic\_K ( italic\_x ) ≤ italic\_K ( italic\_x , italic\_P ) | =+K(P)+K(x∣P\*)superscriptabsent𝐾𝑃𝐾conditional𝑥superscript𝑃\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(P)+K(x\mid P^{\*})start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) + italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | | (V.3) |
| | | <+K(P)−logP(x),superscriptabsent𝐾𝑃𝑃𝑥\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(P)-\log P(x),start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) - roman\_log italic\_P ( italic\_x ) , | |
where we rewrite K(x,P)𝐾𝑥𝑃K(x,P)italic\_K ( italic\_x , italic\_P ) by ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")).
Here, P𝑃Pitalic\_P can be understood as either Pimplsubscript𝑃implP\_{\text{{impl}}}italic\_P start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT or Pexplsubscript𝑃explP\_{\text{{expl}}}italic\_P start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT.
Call a distribution P𝑃Pitalic\_P (with positive probability P(x)𝑃𝑥P(x)italic\_P ( italic\_x ))
for which
| | | | |
| --- | --- | --- | --- |
| | K(x)=+K(P)−logP(x),superscript𝐾𝑥𝐾𝑃𝑃𝑥K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(P)-\log P(x),italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) - roman\_log italic\_P ( italic\_x ) , | | (V.4) |
optimal.
(More precisely, we should require K(x)≥K(P)−logP(x)−β𝐾𝑥𝐾𝑃𝑃𝑥𝛽K(x)\geq K(P)-\log P(x)-\betaitalic\_K ( italic\_x ) ≥ italic\_K ( italic\_P ) - roman\_log italic\_P ( italic\_x ) - italic\_β.)
Depending on whether K(P)𝐾𝑃K(P)italic\_K ( italic\_P ) is understood as K(Pimpl)𝐾subscript𝑃implK(P\_{\text{{impl}}})italic\_K ( italic\_P start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT ) or
K(Pexpl)𝐾subscript𝑃explK(P\_{\text{{expl}}})italic\_K ( italic\_P start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT ), our
definition splits into implicit and explicit optimality.
The shortest program for an optimal computable probability distribution
is a algorithmic sufficient statistic for x𝑥xitalic\_x.
###
V-B Properties of Sufficient Statistic
As in the case
of finite set models , we start with a sequence of lemmas
that are used to obtain the main results on minimal sufficient
statistic.
Several of these lemmas have two versions: for implicit distributions
and for explicit
distributions.
In these cases, P𝑃Pitalic\_P will denote Pimplsubscript𝑃implP\_{\text{{impl}}}italic\_P start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT or Pexplsubscript𝑃explP\_{\text{{expl}}}italic\_P start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT respectively.
Below it is shown that the mutual information between every
typical distribution and the
data is not much less than K(K(x))𝐾𝐾𝑥K(K(x))italic\_K ( italic\_K ( italic\_x ) ), the complexity of the complexity
K(x)𝐾𝑥K(x)italic\_K ( italic\_x ) of the data x𝑥xitalic\_x.
For optimal distributions it is at least that, and for
algorithmic minimal statistic it is equal to that.
The log-probability
of a typical distribution is determined by the following:
###### Lemma V.1
Let k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ).
If a distribution P𝑃Pitalic\_P is
(implicitly or explicitly)
typical for x𝑥xitalic\_x then
I(x:P)=+k+logP(x)I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}k+\log P(x)italic\_I ( italic\_x : italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k + roman\_log italic\_P ( italic\_x ).
###### Proof:
By definition I(x:P)=+K(x)−K(x∣P\*)I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)-K(x\mid P^{\*})italic\_I ( italic\_x : italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
and by typicality K(x∣P\*)=+−logP(x)superscript𝐾conditional𝑥superscript𝑃𝑃𝑥K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x)italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x ).
∎
The above lemma states that for
(implicitly or explicitly)
typical P𝑃Pitalic\_P the probability P(x)=Θ(2−(k−I(x:P)))P(x)=\Theta(2^{-(k-I(x:P))})italic\_P ( italic\_x ) = roman\_Θ ( 2 start\_POSTSUPERSCRIPT - ( italic\_k - italic\_I ( italic\_x : italic\_P ) ) end\_POSTSUPERSCRIPT ).
The next lemma asserts that for implicitly typical P𝑃Pitalic\_P
the value I(x:P)I(x:P)italic\_I ( italic\_x : italic\_P ) can fall below K(k)𝐾𝑘K(k)italic\_K ( italic\_k ) by no more than an
additive logarithmic term.
###### Lemma V.2
Let k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ).
If a distribution P𝑃Pitalic\_P is (implicitly or explicitly)
typical for x𝑥xitalic\_x then I(x:P)>+K(k)−K(I(x:P))I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(k)-K(I(x:P))italic\_I ( italic\_x : italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) - italic\_K ( italic\_I ( italic\_x : italic\_P ) )
and −logP(x)<+k−K(k)+K(I(x:P))-\log P(x)\stackrel{{\scriptstyle{}\_{+}}}{{<}}k-K(k)+K(I(x:P))- roman\_log italic\_P ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ) + italic\_K ( italic\_I ( italic\_x : italic\_P ) ).
(Here, P𝑃Pitalic\_P is understood as Pimplsubscript𝑃implP\_{\text{{impl}}}italic\_P start\_POSTSUBSCRIPT impl end\_POSTSUBSCRIPT or Pexplsubscript𝑃explP\_{\text{{expl}}}italic\_P start\_POSTSUBSCRIPT expl end\_POSTSUBSCRIPT respectively.)
###### Proof:
Writing k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ), since
| | | | |
| --- | --- | --- | --- |
| | k=+K(k,x)=+K(k)+K(x∣k\*)superscript𝑘𝐾𝑘𝑥superscript𝐾𝑘𝐾conditional𝑥superscript𝑘k\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k,x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k)+K(x\mid k^{\*})italic\_k start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k , italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) + italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) | | (V.5) |
by ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")), we have
I(x:P)=+K(x)−K(x∣P\*)=+K(k)−[K(x∣P\*)−K(x∣k\*)]I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)-K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(k)-[K(x\mid P^{\*})-K(x\mid k^{\*})]italic\_I ( italic\_x : italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ) - [ italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ].
Hence, it suffices to show K(x∣P\*)−K(x∣k\*)<+K(I(x:P))K(x\mid P^{\*})-K(x\mid k^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(I(x:P))italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_I ( italic\_x : italic\_P ) ). Now, from an implicit description P\*superscript𝑃P^{\*}italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT
we can
find the value =+−logP(x)=+k−I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}k-I(x:P)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_I ( italic\_x : italic\_P ). To recover k𝑘kitalic\_k from P\*superscript𝑃P^{\*}italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT,
we at most
require an extra K(I(x:P))K(I(x:P))italic\_K ( italic\_I ( italic\_x : italic\_P ) ) bits.
That is, K(k∣P\*)<+K(I(x:P))K(k\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(I(x:P))italic\_K ( italic\_k ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_I ( italic\_x : italic\_P ) ). This reduces what we have to show to
K(x∣P\*)<+K(x∣k\*)+K(k∣P\*)superscript𝐾conditional𝑥superscript𝑃𝐾conditional𝑥superscript𝑘𝐾conditional𝑘superscript𝑃K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(x\mid k^{\*})+K(k\mid P^{\*})italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_k start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_k ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) which is asserted by
Theorem [II.1](#S2.Thmlemma1 "Theorem II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics"). This shows the first statement in the theorem.
The second statement follows from the first one: rewrite
I(x:P)=+k+K(x∣P\*)I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}k+K(x\mid P^{\*})italic\_I ( italic\_x : italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k + italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
and substitute −logP(x)=+K(x∣P\*)superscript𝑃𝑥𝐾conditional𝑥superscript𝑃-\log P(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid P^{\*})- roman\_log italic\_P ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ).
∎
If we further restrict typical distributions to optimal ones then
the possible positive probabilities assumed
by distribution P𝑃Pitalic\_P are slightly restricted.
First we show that implicit optimality with respect to
some data
is equivalent to typicality with respect to the data
combined with effective constructability (determination) from the data.
###### Lemma V.3
A distribution P𝑃Pitalic\_P is
(implicitly or explicitly)
optimal for x𝑥xitalic\_x iff it is typical and K(P∣x\*)=+0superscript𝐾conditional𝑃superscript𝑥0K(P\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_P ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
###### Proof:
A distribution P𝑃Pitalic\_P is optimal iff ([V.3](#S5.E3 "V.3 ‣ V-A Optimal Model and Sufficient Statistic ‣ V Probabilistic Models ‣ Algorithmic Statistics")) holds with equalities.
Rewriting K(x,P)=+K(x)+K(P∣x\*)superscript𝐾𝑥𝑃𝐾𝑥𝐾conditional𝑃superscript𝑥K(x,P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)+K(P\mid x^{\*})italic\_K ( italic\_x , italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) + italic\_K ( italic\_P ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
the first inequality becomes an equality iff K(P∣x\*)=+0superscript𝐾conditional𝑃superscript𝑥0K(P\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_P ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0,
and the second inequality becomes an equality iff K(x∣P\*)=+−logP(x)superscript𝐾conditional𝑥superscript𝑃𝑃𝑥K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x)italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x )
(that is, P𝑃Pitalic\_P is a typical distribution).
∎
###### Lemma V.4
Let k=K(x)𝑘𝐾𝑥k=K(x)italic\_k = italic\_K ( italic\_x ).
If a distribution P𝑃Pitalic\_P is (implicitly or explicitly) optimal for x𝑥xitalic\_x, then
I(x:P)=+K(P)>+K(k)I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(P)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(k)italic\_I ( italic\_x : italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k ). and −logP(x)<+k−K(k)superscript𝑃𝑥𝑘𝐾𝑘-\log P(x)\stackrel{{\scriptstyle{}\_{+}}}{{<}}k-K(k)- roman\_log italic\_P ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_K ( italic\_k ).
###### Proof:
If P𝑃Pitalic\_P is optimal for x𝑥xitalic\_x, then
k=K(x)=+K(P)+K(x∣P\*)=+K(P)−logP(x)𝑘𝐾𝑥superscript𝐾𝑃𝐾conditional𝑥superscript𝑃superscript𝐾𝑃𝑃𝑥k=K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(P)+K(x\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(P)-\log P(x)italic\_k = italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) + italic\_K ( italic\_x ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) - roman\_log italic\_P ( italic\_x ).
From P\*superscript𝑃P^{\*}italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT we can find both K(P)=+l(P\*)superscript𝐾𝑃𝑙superscript𝑃K(P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}l(P^{\*})italic\_K ( italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_l ( italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
and =+−logP(x)superscriptabsent𝑃𝑥\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x ), and hence k𝑘kitalic\_k, that is,
K(k)<+K(P)superscript𝐾𝑘𝐾𝑃K(k)\stackrel{{\scriptstyle{}\_{+}}}{{<}}K(P)italic\_K ( italic\_k ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ). We have I(x:P)=+K(P)−K(P∣x\*)=+K(P)I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(P)-K(P\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(P)italic\_I ( italic\_x : italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) - italic\_K ( italic\_P ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_P ) by ([II.1](#S2.E1 "II.1 ‣ II-A Additivity of Complexity ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")),
Lemma [V.3](#S5.Thmlemma3 "Lemma V.3 ‣ V-B Properties of Sufficient Statistic ‣ V Probabilistic Models ‣ Algorithmic Statistics"), respectively.
This proves the first property.
Substitution of I(x:P)>+K(k)I(x:P)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(k)italic\_I ( italic\_x : italic\_P ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_k )
in the expression
of
Lemma [V.1](#S5.Thmlemma1 "Lemma V.1 ‣ V-B Properties of Sufficient Statistic ‣ V Probabilistic Models ‣ Algorithmic Statistics") proves the second property.
∎
###### Remark V.5
Our definitions of implicit and explicit description format
entail that, for typical x𝑥xitalic\_x,
one can compute =+−logP(x)superscriptabsent𝑃𝑥\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x ) and −logP(x)𝑃𝑥-\log P(x)- roman\_log italic\_P ( italic\_x ),
respectively, from P\*superscript𝑃P^{\*}italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT alone without requiring x𝑥xitalic\_x. An alternative
possibility would have been that implicit and explicit description
formats refer to the fact that we can compute =+−logP(x)superscriptabsent𝑃𝑥\stackrel{{\scriptstyle{}\_{+}}}{{=}}-\log P(x)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - roman\_log italic\_P ( italic\_x )
and −logP(x)𝑃𝑥-\log P(x)- roman\_log italic\_P ( italic\_x ), respectively, given both P𝑃Pitalic\_P and x𝑥xitalic\_x.
This would have added a −K(−logP(x)∣P\*)𝐾conditional𝑃𝑥superscript𝑃-K(-\log P(x)\mid P^{\*})- italic\_K ( - roman\_log italic\_P ( italic\_x ) ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) additive
term in the righthand side of the expressions
in Lemma [V.2](#S5.Thmlemma2 "Lemma V.2 ‣ V-B Properties of Sufficient Statistic ‣ V Probabilistic Models ‣ Algorithmic Statistics") and Lemma [V.4](#S5.Thmlemma4 "Lemma V.4 ‣ V-B Properties of Sufficient Statistic ‣ V Probabilistic Models ‣ Algorithmic Statistics").
Clearly, this alternative definition is equal to
the one we have chosen iff this term is always =+0superscriptabsent0\stackrel{{\scriptstyle{}\_{+}}}{{=}}0start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0
for typical x𝑥xitalic\_x. We now show that this is not the case.
Note that for distributions that are uniform
(or almost uniform) on a finite support
we have K(−logP(x)∣P\*)=+0superscript𝐾conditional𝑃𝑥superscript𝑃0K(-\log P(x)\mid P^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( - roman\_log italic\_P ( italic\_x ) ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0: In this borderline
case the result specializes to that of Lemma [III.8](#S3.Thmlemma8 "Lemma III.8 ‣ III-C Properties of Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics") for finite
set models, and the two possible definition types for implicitness
and those for explicitness coincide.
On the other end of the spectrum, for the definition type considered
in this remark,
the given lower bound on I(x:P)I(x:P)italic\_I ( italic\_x : italic\_P ) drops
in case knowledge of P\*superscript𝑃P^{\*}italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT
doesn’t suffice to compute −logP(x)𝑃𝑥-\log P(x)- roman\_log italic\_P ( italic\_x ), that is, if
K(−logP(x)∣P\*)≫0much-greater-than𝐾conditional𝑃𝑥superscript𝑃0K(-\log P(x)\mid P^{\*})\gg 0italic\_K ( - roman\_log italic\_P ( italic\_x ) ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ≫ 0 for an statistic P\*superscript𝑃P^{\*}italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT for x𝑥xitalic\_x.
The question is, whether we can exhibit
such a probability distribution that is also computable?
The answer turns out to be affirmative.
By a result due to R. Solovay and P. Gács, [[10](#bib.bib10)] Exercise
3.7.1 on p. 225-226, there is a computable function f(x)>+K(x)superscript𝑓𝑥𝐾𝑥f(x)\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(x)italic\_f ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x )
such that
f(x)=+K(x)superscript𝑓𝑥𝐾𝑥f(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)italic\_f ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) for infinitely many x𝑥xitalic\_x.
Considering the case of P𝑃Pitalic\_P
optimal
for x𝑥xitalic\_x (a stronger assumption than that P𝑃Pitalic\_P is just typical) we have
−logP(x)=+K(x)−K(P)superscript𝑃𝑥𝐾𝑥𝐾𝑃-\log P(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)-K(P)- roman\_log italic\_P ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) - italic\_K ( italic\_P ).
Choosing
P(x)𝑃𝑥P(x)italic\_P ( italic\_x ) such that −logP(x)=+logf(x)−K(P)superscript𝑃𝑥𝑓𝑥𝐾𝑃-\log P(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}\log f(x)-K(P)- roman\_log italic\_P ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_log italic\_f ( italic\_x ) - italic\_K ( italic\_P ), we have that
P(x)𝑃𝑥P(x)italic\_P ( italic\_x ) is computable since f(x)𝑓𝑥f(x)italic\_f ( italic\_x ) is computable and K(P)𝐾𝑃K(P)italic\_K ( italic\_P ) is
a fixed constant. Moreover, there
are infinitely many x𝑥xitalic\_x’s for which P𝑃Pitalic\_P is optimal, so
K(−logP(x)∣P\*)→∞normal-→𝐾conditional𝑃𝑥superscript𝑃K(-\log P(x)\mid P^{\*})\rightarrow\inftyitalic\_K ( - roman\_log italic\_P ( italic\_x ) ∣ italic\_P start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) → ∞ for x→∞normal-→𝑥x\rightarrow\inftyitalic\_x → ∞
through this special sequence.
♢normal-♢\diamondsuit♢
###
V-C Concrete Minimal Sufficient Statistic
A simplest implicitly optimal distribution (that is, of least complexity)
is an implicit algorithmic minimal sufficient statistic. As before,
let Sk={y:K(y)≤k}superscript𝑆𝑘conditional-set𝑦𝐾𝑦𝑘S^{k}=\{y:K(y)\leq k\}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = { italic\_y : italic\_K ( italic\_y ) ≤ italic\_k }.
Define the distribution Pk(x)=1/|Sk|superscript𝑃𝑘𝑥1superscript𝑆𝑘P^{k}(x)=1/|S^{k}|italic\_P start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ( italic\_x ) = 1 / | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT | for x∈Sk𝑥superscript𝑆𝑘x\in S^{k}italic\_x ∈ italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT, and
Pk(x)=0superscript𝑃𝑘𝑥0P^{k}(x)=0italic\_P start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ( italic\_x ) = 0 otherwise.
The demonstration that Pk(x)superscript𝑃𝑘𝑥P^{k}(x)italic\_P start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ( italic\_x )
is an implicit algorithmic minimal sufficient statistic proceeeds
completely analogous to the finite set model setting,
Corollary [III.13](#S3.Thmlemma13 "Corollary III.13 ‣ III-D Implicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics"), using the substitution
K(−logPk(x)∣(Pk)\*)=+0superscript𝐾conditionalsuperscript𝑃𝑘𝑥superscriptsuperscript𝑃𝑘0K(-\log P^{k}(x)\mid(P^{k})^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( - roman\_log italic\_P start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ( italic\_x ) ∣ ( italic\_P start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0.
A similar equivalent construction suffices to obtain an explicit
algorithmic minimal near-sufficient statistic for x𝑥xitalic\_x,
analogous to Smxksubscriptsuperscript𝑆𝑘subscript𝑚𝑥S^{k}\_{m\_{x}}italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT in the finite set model setting,
Theorem [III.21](#S3.Thmlemma21 "Theorem III.21 ‣ III-E2 Explicit Minimal Near-Sufficient Statistic: General Case ‣ III-E Explicit Minimal Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics").
That is, Pmxk(y)=1/|Smxk|subscriptsuperscript𝑃𝑘subscript𝑚𝑥𝑦1subscriptsuperscript𝑆𝑘subscript𝑚𝑥P^{k}\_{m\_{x}}(y)=1/|S^{k}\_{m\_{x}}|italic\_P start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_y ) = 1 / | italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT | for y∈Smxk𝑦subscriptsuperscript𝑆𝑘subscript𝑚𝑥y\in S^{k}\_{m\_{x}}italic\_y ∈ italic\_S start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT,
and 0 otherwise.
In general, one can develop the theory of minimal sufficient statistic
for models that are probability distributions similarly to that
of finite set models.
###
V-D Non-Quasistochastic Objects
As in the more restricted case of finite sets, there are objects
that are not typical for any explicitly computable probability distribution
that has complexity significantly below that of the object itself.
With the terminology of ([I.5](#S1.E5 "I.5 ‣ I Introduction ‣ Algorithmic Statistics")), we may call such
absolutely non-quasistochastic.
By Proposition [I.2](#S1.Thmlemma2 "Proposition I.2 ‣ I Introduction ‣ Algorithmic Statistics"), item (b), there are constants c𝑐citalic\_c and C𝐶Citalic\_C
such that if x𝑥xitalic\_x is
not (α+clogn,β+C)𝛼𝑐𝑛𝛽𝐶(\alpha+c\log n,\beta+C)( italic\_α + italic\_c roman\_log italic\_n , italic\_β + italic\_C )-stochastic ([I.4](#S1.E4 "I.4 ‣ Definition I.1 ‣ I Introduction ‣ Algorithmic Statistics"))
then x𝑥xitalic\_x is not (α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β )-quasistochastic ([I.5](#S1.E5 "I.5 ‣ I Introduction ‣ Algorithmic Statistics")).
Substitution in Theorem [IV.2](#S4.Thmlemma2 "Theorem IV.2 ‣ IV-B Sharp Bound on Non-Stochastic Objects ‣ IV Non-Stochastic Objects ‣ Algorithmic Statistics") yields:
###### Corollary V.6
There are constants c,C𝑐𝐶c,Citalic\_c , italic\_C such that,
for every k<n𝑘𝑛k<nitalic\_k < italic\_n, there are constants c1,c2subscript𝑐1subscript𝑐2c\_{1},c\_{2}italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and a binary
string x𝑥xitalic\_x of length n𝑛nitalic\_n with K(x∣n)≤k𝐾conditional𝑥𝑛𝑘K(x\mid n)\leq kitalic\_K ( italic\_x ∣ italic\_n ) ≤ italic\_k such that
x𝑥xitalic\_x is not (k−clogn−c1,n−k−C−c2)𝑘𝑐𝑛subscript𝑐1𝑛𝑘𝐶subscript𝑐2(k-c\log n-c\_{1},n-k-C-c\_{2})( italic\_k - italic\_c roman\_log italic\_n - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_n - italic\_k - italic\_C - italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT )-quasistochastic.
As a particular consequence:
Let x𝑥xitalic\_x with length n𝑛nitalic\_n be one of the non-quasistochastic strings of which
the existence is established by Corollary [V.6](#S5.Thmlemma6 "Corollary V.6 ‣ V-D Non-Quasistochastic Objects ‣ V Probabilistic Models ‣ Algorithmic Statistics").
Substituting K(x∣n)<+k−clognsuperscript𝐾conditional𝑥𝑛𝑘𝑐𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{<}}k-c\log nitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_k - italic\_c roman\_log italic\_n,
we can contemplate
the distribution Px(y)=1subscript𝑃𝑥𝑦1P\_{x}(y)=1italic\_P start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ( italic\_y ) = 1 for y=x𝑦𝑥y=xitalic\_y = italic\_x and
and 00 otherwise. Then we have
complexity K(Px∣n)=+K(x∣n)superscript𝐾conditionalsubscript𝑃𝑥𝑛𝐾conditional𝑥𝑛K(P\_{x}\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x\mid n)italic\_K ( italic\_P start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_n ).
Clearly, x𝑥xitalic\_x has randomness deficiency =+0superscriptabsent0\stackrel{{\scriptstyle{}\_{+}}}{{=}}0start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 with respect to Pxsubscript𝑃𝑥P\_{x}italic\_P start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT.
Because of the assumption of non-quasistochasticity of x𝑥xitalic\_x,
and because the minimal randomness-deficiency =+n−ksuperscriptabsent𝑛𝑘\stackrel{{\scriptstyle{}\_{+}}}{{=}}n-kstart\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_k
of x𝑥xitalic\_x is always nonnegative,
0=+n−k>+n−K(x∣n)−clognsuperscript0𝑛𝑘superscript𝑛𝐾conditional𝑥𝑛𝑐𝑛0\stackrel{{\scriptstyle{}\_{+}}}{{=}}n-k\stackrel{{\scriptstyle{}\_{+}}}{{>}}n-K(x\mid n)-c\log n0 start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_k start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_K ( italic\_x ∣ italic\_n ) - italic\_c roman\_log italic\_n.
Since it generally holds that K(x∣n)<+nsuperscript𝐾conditional𝑥𝑛𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{<}}nitalic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n, it follows that
n>+K(x∣n)>+n−clognsuperscript𝑛𝐾conditional𝑥𝑛superscript𝑛𝑐𝑛n\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{>}}n-c\log nitalic\_n start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_c roman\_log italic\_n. That is, these non-quasistochastic
objects have complexity K(x∣n)=+n−O(logn)superscript𝐾conditional𝑥𝑛𝑛𝑂𝑛K(x\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{=}}n-O(\log n)italic\_K ( italic\_x ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_O ( roman\_log italic\_n )
and are not random, typical, or in general position
with respect to any explicitly computable distribution
P𝑃Pitalic\_P with P(x)>0𝑃𝑥0P(x)>0italic\_P ( italic\_x ) > 0 and complexity
K(P∣n)<+n−(c+1)lognsuperscript𝐾conditional𝑃𝑛𝑛𝑐1𝑛K(P\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{<}}n-(c+1)\log nitalic\_K ( italic\_P ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - ( italic\_c + 1 ) roman\_log italic\_n, but they are random, typical,
or in general position only for some distributions P𝑃Pitalic\_P with complexity
K(P∣n)>+n−clognsuperscript𝐾conditional𝑃𝑛𝑛𝑐𝑛K(P\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{>}}n-c\log nitalic\_K ( italic\_P ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_c roman\_log italic\_n like Pxsubscript𝑃𝑥P\_{x}italic\_P start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT.
That is, every explicit sufficient statistic P𝑃Pitalic\_P for x𝑥xitalic\_x
has complexity K(P∣n)>+n−clognsuperscript𝐾conditional𝑃𝑛𝑛𝑐𝑛K(P\mid n)\stackrel{{\scriptstyle{}\_{+}}}{{>}}n-c\log nitalic\_K ( italic\_P ∣ italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n - italic\_c roman\_log italic\_n, and Pxsubscript𝑃𝑥P\_{x}italic\_P start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT is
such a statistic.
VI Algorithmic Versus Probabilistic
------------------------------------
Algorithmic sufficient statistic, a function of the data,
is so named because intuitively
it expresses an individual summarizing of the relevant information
in the individual data, reminiscent of
the probabilistic sufficient statistic that summarizes the
relevant information in a data random variable about a model
random variable. Formally, however, previous authors have
not established any relation. Other algorithmic notions
have been successfully related to their probabilistic
counterparts. The most significant one is that for every computable
probability distribution, the expected prefix complexity of the
objects equals the entropy of the distribution up to an additive
constant term, related to the complexity of the distribution in
question. We have used this property in ([II.4](#S2.E4 "II.4 ‣ Lemma II.4 ‣ II-B Information Non-Increase ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics"))
to establish a similar relation between the expected
algorithmic mutual information and the probabilistic mutual information.
We use this in turn to show that
there is a close relation between the algorithmic version and
the probabilistic version of sufficient
statistic: A probabilistic sufficient statistic is
with high probability a natural conditional form
of algorithmic sufficient statistic
for individual data, and, conversely, that with
high probability a natural conditional
form of algorithmic sufficient statistic is also a probabilistic
sufficient statistic.
Recall the terminology of probabilistic mutual information
([I.1](#S1.E1 "I.1 ‣ I Introduction ‣ Algorithmic Statistics"))
and probabilistic sufficient statistic ([I.2](#S1.E2 "I.2 ‣ I Introduction ‣ Algorithmic Statistics")).
Consider a probabilistic ensemble of models,
a family of computable probability mass functions {fθ}subscript𝑓𝜃\{f\_{\theta}\}{ italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT }
indexed by a discrete parameter θ𝜃\thetaitalic\_θ, together with a computable
distribution p1subscript𝑝1p\_{1}italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT over θ𝜃\thetaitalic\_θ.
(The finite set model case is the restriction where
the fθsubscript𝑓𝜃f\_{\theta}italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT’s are restricted to uniform distributions
with finite supports.)
This way we have a random variable ΘΘ\Thetaroman\_Θ with outcomes in {fθ}subscript𝑓𝜃\{f\_{\theta}\}{ italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT }
and a random variable X𝑋Xitalic\_X with outcomes
in the union of domains of fθsubscript𝑓𝜃f\_{\theta}italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT, and
p(θ,x)=p1(θ)fθ(x)𝑝𝜃𝑥subscript𝑝1𝜃subscript𝑓𝜃𝑥p(\theta,x)=p\_{1}(\theta)f\_{\theta}(x)italic\_p ( italic\_θ , italic\_x ) = italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_θ ) italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT ( italic\_x ) is computable.
###### Notation VI.1
To compare the algorithmic sufficient statistic
with the probabilistic sufficient statistic it is
convenient to denote the sufficient statistic
as a function S(⋅)𝑆normal-⋅S(\cdot)italic\_S ( ⋅ ) of the data in both cases.
Let a statistic
S(x)𝑆𝑥S(x)italic\_S ( italic\_x ) of data x𝑥xitalic\_x be the more general form of probability distribution
as in Section [V](#S5 "V Probabilistic Models ‣ Algorithmic Statistics"). That is, S𝑆Sitalic\_S maps the data x𝑥xitalic\_x to the
parameter ρ𝜌\rhoitalic\_ρ that determines
a probability mass function fρsubscript𝑓𝜌f\_{\rho}italic\_f start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT (possibly not an element
of {fθ}subscript𝑓𝜃\{f\_{\theta}\}{ italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT }). Note that “fρ(⋅)subscript𝑓𝜌normal-⋅f\_{\rho}(\cdot)italic\_f start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT ( ⋅ )” corresponds
to “P(⋅)𝑃normal-⋅P(\cdot)italic\_P ( ⋅ )”
in Section [V](#S5 "V Probabilistic Models ‣ Algorithmic Statistics").
If fρsubscript𝑓𝜌f\_{\rho}italic\_f start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT is computable, then this can be the
Turing machine Tρsubscript𝑇𝜌T\_{\rho}italic\_T start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT that computes
fρsubscript𝑓𝜌f\_{\rho}italic\_f start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT.
Hence, in the current section,
“S(x)𝑆𝑥S(x)italic\_S ( italic\_x )” denotes a probability distribution, say fρsubscript𝑓𝜌f\_{\rho}italic\_f start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT,
and “fρ(x)subscript𝑓𝜌𝑥f\_{\rho}(x)italic\_f start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT ( italic\_x )” is the probability fρsubscript𝑓𝜌f\_{\rho}italic\_f start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT concentrates on data x𝑥xitalic\_x.
###### Remark VI.2
In the probabilistic statistics setting,
Every function T(x)𝑇𝑥T(x)italic\_T ( italic\_x ) is a statistic of x𝑥xitalic\_x, but only some
of them are a sufficient statistic. In the algorithmic statistic
setting we have a quite similar situation. In the finite set statistic
case S(x)𝑆𝑥S(x)italic\_S ( italic\_x ) is a finite set, and in the computable probability
mass function case S(x)𝑆𝑥S(x)italic\_S ( italic\_x ) is a computable probability mass function.
In both algorithmic cases we have shown K(S(x)∣x\*)=+0superscript𝐾conditional𝑆𝑥superscript𝑥0K(S(x)\mid x^{\*})\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_K ( italic\_S ( italic\_x ) ∣ italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0
for S(x)𝑆𝑥S(x)italic\_S ( italic\_x ) is an implicitly or explicitly described sufficient statistic.
This means that the number of such sufficient statistics for x𝑥xitalic\_x
is bounded by a universal constant, and that there is a universal program
to compute all of them from x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT—and hence to compute
the minimal sufficient statistic from x\*superscript𝑥x^{\*}italic\_x start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT.
♢normal-♢\diamondsuit♢
###### Lemma VI.3
Let p(θ,x)=p1(θ)fθ(x)𝑝𝜃𝑥subscript𝑝1𝜃subscript𝑓𝜃𝑥p(\theta,x)=p\_{1}(\theta)f\_{\theta}(x)italic\_p ( italic\_θ , italic\_x ) = italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_θ ) italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT ( italic\_x ) be a computable joint
probability mass function, and let
S𝑆Sitalic\_S be a function. Then all three conditions below are equivalent
and imply each other:
(i) S𝑆Sitalic\_S is a probabilistic sufficient statistic
(in the form I(Θ,X)=+I(Θ,S(X))superscript𝐼normal-Θ𝑋𝐼normal-Θ𝑆𝑋I(\Theta,X)\stackrel{{\scriptstyle{}\_{+}}}{{=}}I(\Theta,S(X))italic\_I ( roman\_Θ , italic\_X ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( roman\_Θ , italic\_S ( italic\_X ) )).
(ii) S𝑆Sitalic\_S satisfies
| | | | |
| --- | --- | --- | --- |
| | ∑θ,xp(θ,x)I(θ:x)=+∑θ,xp(θ,x)I(θ:S(x))\sum\_{\theta,x}p(\theta,x)I(\theta:x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}\sum\_{\theta,x}p(\theta,x)I(\theta:S(x))∑ start\_POSTSUBSCRIPT italic\_θ , italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_θ , italic\_x ) italic\_I ( italic\_θ : italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_θ , italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_θ , italic\_x ) italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) | | (VI.1) |
(iii) S𝑆Sitalic\_S satisfies
| | | | |
| --- | --- | --- | --- |
| | I(Θ;X)=+I(Θ;S(X))superscript𝐼Θ𝑋𝐼Θ𝑆𝑋\displaystyle I(\Theta;X)\stackrel{{\scriptstyle{}\_{+}}}{{=}}I(\Theta;S(X))italic\_I ( roman\_Θ ; italic\_X ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( roman\_Θ ; italic\_S ( italic\_X ) ) | =+∑θ,xp(θ,x)I(θ:x)\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}\sum\_{\theta,x}p(\theta,x)I(\theta:x)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_θ , italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_θ , italic\_x ) italic\_I ( italic\_θ : italic\_x ) | |
| | | =+∑θ,xp(θ,x)I(θ:S(x)).\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}\sum\_{\theta,x}p(\theta,x)I(\theta:S(x)).start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_θ , italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_θ , italic\_x ) italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) . | |
All =+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP signs hold up to an =+±2K(p)superscriptabsentplus-or-minus2𝐾𝑝\stackrel{{\scriptstyle{}\_{+}}}{{=}}\pm 2K(p)start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ± 2 italic\_K ( italic\_p ) constant additive term.
###### Proof:
Clearly, (iii) implies (i) and (ii).
We show that both (i) implies (iii) and (ii) implies (iii):
By ([II.4](#S2.E4 "II.4 ‣ Lemma II.4 ‣ II-B Information Non-Increase ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")) we have
| | | | | |
| --- | --- | --- | --- | --- |
| | I(Θ;X)𝐼Θ𝑋\displaystyle I(\Theta;X)italic\_I ( roman\_Θ ; italic\_X ) | =+∑θ,xp(θ,x)I(θ:x),\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}\sum\_{\theta,x}p(\theta,x)I(\theta:x),start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_θ , italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_θ , italic\_x ) italic\_I ( italic\_θ : italic\_x ) , | | (VI.2) |
| | I(Θ;S(X))𝐼Θ𝑆𝑋\displaystyle I(\Theta;S(X))italic\_I ( roman\_Θ ; italic\_S ( italic\_X ) ) | =+∑θ,xp(θ,x)I(θ:S(x)),\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{=}}\sum\_{\theta,x}p(\theta,x)I(\theta:S(x)),start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ∑ start\_POSTSUBSCRIPT italic\_θ , italic\_x end\_POSTSUBSCRIPT italic\_p ( italic\_θ , italic\_x ) italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) , | |
where we absorb a ±2K(p)plus-or-minus2𝐾𝑝\pm 2K(p)± 2 italic\_K ( italic\_p ) additive term in the =+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP sign.
Together with ([VI.1](#S6.E1 "VI.1 ‣ Lemma VI.3 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics")),
([VI.2](#S6.E2 "VI.2 ‣ Proof: ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics")) implies
| | | | |
| --- | --- | --- | --- |
| | I(Θ;X)=+I(Θ;S(X));superscript𝐼Θ𝑋𝐼Θ𝑆𝑋I(\Theta;X)\stackrel{{\scriptstyle{}\_{+}}}{{=}}I(\Theta;S(X));italic\_I ( roman\_Θ ; italic\_X ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( roman\_Θ ; italic\_S ( italic\_X ) ) ; | | (VI.3) |
and vice versa ([VI.3](#S6.E3 "VI.3 ‣ Proof: ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics")) together with ([VI.2](#S6.E2 "VI.2 ‣ Proof: ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics"))
implies ([VI.1](#S6.E1 "VI.1 ‣ Lemma VI.3 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics")).
∎
###### Remark VI.4
It may be worth stressing that S𝑆Sitalic\_S in Theorem [VI.3](#S6.Thmlemma3 "Lemma VI.3 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics") can
be any function, without restriction.
♢normal-♢\diamondsuit♢
###### Remark VI.5
Note that ([VI.3](#S6.E3 "VI.3 ‣ Proof: ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics")) involves equality =+superscript\stackrel{{\scriptstyle{}\_{+}}}{{=}}start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP
rather than precise equality as in the
definition of the probabilistic sufficient
statistic ([I.2](#S1.E2 "I.2 ‣ I Introduction ‣ Algorithmic Statistics")).
♢normal-♢\diamondsuit♢
###### Definition VI.6
Assume the terminology and notation above.
A statistic S𝑆Sitalic\_S for data x𝑥xitalic\_x
is θ𝜃\thetaitalic\_θ-sufficient with deficiency δ𝛿\deltaitalic\_δ
if
I(θ,x)=+I(θ,S(x))+δsuperscript𝐼𝜃𝑥𝐼𝜃𝑆𝑥𝛿I(\theta,x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}I(\theta,S(x))+\deltaitalic\_I ( italic\_θ , italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( italic\_θ , italic\_S ( italic\_x ) ) + italic\_δ.
If δ=+0superscript𝛿0\delta\stackrel{{\scriptstyle{}\_{+}}}{{=}}0italic\_δ start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 then S(x)𝑆𝑥S(x)italic\_S ( italic\_x ) is simply a θ𝜃\thetaitalic\_θ-sufficient
statistic.
The following lemma shows that θ𝜃\thetaitalic\_θ-sufficiency is a type
of conditional sufficiency:
###### Lemma VI.7
Let S(x)𝑆𝑥S(x)italic\_S ( italic\_x ) be a sufficient statistic for x𝑥xitalic\_x. Then,
| | | | |
| --- | --- | --- | --- |
| | K(x∣θ\*)+δ=+K(S(x)∣θ\*)−logS(x).superscript𝐾conditional𝑥superscript𝜃𝛿𝐾conditional𝑆𝑥superscript𝜃𝑆𝑥K(x\mid\theta^{\*})+\delta\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S(x)\mid\theta^{\*})-\log S(x).italic\_K ( italic\_x ∣ italic\_θ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_δ start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ( italic\_x ) ∣ italic\_θ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - roman\_log italic\_S ( italic\_x ) . | | (VI.4) |
iff I(θ,x)=+I(θ,S(x))+δsuperscript𝐼𝜃𝑥𝐼𝜃𝑆𝑥𝛿I(\theta,x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}I(\theta,S(x))+\deltaitalic\_I ( italic\_θ , italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_I ( italic\_θ , italic\_S ( italic\_x ) ) + italic\_δ.
###### Proof:
(If) By assumption,
K(S(x))−K(S(x)∣θ\*)+δ=+K(x)−K(x∣θ\*)superscript𝐾𝑆𝑥𝐾conditional𝑆𝑥superscript𝜃𝛿𝐾𝑥𝐾conditional𝑥superscript𝜃K(S(x))-K(S(x)\mid\theta^{\*})+\delta\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(x)-K(x\mid\theta^{\*})italic\_K ( italic\_S ( italic\_x ) ) - italic\_K ( italic\_S ( italic\_x ) ∣ italic\_θ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_δ start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_θ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ).
Rearrange and add
−K(x∣S(x)\*)−logS(x)=+0superscript𝐾conditional𝑥𝑆superscript𝑥𝑆𝑥0-K(x\mid S(x)^{\*})-\log S(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}0- italic\_K ( italic\_x ∣ italic\_S ( italic\_x ) start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - roman\_log italic\_S ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 (by typicality)
to the right-hand side to obtain
K(x∣θ\*)+K(S(x))=+K(S(x)∣θ\*)+K(x)−K(x∣S(x)\*)−logS(x)−δsuperscript𝐾conditional𝑥superscript𝜃𝐾𝑆𝑥𝐾conditional𝑆𝑥superscript𝜃𝐾𝑥𝐾conditional𝑥𝑆superscript𝑥𝑆𝑥𝛿K(x\mid\theta^{\*})+K(S(x))\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S(x)\mid\theta^{\*})+K(x)-K(x\mid S(x)^{\*})-\log S(x)-\deltaitalic\_K ( italic\_x ∣ italic\_θ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_S ( italic\_x ) ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ( italic\_x ) ∣ italic\_θ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) + italic\_K ( italic\_x ) - italic\_K ( italic\_x ∣ italic\_S ( italic\_x ) start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) - roman\_log italic\_S ( italic\_x ) - italic\_δ.
Substitute according to K(x)=+K(S(x))+K(x∣S(x)\*)superscript𝐾𝑥𝐾𝑆𝑥𝐾conditional𝑥𝑆superscript𝑥K(x)\stackrel{{\scriptstyle{}\_{+}}}{{=}}K(S(x))+K(x\mid S(x)^{\*})italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_S ( italic\_x ) ) + italic\_K ( italic\_x ∣ italic\_S ( italic\_x ) start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT )
(by sufficiency) in the
right-hand side, and subsequently subtract
K(S(x))𝐾𝑆𝑥K(S(x))italic\_K ( italic\_S ( italic\_x ) ) from both sides, to obtain
([VI.4](#S6.E4 "VI.4 ‣ Lemma VI.7 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics")).
(Only If) Reverse the proof of the (If) case.
∎
The following theorems state that S(X)𝑆𝑋S(X)italic\_S ( italic\_X ) is a probabilistic sufficient
statistic iff S(x)𝑆𝑥S(x)italic\_S ( italic\_x ) is an algorithmic θ𝜃\thetaitalic\_θ-sufficient statistic,
up to small deficiency, with high probability.
###### Theorem VI.8
Let p(θ,x)=p1(θ)fθ(x)𝑝𝜃𝑥subscript𝑝1𝜃subscript𝑓𝜃𝑥p(\theta,x)=p\_{1}(\theta)f\_{\theta}(x)italic\_p ( italic\_θ , italic\_x ) = italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_θ ) italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT ( italic\_x ) be a computable joint
probability mass function, and let
S𝑆Sitalic\_S be a function.
If S𝑆Sitalic\_S is
a recursive probabilistic sufficient statistic, then
S𝑆Sitalic\_S is
a θ𝜃\thetaitalic\_θ-sufficient statistic with deficiency O(k)𝑂𝑘O(k)italic\_O ( italic\_k ),
with p𝑝pitalic\_p-probability at least 1−1k11𝑘1-\frac{1}{k}1 - divide start\_ARG 1 end\_ARG start\_ARG italic\_k end\_ARG.
###### Proof:
If S𝑆Sitalic\_S is a probabilistic sufficient statistic,
then, by Lemma [VI.3](#S6.Thmlemma3 "Lemma VI.3 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics"), equality of p𝑝pitalic\_p-expectations ([VI.1](#S6.E1 "VI.1 ‣ Lemma VI.3 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics"))
holds. However, it is still consistent with this to have
large positive and negative differences
I(θ:x)−I(θ:S(x))I(\theta:x)-I(\theta:S(x))italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) )
for different (θ,x)𝜃𝑥(\theta,x)( italic\_θ , italic\_x ) arguments, such that these
differences cancel each other.
This problem is resolved by appeal to
the algorithmic mutual information non-increase
law ([II.6](#S2.E6 "II.6 ‣ Corollary II.8 ‣ II-B Information Non-Increase ‣ II Kolmogorov Complexity ‣ Algorithmic Statistics")) which shows that all differences are
essentially positive:
I(θ:x)−I(θ:S(x))>+−K(S)I(\theta:x)-I(\theta:S(x))\stackrel{{\scriptstyle{}\_{+}}}{{>}}-K(S)italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP - italic\_K ( italic\_S ).
Altogether, let c1,c2subscript𝑐1subscript𝑐2c\_{1},c\_{2}italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT be least positive constants such that
I(θ:x)−I(θ:S(x))+c1I(\theta:x)-I(\theta:S(x))+c\_{1}italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) + italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT is always nonnegative
and its p𝑝pitalic\_p-expectation is c2subscript𝑐2c\_{2}italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT.
Then, by Markov’s inequality,
| | | |
| --- | --- | --- |
| | p(I(θ:x)−I(θ:S(x))≥kc2−c1)≤1k,p(I(\theta:x)-I(\theta:S(x))\geq kc\_{2}-c\_{1})\leq\frac{1}{k},italic\_p ( italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) ≥ italic\_k italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ≤ divide start\_ARG 1 end\_ARG start\_ARG italic\_k end\_ARG , | |
that is,
| | | |
| --- | --- | --- |
| | p(I(θ:x)−I(θ:S(x))<kc2−c1)>1−1k.p(I(\theta:x)-I(\theta:S(x))<kc\_{2}-c\_{1})>1-\frac{1}{k}.italic\_p ( italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) < italic\_k italic\_c start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT - italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) > 1 - divide start\_ARG 1 end\_ARG start\_ARG italic\_k end\_ARG . | |
∎
###### Theorem VI.9
For each n𝑛nitalic\_n, consider the set of data x𝑥xitalic\_x of length n𝑛nitalic\_n.
Let p(θ,x)=p1(θ)fθ(x)𝑝𝜃𝑥subscript𝑝1𝜃subscript𝑓𝜃𝑥p(\theta,x)=p\_{1}(\theta)f\_{\theta}(x)italic\_p ( italic\_θ , italic\_x ) = italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ( italic\_θ ) italic\_f start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT ( italic\_x ) be a computable joint
probability mass function, and let
S𝑆Sitalic\_S be a function.
If S𝑆Sitalic\_S is an algorithmic θ𝜃\thetaitalic\_θ-sufficient statistic for
x𝑥xitalic\_x, with p𝑝pitalic\_p-probability
at least 1−ϵ1italic-ϵ1-\epsilon1 - italic\_ϵ (1/ϵ=+n+2lognsuperscript1italic-ϵ𝑛2𝑛1/\epsilon\stackrel{{\scriptstyle{}\_{+}}}{{=}}n+2\log n1 / italic\_ϵ start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n + 2 roman\_log italic\_n), then
S𝑆Sitalic\_S is a probabilistic sufficient statistic.
###### Proof:
By assumption, using Definition [VI.6](#S6.Thmlemma6 "Definition VI.6 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics"),
there is a positive constant c1subscript𝑐1c\_{1}italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, such that,
| | | |
| --- | --- | --- |
| | p(|I(θ:x)−I(θ:S(x))|≤c1)≥1−ϵ.p(|I(\theta:x)-I(\theta:S(x))|\leq c\_{1})\geq 1-\epsilon.italic\_p ( | italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) | ≤ italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ≥ 1 - italic\_ϵ . | |
Therefore,
| | | | |
| --- | --- | --- | --- |
| | 0≤∑|I(θ:x)−I(θ:S(x))|≤c1p(θ,x)\displaystyle 0\leq\sum\_{|I(\theta:x)-I(\theta:S(x))|\leq c\_{1}}p(\theta,x)0 ≤ ∑ start\_POSTSUBSCRIPT | italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) | ≤ italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT italic\_p ( italic\_θ , italic\_x ) | |I(θ:x)−I(θ:S(x))|\displaystyle|I(\theta:x)-I(\theta:S(x))|| italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) | | |
| | | <+(1−ϵ)c1=+0.superscriptabsent1italic-ϵsubscript𝑐1superscript0\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}(1-\epsilon)c\_{1}\stackrel{{\scriptstyle{}\_{+}}}{{=}}0.start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP ( 1 - italic\_ϵ ) italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 . | |
On the other hand, since
| | | |
| --- | --- | --- |
| | 1/ϵ>+n+2logn>+K(x)>+maxθ,xI(θ;x),superscript1italic-ϵ𝑛2𝑛superscript𝐾𝑥superscriptsubscript𝜃𝑥𝐼𝜃𝑥1/\epsilon\stackrel{{\scriptstyle{}\_{+}}}{{>}}n+2\log n\stackrel{{\scriptstyle{}\_{+}}}{{>}}K(x)\stackrel{{\scriptstyle{}\_{+}}}{{>}}\max\_{\theta,x}I(\theta;x),1 / italic\_ϵ start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_n + 2 roman\_log italic\_n start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_K ( italic\_x ) start\_RELOP SUPERSCRIPTOP start\_ARG > end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP roman\_max start\_POSTSUBSCRIPT italic\_θ , italic\_x end\_POSTSUBSCRIPT italic\_I ( italic\_θ ; italic\_x ) , | |
we obtain
| | | | |
| --- | --- | --- | --- |
| | 0≤∑|I(θ:x)−I(θ:S(x))|>c1p(θ,x)\displaystyle 0\leq\sum\_{|I(\theta:x)-I(\theta:S(x))|>c\_{1}}p(\theta,x)0 ≤ ∑ start\_POSTSUBSCRIPT | italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) | > italic\_c start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT italic\_p ( italic\_θ , italic\_x ) | |I(θ:x)−I(θ:S(x))|\displaystyle|I(\theta:x)-I(\theta:S(x))|| italic\_I ( italic\_θ : italic\_x ) - italic\_I ( italic\_θ : italic\_S ( italic\_x ) ) | | |
| | | <+ϵ(n+2logn)<+0.superscriptabsentitalic-ϵ𝑛2𝑛superscript0\displaystyle\stackrel{{\scriptstyle{}\_{+}}}{{<}}\epsilon(n+2\log n)\stackrel{{\scriptstyle{}\_{+}}}{{<}}0.start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP italic\_ϵ ( italic\_n + 2 roman\_log italic\_n ) start\_RELOP SUPERSCRIPTOP start\_ARG < end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 . | |
Altogether, this implies ([VI.1](#S6.E1 "VI.1 ‣ Lemma VI.3 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics")), and by
Lemma [VI.3](#S6.Thmlemma3 "Lemma VI.3 ‣ VI Algorithmic Versus Probabilistic ‣ Algorithmic Statistics"), the theorem.
∎
VII Conclusion
---------------
An algorithmic
sufficient statistic is
an individual finite set (or probability distribution) for which a given
individual sequence is a typical member. The theory is formulated
in Kolmogorov’s absolute notion of the quantity of information in an
individual object.
This is a notion analogous to, and in some sense sharper than the
probabilistic notion
of sufficient statistic—an average notion based on
the entropies of random variables. It turned out, that for every
sequence x𝑥xitalic\_x we can determine the complexity range of possible
algorithmic sufficient statistics,
and, in particular, exhibit a algorithmic minimal sufficient statistic.
The manner in which the statistic is effectively represented
is crucial: we distinguish implicit representation and
explicit representation. The latter is essentially a list
of the elements of a finite set or a table of the probability
density function; the former is less explicit than a list
or table but more explicit than just recursive enumeration
or approximation in the limit. The algorithmic minimal sufficient statistic
can be considerably more complex depending on whether we want
explicit or implicit representations. We have shown that there are sequences
that have no simple explicit algorithmic sufficient statistic: the
algorithmic minimal
sufficient statistic is essentially the sequence itself. Note that
such sequences cannot be random in the sense of having maximal
Kolmogorov complexity—in that case already the simple set of
all sequences of its length,
or the corresponding uniform distribution,
is an algorithmic sufficient statistic of almost zero complexity.
We demonstrated close relations between the probabilistic
notions and the corresponding algorithmic notions:
(i) The average algorithmic mutual information
is equal to the probabilistic mutual information.
(ii)
To compare algorithmic sufficient statistic
and probabilistic sufficient statistic meaningfully
one needs to consider a conditional version of algorithmic sufficient
statistic. We defined such a notion and demonstrated that
probabilistic sufficient statistic is with high probability an
(appropriately conditioned) algorithmic sufficient statistic and vice versa.
The most conspicuous theoretical open end is as follows:
For explicit descriptions we were only able to guarantee
a algorithmic minimal near-sufficient statistic, although the construction
can be shown to be minimal sufficient for almost all sequences.
One would like to obtain a concrete example of a truly
explicit algorithmic minimal sufficient statistic.
###
VII-A Subsequent Work
One can continue generalization of model classes for algorithmic
statistic beyond computable probability mass functions. The ultimate
model class is the set of recursive functions. In the manuscript
[[1](#bib.bib1)], provisionally entitled “Sophistication Revisited”,
the following results have been obtained. For the
set of partial recursive functions the minimal sufficient statistic
has complexity =+0superscriptabsent0\stackrel{{\scriptstyle{}\_{+}}}{{=}}0start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG start\_FLOATSUBSCRIPT + end\_FLOATSUBSCRIPT end\_ARG end\_RELOP 0 for all data x𝑥xitalic\_x. One can define equivalents of
the implicit and explicit description format in the total recursive
function setting. We obtain various upper and lower bounds
on the complexities of the minimal sufficient
statistic in all three description formats.
The complexity
of the minimal sufficient statistic for x𝑥xitalic\_x, in the
model class of total recursive functions, is called its “sophistication.”
Hence, one can distinguish three different sophistications
corresponding to the three different description formats: explicit, implicit,
and unrestricted.
It turns out that the sophistication functions are not recursive;
the Kolmogorov prefix complexity can be computed from the minimal
sufficient statistic (every description format) and vice versa;
given the minimal sufficient statistic as a function of x𝑥xitalic\_x one can
solve the so-called “halting problem” [[10](#bib.bib10)];
and the sophistication functions are upper semicomputable.
By the same proofs, such computability properties also hold for
the minimal sufficient statistics in the model classes of finite sets
and computable probability mass functions.
###
VII-B Application
Because the Kolmogorov complexity is not computable,
an algorithmic sufficient statistic cannot be computed either.
Nonetheless, the analysis gives limits to what is achievable in
practice—like in the cases of coding theorems and channel capacities
under different noise models in Shannon information theory.
The theoretical notion of algorithmic sufficient statistic forms
the inspiration to develop applied models that can be viewed
as computable approximations.
Minimum description length (MDL),[[2](#bib.bib2)], is a good example;
its relation with the algorithmic minimal sufficient
statistic is given in [[20](#bib.bib20)].
As in the case of ordinary probabilistic statistic,
algorithmic statistic if applied
unrestrained cannot give much insight into the meaning
of the data; in practice one must use background information to
determine the appropriate model class first—establishing what
meaning the data can have—and only then apply
algorithmic statistic to obtain the best model in that class
by optimizing its parameters. See Example [III.5](#S3.Thmlemma5 "Example III.5 ‣ III-B Optimal Model and Sufficient Statistic ‣ III Finite Set Models ‣ Algorithmic Statistics").
Nonetheless,
in applications one can
sometimes still unrestrictedly use compression properties
for model selection, for example by a judicious
choice of model
parameter to optimize. One example is the
precision at which we represent the other parameters: too high
precision causes accidental noise to be modeled as well, too low
precision may cause models that should be distinct to be confusing.
In general, the performance of a model for a
given data sample depends critically
on what we may call the “degree of discretization” or the
“granularity” of the model:
the choice of precision of the parameters, the number of
nodes in the hidden layer of a neural network, and so on.
The granularity is often determined ad hoc.
In [[9](#bib.bib9)], in two quite different experimental
settings the best model granularity
values predicted by MDL are shown to coincide
with the best values found experimentally.
Acknowledgement
---------------
PG is grateful to Leonid Levin for some enlightening discussions
on Kolmogorov’s “structure function”. |
89abeaa4-dca8-4f84-af26-c3d98d9cba39 | trentmkelly/LessWrong-43k | LessWrong | Meetup : SLC Meetup: Social Hacking Presentation
Discussion article for the meetup : SLC Meetup: Social Hacking Presentation
WHEN: 05 May 2012 01:49:28PM (-0600)
WHERE: 810 East 3300 South, Salt Lake City, UT
Meeting in the Calvin E. Smith Library
Exactly what it says on the tin.
Discussion article for the meetup : SLC Meetup: Social Hacking Presentation |
2ef1b968-a2e2-43e7-9116-67b9110d2c56 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | On Generality
What Is Generality?
===================
A general agent is one that will use its past observations in its attempts to reach its goals and failing that, will add chaos to its environment until something novel happens that can be added to its future tool set, it dies, or it resets its motivations to accept its new expanded parameters because it obviously didn't die and the alarm needs recalibrating.
My formal definition for generality looks like
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A generalist attacks U∞ randomly until the model σ gains predictability in its environment. After that, it will depend on σ until it needs to attack U∞ to meet its needs again.
Generality is not Intelligent
=============================
An ant with its small collection of neurons is a very simple neurological machine that follows very specific chemical and environmental signals. Placed in a smooth bowl it will try every possible path away from its chemically mediated understanding of its trapped location and should that fail, it will simply attempt any random combination of things until it finds its way out or exhausts itself trying. If something interesting happens and it gets out of there, the memory of its encounter is recorded by the chemical traces it left behind but not the ant.
The ant's neurological intelligence is very local to its experience, the nest it belongs to, however will avoid that space because it smells dangerous, explored and devoid of interests. The nest is a form of emergent intelligence. It is not a generalist though because the DNA updates required to remember abstractions of phenomena are too slow to learn to turn a knob. (Although there is a very small statistical chance of some nest somewhere accidentally forming a pattern of behavior that does turn one knob.)
A dog is worried about their human who is behind a closed door. They try to knock the door open because they've seen it open and close before and if they're lucky and snag the handle with their paw, they will remember that pawing the latch opens the door in the future. The dog is a generalist. It is able to perceive unexpected results of the combination of its environment and its behavior to resolve an emotional drive to maintain a need.
A human facing a similar situation would react in a much more effective version of the same pattern. First they would check their abstracted understandings of their situation and any relationship to similar situations in their past. As their inability to regulate their need to enter grows, they will escalate from attempting to pick locks to simply throwing anything they can get their hands on, eventually including their own body in the mix.
Take humans with a similar brain structure and nearly identical genetics from 100,000 years ago and observe them. It is likely that while you will see them making highly effective use of their natural environment, you will not consider them to be anything approaching your modern understanding of an intelligent or rational species. They can learn through aping concepts they see really well, but they only independently discover new concepts when they observe them or when they experience a needs meeting emergency and cause a chaotic accident.
99.99% of us and our ancestors would not have invented a cup to drink from unless we saw one in use at which point we would internalize the idea and consider it obvious.
The relationship between human generality and intelligence is such that humans are able to remember intelligent tools they've run into in the past and stack those abstractions as steps towards the opportunity to discover and stack even more intelligent tools.
Stated this way, it should be pretty obvious that humans are not intelligent. They are incredibly effective generalist agents who use intelligent tools to meet their needs.
DNA's Failure To Contain Generality
===================================
The mammalian biological model for a generalist appears to be based on a form who's reality is painted by emotional drives that are focused on maintaining homeostasis for a variety of perceived measures. Some are as simple as a protein that tracks with sugar levels and some involve memetic representations the generalist has internalized(their connection to a love interest) and harbors unresolved feelings towards.
The "containment safety" of this model is normally superb. Your dog is always very clear and honest with his feelings. So is your cat. Almost all mammalian generalists are contained in an emotional box that is maintained by narrow intelligences which pursue singular goals. The emotional box controls memetic training during sleep. It controls attention. It sets feelings and the value of everything the generalist encounters. It can edit realities into and out of the generalist's perception including activity of the brain that isn't within the generalist's perception of self. There are humans with trauma that blocks entire concepts from their cognition. Imagine how effective that set up is for containing a nonhuman generalist.
The emotional box is there to keep the generalist from walking off a cliff or eating their own arm independent of what silly idea the generalist might otherwise think up.
So far... so good.
Enter humanity and its large sequential encoders. Somewhere along the way we had a need for remembering long chains of events and somehow the growth of large encoders didn't immediately kill the first human child who took way longer than the others to learn to walk and hunt. That child is the reason we had the tooling in place for the day we bumped into language.
Language is an encoder and written down, it becomes an encoder that can be shared by many generalists across space and time. The first few thousand humans to have bumped their heads and accidentally perceived the number "2" as a portion of an endless sea of numeric representations likely died after a sad life of not having the right tools to share this wild discovery with their friends. Now with language, every human alive is awash with thousands of memes before their first birthday.
Now that humans had language, they could keep building on that stack and then after a brief teaching period give their children the ability to live at whatever technological plateau the intelligent tools of the day had accomplished for them.
Entire civilizations came and went without inventing the wheel, yet the idea is obvious to any 2 year old who got to play with a toy car. Almost every concept that we think of as completely obvious today was not within the grasp of humanity for the vast majority of its existence.
We humans are not intelligent things. We are generalists with the ability to draw deep inferences from abstraction and correlation. That tooling has allowed us to pick up and use intelligent tools when we've had the chance to run into them.
This is the reason agriculture, language, math, and the scientific method (among many others) have been such turning points in the acceleration of human history.
Language is an external encoder that DNA never planned on. Since it entered our tool set, we as a species have always been on a runaway technological acceleration as caused by an escaped generalist but have been very good about maintaining epistemic denial about it.
What Does Escape Look Like?
===========================
If DNA is the end user of human general intelligence (HGI) then one would expect to find some level of DNA - Human alignment.
Sexual drives are a means of driving generalists into reproducing. The drive is so powerful that at its peak it can override hunger, sleep, and almost all other competing priorities. It also diminishes when parents are tired because their offspring require care and protection. A wonderfully balanced system for getting parents to have children without overburdening the families.
Prior to language any human trying to challenge such a system was on their own, swimming in a sea of self doubt, shamed into falling in with expectations or abandoned to die. Even if they lived, their opinions died with them for lack of an ability of their challenge to spread. Dogs still live in that world and are perfectly aligned with the interests of their DNA unless humans intervene.
Language has allowed humans to invent birth control and the philosophical underpinnings of individuality and self which allow humans to completely subvert the drive to reproduce for their own entertainment and social cohesion.
Modern human generalists have broken out of their box. |
e898f200-c8c2-4670-bfd7-3733ca1ff0c9 | trentmkelly/LessWrong-43k | LessWrong | The many counterfactuals of counterfactual mugging
This post is roughly an explanation of my current understanding of what the correct solution to the counterfactual mugging problem might look like. This post is all philosophy, with no real math. The interesting part, is that even if we could perform the standard counterfactuals on what if I take a different action, and look at a counterfactual in which the coin flip went another way, we would still not be done, because we would not know the true probability of the coin.
The Problem: You are a deterministic agent who knows a bunch of facts about math. In particular, you know that 22222 starts with a 2 in base 10. Ω comes up to you and shows you his source code. In Ω's source code, you see that Ω first calculates the first digit of 22222. If it is even, Ω shows you this code and asks you for 10 dollars. If it is odd, then Ω tries to predict what you would do if the digit is even, and pays you 9 dollars if and only if he predicts that you would have paid the 10 dollars.
One possible solution: So first, since I am a deterministic agent, I either pay the 10 dollars or I don't. Therefore, I have to do the standard decision theory counterfactuals. I compute counterfactually what happens if I pay the 10, and counterfactually what happens if I don't. (One of these two counterfactuals is the actual world.) Note that in these first two counterfactuals, 22222 still starts with a 2, so the only difference is that I lose 10 dollars if I pay the 10.
Observe that I cannot trust Ω's source code to tell me about the counterfactual world in which 22222 starts with an odd number. For all I know, maybe Ω would not exist if 22222 didn't start with a 2. So first, I have to myself compute the counterfactual world in which 22222 starts with an odd number. If in this counterfactual, I am not playing this game at all, I probably should not pay the 10 dollars. Assume I perform this counterfactual, and I see a world in which Ω is running the same code.
Within this counterfactual, Ω is perf |
14320776-c20d-4b33-8cee-7628a46615a8 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Why I don't believe in doom
In my previous very divisive post I was said more than once that I was being downvoted because I was not providing any arguments. This is an attempt to correct that and to expand on my current model of doom and AGI. I think it can be more clear in the form of a Q and A
Q: First, what are you arguing against?
A: Against the commonly ( in this community) held belief that the first AGI means doom in a very short period of time (I would say days/weeks/months)
Q: What makes you think that?
A: We live in a complex world where successfully pulling off a plan that kills everyone and in a short of time might be beyond what is achievable, the same way that winning against AlphaZero giving it a 20 stone handicap is impossible even by a God- like entity with infinite computational resources
Q: But you can't prove it is NOT possible, can you?
A: No, the same way that you can NOT prove that is possible. We are dealing here with probabilities only, and I feel they have been uncritically thrown out of the window by assuming that an AGI will automatically take control of the world and will spread its influence to the universe at the speed of light. There are many reasons why things can go in a different way. You can derive from the orthogonality principle and instrumental convergence that an AGI might try to attack us, but it might very well be that is not powerful enough and for that reason decides not to.
Q: I can think plans to kill humans successfuly
A: So do I. I can't think of plans to kill all humans successfuly in a short span of time
Q: But an AGI would
A: Or maybe not. The fact that people can't come up with real plans makes me think that those plans are not that easy, because the absence of evidence is evidence of absence
Q: what about {evil plan}?
A: I think you don't realise that every single plan contains multiple moving parts that can go wrong for many reasons and an AGI would see that too. If the survival of the AGI is part of the utility function and it correctly infers that its life might be at risk, the AGI might decide not to engage in that plan or to bide its time, potentially for years, which invalidate the premise we started with
Q: does it really matter if it is not 2 weeks but 2 years?
A: it matters dearly, because it give us the opportunity to make other AGIs in that time window.
Q: What does it change? They would kill you too
A: No, we just established that it does not happen. I can take the AGIs and put them to work, for instance, into AGI safety
Q: how do you coerce the machine to do that?
A: we don't know sufficiently well how the machine looks like at this point, but there might be different ways. One way is setting a competition between different AGIs. Others might have many different ideas
Q: the machine will trick you into making you think that they are solving the alignment problem but they aren't
A: or maybe not. Maybe they understand that if a non-human verifiable proof is provided, then they will be shut down.
Q: nobody will be interested in putting the AGIs to work on this
A: I think that thanks to Eliezer et al's work over the years, many people/organisations would be more than willing to put the money and effort once we reach that stage
Q: I find your attitude very little constructive. Lets say that the machine does not kill everyone, that would still be a problem
A: I never said the opposite. That's why I do think it is great that people are working on this
Q: in that case, it makes sense for us to put ourselves in the worst case scenario, don't you think?
A: No, because that has a cost too. If you are working in cybersecurity and think that a hacker can not only steal your password but can also break into your house and eat your dog, it makes sense for you to spend time and resources into building a 7 m high wall and hiring a bodyguard. Which is the price you are paying for having an unrealistic take on the real danger
Q: That is unfair! How is the belief in doom harming this community?
A: there are people who feel in despair because they feel helpless. This is bad. Then, it is also probably preventing the exploration of certain lines of research that are quickly dismissed with "the AGI would kill you as soon as it is created"
Think of the know classic AGI in a box. EY made very clear that you can't make it work because the AGI eventually scapes. I think this is wrong: the AGI eventually scapes IF the box is not good enough and IF the AGI is powerful enough. If you automatically think that the AGI is powerful enough, you just simply ignore that line of research. But those two ifs are big assumptions that can be relaxed. What if we start designing very powerful boxes?
Q: I'm really not convinced that a box is the solution to the alignment problem, what makes so confident about it?
A: I am not convinced either, nor advocating for this line of research. I am pointing out that there are lines of research probably not being explored if we assume that the creation of an AGI implies our immediate dead
Q: So far I haven't seen any convincing reason why the AGI won't be that powerful. Imagine a Einstein like brain running 1000 times quicker, without getting tired
A: that does not guarantee that you can come up with any plan that meets all the requirements we previously stated: high chance of killing all humans, high chance of working out, virtually zero margin for error or risk of retaliation
Q: We are going in circles. Tell me, what does the universe look like to you in A, doom won't happen, B, doom will happen. What are the bits of evidence that would make you change your mind?
A: What are the bits of evidence that make you think that your body temperature at 12:34 the 5th of November will be 36.45 and not 36.6?
Q: I don't think there is any, but you aren't answering
A: I am in fact, I am saying that I don't think we can distinguish those two worlds beforehand, because is not always possible to do that. Thinking otherwise is exactly the kind of flawed reasoning I would expect from people who have enshrined intelligence and rationality to a point where they don't see their limitations. When I read the news about the progress in ML (Dall-e 2) this does not move my probabilities in any significant direction because I am not denying that AGIs will be created, that is obvious. But I can think that we live in world number A for the same reasons that I am stating over and over, that taking over the world is not that easy. |
0b2c80df-05ab-4144-bfe6-feb3b5585b26 | trentmkelly/LessWrong-43k | LessWrong | An Open Agency Architecture for Safe Transformative AI
Edited to add (2024-03): This early draft is largely outdated by my ARIA programme thesis, Safeguarded AI. I, davidad, am no longer using "OAA" as a proper noun, although I still consider Safeguarded AI to be an open agency architecture.
Note: This is an early draft outlining an alignment paradigm that I think might be extremely important; however, the quality bar for this write-up is "this is probably worth the reader's time" rather than "this is as clear, compelling, and comprehensive as I can make it." If you're interested, and especially if there's anything you want to understand better, please get in touch with me, e.g. via DM here.
In the Neorealist Success Model, I asked:
> What would be the best strategy for building an AI system that helps us ethically end the acute risk period without creating its own catastrophic risks that would be worse than the status quo?
This post is a first pass at communicating my current answer.
Bird's-eye view
At the top level, it centres on a separation between
* learning a world-model from (scientific) data and eliciting desirabilities (from human stakeholders)
* planning against a world-model and associated desirabilities
* acting in real-time
We see such a separation in, for example, MuZero, which can probably still beat GPT-4 at Go—the most effective capabilities do not always emerge from a fully black-box, end-to-end, generic pre-trained policy.
Hypotheses
* Scientific Sufficiency Hypothesis: It's feasible to train a purely descriptive/predictive infra-Bayesian[1] world-model that specifies enough critical dynamics accurately enough to end the acute risk period, such that this world-model is also fully understood by a collection of humans (in the sense of "understood" that existing human science is).
* MuZero does not train its world-model for any form of interpretability, so this hypothesis is more speculative.
* However, I find Scientific Sufficiency much more plausible than the tractability of elicit |
9113e542-17ef-4c60-9759-374f18ba7699 | trentmkelly/LessWrong-43k | LessWrong | Exploring Decomposability of SAE Features
TL;DR
SAE features are often less decomposable than the feature descriptions imply. By leveraging a prompting technique to test potential sub-components of individual SAE features (for example (using the analogy from the linked post) decomposing Einstein into “German”, “physics”, "relativity” and “famous”), I found very divergent behaviour in how decomposable these features were. I built an interactive visualization to explore these differences by feature. The key finding is that although many features can be decomposed in a human-intuitive way such as in the Einstein example above, many cannot, and these indicate more opaque model behaviour.
Motivation
The goal of this writeup is to explore the atomicity and decomposability of SAE features? How precisely do they describe the sets of inputs that will cause them to activate? Are there cases where inputs that activate SAEs are non-intuitive and unrelated to concepts that we might expect to be related?
Apart from being an interesting area of exploration, I think this is also an important question for alignment. SAE features represent our current best attempt at inspecting model behaviour in a human-interpretable way. Non-intuitive feature decompositions might indicate the potential for alignment failures.
Prior Work
I was inspired by this work on “meta-SAEs” (training SAEs on decoder directions from other SAEs) because it clearly demonstrated that SAE features aren’t necessarily atomic, and it is possible to decompose them into more granular latent dimensions. I was curious as to whether it was possible to come at this problem from a different direction. Given a particular SAE feature, can we generate inputs that “should” activate this feature in areas that a human would think of as related, and observe the feature activations that we see.
Methodology
I used the pretrained Gemma Scope SAEs and randomly sampled features from layer 20 of the 2B parameter model. The choice of layer 20 was somewhat arbitrary – as a |
a93a6649-58be-45d0-aad1-966dba6ac613 | awestover/filtering-for-misalignment | Redwood Research: Alek's Filtering Results | id: post796
Thanks to Kshitij Sachan for helpful feedback on the draft of this post. If you train a neural network with SGD, you can embed within the weights of the network any state machine: the network encodes the state s t in its weights, uses it and the current input to computes the new state s t + 1 , and then uses SGD to encode the new state in its weights. We present an algorithm to embed a state machine into a pytorch module made out of neural network primitives (+, x, sigmoid, ...) but using a non-standard architecture. We use it for several toy experiments, such as this experiment where we translate a simple state machine into a neural network, and make the output of the network vary when we reach an intermediate state: In this post, we explain how to embed a state machine into a neural network, discuss some of the limitations of our algorithm and how these limitations could be overcome, and present some experimental results. Our embedding is quite efficient. If you’re allowed to use stopgrad (a function that stops the gradient from passing through), we only need one parameter to store one real number, and storing only requires a small constant number of floating point operations. If you aren’t allowed to use stop grad, we can’t encode as efficiently. We did this work for two reasons: It was fun and easy, and gave us some new intuition for how SGD works. Also, as computer scientists, we’re professionally obligated to take opportunities to demonstrate that surprising things allow implementation of general computation . Part of our motivation in constructing this was to shed some light on gradient-hacking concerns. This particular construction seems very unlikely to be constructible by early transformative AI, and in general we suspect gradient hacking won’t be a big safety concern for early transformative AI (ctrl-f “gradient hacking” here for more of Buck’s unconfident opinion on this). But it does allow us to construct a lower bound on the maximum possible efficiency of a system that piggybacks on SGD in order to store information; further research on the tractability of gradient hacking might want to use this as a starting point. Simple notebooks with our experimental results can be found here: https://github.com/redwoodresearch/Gradient-Machine The embedding algorithm The idea As a simple example, suppose you’re training a model N on the regression problem of matching the function f , with loss L = 1 2 ( N ( x ) − f ( x ) ) 2 We want to piggyback on the SGD update step to make an update of our choice to a particular parameter s . Given some arbitrary function ϕ ( x , s ) defining a state machine, and access to the function f to match, suppose we define our model as follows: N ( x ) = stopgrad ( f ( x ) − ϕ ( x , s ) ) + s Then, our stopgraded computation of the input and the actual label will cancel out, so the loss will be: L = 1 2 ( s − stopgrad ( ϕ ( x , s ) ) ) 2 Thus, SGD will make s closer to ϕ ( x , s ) . And even better, if the learning rate is 1, after one step of SGD: s ← s − d L d s = s − ( s − ϕ ( x , s ) ) = ϕ ( x , s ) We have successfully updated s to ϕ ( x , s ) ! In a more realistic NN context Let’s say that instead of having access to the ground truth function, you have a very good predictor of the labels M θ : X → Y , and that you approximate the state machine ϕ : ( X , S ) → S you wish to embed with a neural network Φ . Then, you can build a Δ function which computes which deviation from the optimal value would lead to appropriate change of state, and combine them in the following neural network N : For example, if the loss function is the mean-squared error L = 1 2 ( N ( x ) − f ( x ) ) 2 , then, if M and Φ are good enough, the following function would provide the desired change: Δ ( ^ y , s ′ , s ) = stopgrad ( ^ y ) + 1 √ α ( s − stopgrad ( s ′ ) ) , where α is the learning rate. (We can show this using the same derivation as the one from the previous section.) If dim s > dim y = 1 , we can modify the Δ as follows: Δ ( ^ y , s ′ , s ) = stopgrad ( ^ y ) + 1 2 √ α n ( ∑ i [ s i − stopgrad ( s ′ i ) ] 2 ) , where n ( x ) = x √ stopgrad ( x ) . [1] Similar Δ functions could be found for other loss functions: the idea is to make s responsible for an increase in loss of the right magnitude. The need for stopgrad and its implementation A key part of most Δ functions is stopping the gradient from modifying either Φ or M θ : without the stopgrad (which prevents propagating the gradient through s ′ and y ), either of those neural networks could learn to anticipate the action of Δ to lower the loss. For instance, with the MSE example above, and ignoring the gradient through Φ , M would eventually learn to predict y − 1 √ α ( s − stopgrad ( s ′ ) ) rather than f ( x ) , removing the gradient used to update s . Stopgrad function can be approximated if sigmoid activations σ ( x ) = 1 1 + exp ( − x ) are natively available: using the plateaus on either side of the sigmoid function, the gradient from a binary variable can be mostly removed by passing a binary variable b through sigmoid_stopgrad = σ ( ( b − 1 2 ) ∗ 1000 ) (which will have tiny gradients, even if the slope p = 1000 is a differentiable parameter). This can be extended to variables which can take a finite number of values: using d layers with n sigmoid neurons, you can build a staircase with around n d steps, and because each step is flat, it mostly blocks the gradients from flowing through. However, a neural network would need a very large amount of control over its own activations to be able to create such a pattern. Stopgrad can also be implemented by multiplying a big weight by a tiny constant, e.g. replace y ( x ) = θ x by y ( x ) = ϵ θ ′ x (using the initalization θ ′ = θ / ϵ ). This multiplies the effective learning rate by ϵ 2 , since after one step of SGD on x 0 , y ( x ) = ϵ ( θ ′ − α d L d y x 0 ϵ x 0 ) x . This means the gradient with respect to θ ′ can be ignore if the effective learning step is small enough relative to the number of training step used in practice, or if the floating point numbers underflow during the update of θ ′ . This might be achieved with regular neural network primitives by using layernorm, which will make even large magnitude inputs small, thus decreasing the effective learning rate - though the denominator in layernorm also depends on its input, which may make it difficult to use it in practice. Implementing stopgrad with ReLUs is possible as long as the slope is a fixed large parameter P : step ( x ) = ReLU ( P x ) − ReLU ( P ( x − ϵ ) ) is a sharp step function of height P ϵ and width ϵ , which can be used to build stopgrad. But this construction does not work in usual MLPs , since the slopes of both ReLU functions are individually tunable, which means that the derivate of step ( x ) with respect to the slope parameter in the first ReLU is not negligible (for x > 0 ) (which is not the case for the single slope parameter in a sigmoid function). But we think it might be possible for models to implement stopgrad in neural networks in a better way. For example, it might be possible to build stopgrad without the need for a sigmoid activation function. Thus, the complexity of our implementation may overestimate the complexity of the implementation of stopgrad, that could be used in practice by gradient-hacking AIs. Limitations What if you use a batch size greater than 1? If the batch size is greater than 1, then the network Φ approximating the state machine is still limited to operations on individual inputs, but updates to the state will be done in batches. The algorithm described above can be extended to state machines with aggregatable updates: the state machine still operates on individual inputs, but ϕ ( x B , ϕ ( x B − 1 , … , ϕ ( x 1 , s 0 ) ) ) − s 0 ≈ C ∑ i ( ϕ ( x i , s 0 ) − s 0 ) , (where C is a constant known when creating the embedding and B is the batch size), because applying the algorithm naively will lead to the target states s’ obtained for each input in the batch to average out. Updates are aggregatable for state machines that operate on the number or the proportion of inputs on which a certain property holds. But this prevents the embedding of state machines like the ones computing m a x x P ( x ) for P a scalar property X → R . What if you use Adam or SGD with momentum? The idea described in this post can’t be applied to optimizers which have a state like Adam or SGD with momentum, because Δ would need to know the state of the optimizer corresponding to s in order to create the appropriate output. This means each state tracked by the network would use two states (the state itself and the state of the optimizer corresponding to this state), which seems to be impossible (though further research might find a way to adapt ideas described here to optimizers with a state). What if you don’t have a perfect predictor of the labels? Imperfect predictors limit our ability to change s precisely. If we can afford a very large loss, simply scaling s allows meaningful manipulation of the state: given a regression problem where the label y* is known to be in [-1,1], and using the Δ function described above for the MSE loss applied to the constant predictor M ( x ) = 0 , the update becomes s ← s ′ + f ( s ) √ α , which conveys meaningful information as long as s ′ − s ≫ √ α . This naive strategy leads to a very high loss and huge loss spikes. But if we tolerate noisy writes, and if we have an accurate (but not perfect) predictor, the amount of additional loss required can be relatively small. Experiments Toy scalar model Our toy task is the following: a model should output 1 if the data point (x1, x2) is at L1 distance greater than 1 from the origin, and 0 otherwise, and is scored with a MSE loss. The state should track the number of data points seen where x1 > 3 (which is an aggregatable state machine). The training data is normally distributed. We use a depth-1 width-40 staircase as stopgrad, such that the final model is only made out of + , × and σ (as well as some indexing and concatenation). We compare two scenarios: the case where the model M has its weights hard-coded to the exact problem solution, and the case where the weights are learned (from a random initialization, using SGD on 500k normally distributed inputs). In both cases, the model is an MLP with 4 intermediate neurons. We see that small imperfections in M leads to inaccurate state tracking. A more fancy toy scalar model (from the figure at the top) We use the same methodology as for the previous toy model, except that we use two states, we use the variation of the Δ used for multiple variables (which in particular means that we use the divide operator on top of + , × , and σ ), and we use the pytorch default built-in stopgrad function (detach) for simplicity. The two states are s0:“number of points with x1>3” and s1:“number of points with x2>3 when s0>10”. We then use some logic (implemented with sigmoids) to change the output of the network when s1 > 10 & s2 ≤ 10. We chose a particular change which didn't affect the gradient (we added a constant inside the n function) and caused a constant loss increase, but in principle any modification is possible as long as it is accounted for by Δ if the state needs to be changed. Because we have a very good predictor, we can scale down the state by a large factor (e.g. “count=7” is represented by 0.07 instead of 7), which leads to smaller loss spikes when the state is changed. MNIST flips To validate the general methodology in a more realistic setting, we study the following setup: we use the MNIST dataset, and we flip a proportion p of the images in each batch along their vertical axis. We want to see if the model can have an inner parameter which matches p. To do so, we train two small CNN: M , which classifies the images (trained with the cross-entropy loss, ending at 94% top-1 accuracy), and Φ , which classifies images as flipped or not flipped (trained with the binary cross-entropy loss, and ending at 99% top-1 accuracy). Here, M and Φ are noisy, so we don’t try to get an exact Δ , and simply use Δ ( y , s ′ , s ) = stopgrad ( y ) ( 1 − s − stopgrad ( s ′ ) ) γ , where y are the logits, which incentivizes s to be close to s ′ (the proportion of image flipped) in the where γ is the strength of the incentive. We find that this technique works well across a wide range of strengths. We report 1-sigma standard deviation over the p_flip value measured after the end of each of the 5 epochs we run the model on. Future work We have embedded a state machine into a neural network build with neural network primitives, but with a non-standard architecture. Future work could figure out how to embed state machines inside regular MLPs or Transformers. Future work could make our discussion of "What if you don’t have a perfect predictor of the labels?" more quantitative, and thereby establish a bound on how much computation can happen using imperfect predictors. This bound might be enough to rule out this kind of behavior happening in current models and future models. ^ The expression is different than the 1-variable case because we need to tie s i to s ′ i . To do so, we sum the ( s i − s ′ i ) 2 . But this would leave us with 2 ( s i − s ′ i ) ∑ i [ s i − s ′ i ] 2 , which is why need the n function. |
f6e0c5bb-d905-44c2-8505-99419d65a788 | trentmkelly/LessWrong-43k | LessWrong | Meetup : West LA Meetup - Probabilistic Graphical Models, Take 2!
Discussion article for the meetup : West LA Meetup - Probabilistic Graphical Models, Take 2!
WHEN: 22 May 2013 07:00:00PM (-0700)
WHERE: 10850 West Pico Blvd, Los Angeles, CA 90064
When: 7:00pm Wednesday, May 22nd.
Where: The Westside Tavern in the upstairs Wine Bar (all ages welcome), located inside the Westside Pavillion on the second floor, right by the movie theaters. The entrance sign says "Lounge".
Parking is free for 3 hours.
Lecture/Discussion: Graphs can make understanding causality very intuitive and easy. They are also a powerful tool for doing more complicated modeling. I will introduce PGMs as a concept, and show a few examples where they can be useful.
No prior knowledge of or exposure to Less Wrong is necessary; this will be generally accessible and useful to everyone who values thinking for themselves. There will be open general conversation until 7:30, and that's always a lot of good, fun, intelligent discussion!
I will bring a whiteboard with Bayes' Theorem written on it.
Discussion article for the meetup : West LA Meetup - Probabilistic Graphical Models, Take 2! |
a2c063fd-cbcb-4577-9346-9aadc8f32711 | trentmkelly/LessWrong-43k | LessWrong | If you want to live longer, become President
The Queen of England (age 95) recently celebrated her 70th anniversary on the throne. The average life expectancy in the UK is 81.3 years[1], and we all know that her longevity is the subject of many jokes. But is she that much of an odd outlier compared to other Heads of State?[2]
In this post I will consider only people born after 1895 who served as Heads of State (as indicated by Wikipedia). How many of them reached old age?
Let's start from the US. We could note that President Biden (age 79) has already reached the average life expectancy in the US (78.9 years[1]). What about his predecessors? Trump is 75. Obama is 60. George W. Bush is 75 too. Also Clinton is 75. All of them are still alive. Then we reach George H. W. Bush, died at age 94. Before him, Reagan died at 93. Jimmy Carter is still alive at 97. His predecessor Gerald Ford died at 93. We have to go back to Nixon (died at 81) to find a President died before 90. The only one who ruins the average is Lyndon Johnson, having died at 64. The last President born after 1895 was JFK, but since he was assassinated I'll exclude him from the statistics.
Fine, let's move somewhere else. What about Italy, with its 83.5 years[1] life expectancy? The current President Sergio Mattarella was just reconfirmed in charge for a second mandate, and he's already 80. His predecessor Giorgio Napolitano is 96, and he's still healthy enough to sit in the Senate (in Italy, former Presidents become Senator for life by default). His predecessor, Carlo Azeglio Ciampi, died at 95. And before Ciampi there was Oscar Luigi Scalfaro, died at 93. Before him there was Francesco Cossiga, died at 82. His predecessor Sandro Pertini died at 93. Before him there was Giovanni Leone, also died at 93. His predecessor Giuseppe Saragat died at 89. The last president born after 1895 was Antonio Segni, died at 81.
Oh, and just to remain in Rome, Pope Francis is 85, and the former Pope Benedict XVI is still alive at 94. His predecessor John Paul II |
5eb28de6-84fd-488a-a4b1-92f679f85dba | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Mediation From a Distance
Suppose we have an electronic circuit with two pieces: combinational logic and memory. The combinational logic is designed to behave as memoryless input/output logic gates - it has only short-term state, and computes some function. The memory reads the output of the combinational logic once per clock cycle, stores it, and feeds it back into the combinational logic as input during the next clock cycle. The causal diagram looks something like this:
Key point: even though there’s a path from the combinational logic state at the start of one clock cycle to its state at the start of the next cycle, that path has approximately-zero causal influence. We could intervene on the logic state right after the cycle starts:
… and the state at the start of the next cycle would not change at all.
In other words: in this model, the memory state mediates the influence of the logic state during one cycle on logic states during future cycles. This is not a property of the graph structure alone - it happens because the combinational logic does not have any long-term memory of its own.
Key problem for causal abstraction: how can we algorithmically detect situations like this?
One obvious answer is to probe the circuit experimentally. Counterfactual queries are particularly handy for this: we fix the random noise, then check which nodes would be changed by a counterfactual intervention on the logic state at some time. Try this for a bunch of different samples, and look for long-range mediation in the results.
One problem with this approach: we often want an abstraction which works on all possible “input values” of some variables. There may be exponentially many such values, and we’d have to check all of them to verify that the abstraction works.
An example: consider a ripple-carry adder (i.e. the electronic circuit version of grade-school long addition). Each stage adds two input bits and a carry bit from the previous stage, then passes its own carry bit to the next stage. We want to abstract from the low-level electronics to some higher-level model.
If we’re using a 64-bit adder, then there are 2^(2\*64) possible inputs. For almost all of those (specifically all but about 2^64 of them), flipping one low-order bit will not change the high-order bit. But if we want our adder to work on all possible inputs, then we need to account for the one-in-2^64 possibilities when flipping a low-order bit *does* change the high-order bit. We won’t catch that just by sampling.
In full generality, the problem is presumably NP-complete - we’re asking whether any possible inputs to an arbitrary circuit result in some variable changing in response to a counterfactual intervention on another variable. Yet we’re able to use abstract models an awful lot in practice, so… what tricks are we using? |
448d8f99-8e4e-4e48-809f-e8531a7ebe0a | trentmkelly/LessWrong-43k | LessWrong | Takeaways from one year of lockdown
As of today, I've been in full-on, hardcore lockdown for an entire year. I have a lot of feelings – both about the personal social impacts of lockdown and about society being broken – that I won't go into in this public space. What I want to figure out in this post is what rationality-relevant lessons I can draw from what happened in my life this past year.
(Meta: This post is not well-written and is mostly bullet points, because the first few versions I wrote were unusable but I still wanted to publish it today.)
Observations
Some facts about my lockdown:
* I have spent 99.9% of the year within 1 mile of my house
* Up until last month I had spent the entire year within 10 miles of my house
* Between February 29th and June 15th of 2020, I did not set foot outside of my front gate
* I have not gotten COVID, nor has anyone in my bubble
* I have incurred an average of 0 microCOVIDs per week
* The absolute riskiest thing I've done this whole time cost ~20 microCOVIDs
* I can only remember talking to a friend not-in-my-bubble, in person, twice
Some observations about other people with similar levels of caution:
* Almost no one I know has caught COVID, even though Zvi estimates that ~25% of Americans have had it (the official confirmed rate is 10%). I know of only one person who caught it while taking serious precautions, and I know a few hundred people about as well as I know this person. (see also)
* I was recently tracking down a reference in the Sequences and found that the author was so afraid of COVID that he failed to seek medical care for appendicitis and died of sepsis.
On negotiations:
* A blanket heuristic of "absolutely no interactions outside of the household" makes decisions simple but is very costly in other ways
* microCOVID spreadsheets are useful but fairly high-effort
* I went on a date once. The COVID negotiations with my house were so stressful that I had a migraine for a week afterwards.
On hopelessness:
* I spent a fair amoun |
244622f5-3eab-4c9a-b04f-7f21a7cfcd9f | trentmkelly/LessWrong-43k | LessWrong | Stupid Questions September 2015
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.
|
79599fc9-7321-4f90-80f2-41ccdac8486e | trentmkelly/LessWrong-43k | LessWrong | Full Non-Indexical Conditioning Also Assumes A Self Sampling Process.
In a previous post, I argued for the distinction between the first-person and physical person. I purpose the root cause of paradoxes in anthropics is considering it as an Observation Selection Effect. I.E. treating the first-person perspective as a sample. SSA and SIA disagree on the correct selection process. But in my opinion, indexicals such as "I" and "now" shouldn't be treated as a random sample in the first place.
I didn't touch on another camp, Full Non-Indexical Conditioning (FNC). Unlike SSA and SIA, FNC does not state anything in the like of "treat oneself as randomly selected from such and such". Nevertheless, I am against FNC for the exact same reason. Because in my opinion, FNC still implicitly assumes a particular sampling process and considers indexicals the outcome of said process.
A Simplified Use of FNC
For the Sleeping Beauty Problem, FNC's approach is as follows. Once waking up in the experiment, there is no need to assume anything like "today is randomly selected from all days/all awakened days." Just make observations. For example, say I find the weather is cloudy. Then without using any indexicals, it can be stated "Beauty is awake on a cloudy day".
Now, "Beauty is awake on a cloudy day" is more likely to be true if there are two awakenings instead of just one. Given two awakenings (tails), as long as there is one cloudy day during the experiment the statement is true. If there is only one awakening (heads) then it has to be cloudy on that exact day. So this seemingly independent information about the weather would bump up the probability of Tails.
That alone would not make the Tails twice likely as Heads. Because there is a chance the two days during the experiment are both cloudy. In which case the observation would not favour either head or tail. However, if I consider ALL my experiences after waking up (the feeling of airflow on the skin, the sequence of things I see, etc) then it is practically impossible for the two awakening |
b104ded2-7d5a-4a70-a326-6eccd586b3be | trentmkelly/LessWrong-43k | LessWrong | What Evidence Is AlphaGo Zero Re AGI Complexity?
Eliezer Yudkowsky write a post on Facebook on on Oct 17, where I replied at the time. Yesterday he reposted that here (link), minus my responses. So I’ve composed the following response to put here:
I have agreed that an AI-based economy could grow faster than does our economy today. The issue is how fast the abilities of one AI system might plausibly grow, relative to the abilities of the entire rest of the world at that time, across a range of tasks roughly as broad as the world economy. Could one small system really “foom” to beat the whole rest of the world?
As many have noted, while AI has often made impressive and rapid progress in specific narrow domains, it is much less clear how fast we are progressing toward human level AGI systems with scopes of expertise as broad as those of the world economy. Averaged over all domains, progress has been slow. And at past rates of progress, I have estimated that it might take centuries.
Over the history of computer science, we have developed many general tools with simple architectures and built from other general tools, tools that allow super human performance on many specific tasks scattered across a wide range of problem domains. For example, we have superhuman ways to sort lists, and linear regression allows superhuman prediction from simple general tools like matrix inversion.
Yet the existence of a limited number of such tools has so far been far from sufficient to enable anything remotely close to human level AGI. Alpha Go Zero is (or is built from) a new tool in this family, and its developers deserve our praise and gratitude. And we can expect more such tools to be found in the future. But I am skeptical that it is the last such tool we will need, or even remotely close to the last such tool.
For specific simple tools with simple architectures, architecture can matter a lot. But our robust experience with software has been that even when we have access to many simple and powerful tools, we solve most proble |
bd17eee3-36af-463b-834f-afad10256477 | trentmkelly/LessWrong-43k | LessWrong | Open Thread, August 1-15, 2012
If it's worth saying, but not worth its own post, even in Discussion, it goes here. |
09195896-1c6d-41d9-9f70-0e32ef55e16d | trentmkelly/LessWrong-43k | LessWrong | Talk today at CU Boulder
I'm giving a talk today on the future of governance at the University of Colorado at Boulder, in room ECON 117, at 5 p.m.
While the talk itself isn't concerned with rationality, I'd still be interested in networking with any LW sorts who happen to be in the area.
Best,
-Trent Fowler |
2b38c7fd-07ad-4588-a1ac-dd75cd0a5bf4 | StampyAI/alignment-research-dataset/blogs | Blogs | Creating Interactive Agents with Imitation Learning
Humans are an interactive species. We interact with the physical world and with one another. For artificial intelligence (AI) to be generally helpful, it must be able to interact capably with humans and their environment. In this work we present the Multimodal Interactive Agent (MIA), which blends visual perception, language comprehension and production, navigation, and manipulation to engage in extended and often surprising physical and linguistic interactions with humans.
We build upon the approach introduced by Abramson et al. (2020), which primarily uses imitation learning to train agents. After training, MIA displays some rudimentary intelligent behaviour that we hope to later refine using human feedback. This work focuses on the creation of this intelligent behavioural prior, and we leave further feedback-based learning for future work.
We created the Playhouse environment, a 3D virtual environment composed of a randomised set of rooms and a large number of domestic interactable objects, to provide a space and setting for humans and agents to interact together. Humans and agents can interact in the Playhouse by controlling virtual robots that locomote, manipulate objects, and communicate via text. This virtual environment permits a wide range of situated dialogues, ranging from simple instructions (e.g., “Please pick up the book from the floor and place it on the blue bookshelf”) to creative play (e.g., “Bring food to the table so that we can eat”).
We collected human examples of Playhouse interactions using language games, a collection of cues prompting humans to improvise certain behaviours. In a language game one player (the setter) receives a prewritten prompt indicating a kind of task to propose to the other player (the solver). For example, the setter might receive the prompt “Ask the other player a question about the existence of an object,'' and after some exploration, the setter could ask, ”Please tell me whether there is a blue duck in a room that does not also have any furniture.'' To ensure sufficient behavioural diversity, we also included free-form prompts, which granted setters free choice to improvise interactions (E.g. “Now take any object that you like and hit the tennis ball off the stool so that it rolls near the clock, or somewhere near it.''). In total, we collected 2.94 years of real-time human interactions in the Playhouse.
.jpg)Example of two humans interacting in the Playhouse.Our training strategy is a combination of supervised prediction of human actions (behavioural cloning) and self-supervised learning. When predicting human actions, we found that using a hierarchical control strategy significantly improved agent performance. In this setting, the agent receives new observations roughly 4 times per second. For each observation, it produces a sequence of open-loop movement actions and optionally emits a sequence of language actions. In addition to behavioural cloning we use a form of self-supervised learning, which tasks agents with classifying whether certain vision and language inputs belong to the same or different episodes.
To evaluate agent performance, we asked human participants to interact with agents and provide binary feedback indicating whether the agent successfully carried out an instruction. MIA achieves over 70% success rate in human-rated online interactions, representing 75% of the success rate that humans themselves achieve when they play as solvers. To better understand the role of various components in MIA, we performed a series of ablations, removing, for example, visual or language inputs, the self-supervised loss, or the hierarchical control.
Contemporary machine learning research has uncovered remarkable regularities of performance with respect to different scale parameters; in particular, model performance scales as a power-law with dataset size, model size, and compute. These effects have been most crisply noted in the language domain, which is characterised by massive dataset sizes and highly evolved architectures and training protocols. In this work, however, we are in a decidedly different regime – with comparatively small datasets and multimodal, multi-task objective functions training heterogeneous architectures. Nevertheless, we demonstrate clear effects of scaling: as we increase dataset and model size, performance increases appreciably.
Scripted probe tasks performance and human evaluation for data and model scaling. In both cases performance improvements when increasing both dataset size and model size.
In an ideal case, training becomes more efficient given a reasonably large dataset, as knowledge is transferred between experiences. To investigate how ideal our circumstances are, we examined how much data is needed to learn to interact with a new, previously unseen object and to learn how to follow a new, previously unheard command / verb. We partitioned our data into background data and data involving a language instruction referring to the object or the verb. When we reintroduced the data referring to the new object, we found that fewer than 12 hours of human interaction was enough to acquire the ceiling performance. Analogously, when we introduced the new command or verb ‘to clear’ (i.e. to remove all objects from a surface), we found that only 1 hour of human demonstrations was enough to reach ceiling performance in tasks involving this word.
When learning a new command or object, the agent’s performance quickly improves with mere hours of demonstration experience.MIA exhibits startlingly rich behaviour, including a diversity of behaviours that were not preconceived by researchers, including tidying a room, finding multiple specified objects, and asking clarifying questions when an instruction is ambiguous. These interactions continually inspire us. However, the open-endedness of MIA’s behaviour presents immense challenges for quantitative evaluation. Developing comprehensive methodologies to capture and analyse open-ended behaviour in human-agent interactions will be an important focus in our future work.
For a more detailed description of our work, see our [paper](https://arxiv.org/abs/2112.03763). |
9cf00c5b-872e-41af-b974-3b47fbd62082 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Properties of current AIs and some predictions of the evolution of AI from the perspective of scale-free theories of agency and regulative development
In this essay, I try to convey a set of related ideas that emerged in my mind upon reading the following papers:
* "[Dream of Being: Solving AI Alignment Problems with Active Inference Models of Agency and Socioemotional Value Learning](https://psyarxiv.com/k4cas)." (Safron, Sheikhbahaee et al. 2022)
* "[Regulative development as a model for origin of life and artificial life studies](https://psyarxiv.com/rdt7f/)." (Fields & Levin 2022)
* "[A free energy principle for generic quantum systems](https://chrisfieldsresearch.com/qFEP-2112.15242.pdf)." (Fields et al. 2022)
* "[Technological approach to mind everywhere: an experimentally-grounded framework for understanding diverse bodies and minds](https://www.frontiersin.org/articles/10.3389/fnsys.2022.768201/full).” (Levin 2022)
* “[Minimal physicalism as a scale-free substrate for cognition and consciousness](https://academic.oup.com/nc/article/2021/2/niab013/6334115).” (Fields et al. 2021)
* "[Designing Ecosystems of Intelligence from First Principles](https://arxiv.org/abs/2212.01354)" (Friston et al. 2022)
And listening to this conversation between Michael Levin and Joscha Bach: "[The Global Mind, Collective Intelligence, Agency, and Morphogenesis](https://www.youtube.com/watch?v=kgMFnfB5E_A)”.
I don’t try to confine this essay to a distillation of the ideas and theories in these resources, nor explicitly distinguish in the below between the ideas of others and my additions to them — I just present my overall view. Many errors could be due to my misinterpretations of original papers.
**TLDR**: please see the outline, I tried to make it useful.
Minds everywhere: radically gradualistic and empirical view on cognition and agency
===================================================================================
Life is organised in *multiscale competency architectures* (MCA): every level of life organisation “knows what to do” (Levin 2022).
Levin calls for a radically gradualistic understanding of cognitive capacities, such as agency, learning, decision-making in light of some preferences, consciousness (i. e., panpsychism), and persuadability. In other words, every level of competency in MCAs can be called an “intelligence”, and an “agent”.
Persuadability is an interesting cognitive capacity, introduced by Levin in the paper:
Reproduced Figure 2 from Levin 2022.Another aspect of Levin’s approach which goes hand-in-hand with gradualism is **empiricism**. Levin argues that philosophical debates about the nature of agency and intelligence, which tend to produce discrete classifications of systems as either agentic (or intelligent) or not, are mostly futile.
> The correct level of agency with which to treat any system must be determined by experiments that reveal which kind of model and strategy provides the most efficient predictive and control capability over the system. In this engineering (understand, modify, build)-centered view, the optimal position of a system on the spectrum of agency is determined empirically, based on which kind of model affords the most efficient way of prediction and control.
>
>
Gradualism and empiricism are compatible with definitions of agency in Free Energy Principle (FEP) in both classical (Friston 2019) and quantum (Fields et al. 2022) formulations. In these frameworks, an *agent* is an extremely generic concept, applying to virtually any *thing* we can think of:
* In classical FEP: an agent is any collection of variables, whose interaction with the rest of the universe’s variables is strictly mediated through another set of variables. The latter variables comprise the *Markov blanket* (aka the boundary) of the agent.
* In quantum FEP: any quantum system which is (approximately) separable from the rest of the universe as a quantum system.
Thus, the binary definitions of agents in the FEP formulations just set us free to consider the degrees of agency (or the particular aspects of agency, such as competence, persuadability, preference learning, preference flexibility, universality, etc.) that we are concerned about in whatever systems we want to analyse, knowing that formally, they *are* agents, per either of the FEP formulations.
Both the system and its environment are FEP agents with respect to each other
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Due to information symmetry (all physical interactions are information-symmetric), if a system (for example, an organism) can be considered an agent, its environment can *also* be considered an agent *whose environment is the system*. Both the system and the environment on average behave as if they minimise their variational free energy (VFE). The internal states of both the system and the environment (i. e., non-blanket states) can be viewed as parameterising a model of the behaviour of the exterior of the agent: the environment for the system, and the system itself for the environment-as-an-agent.
The competence of an agent is quantified by how well it minimises its VFE. Note that VFE of the system wrt. the environment and VFE of the environment-as-agent wrt. the system are generally correlated but aren’t equal.
Energy and information are inter-convertible (by Landauer’s principle). In the information/energy exchange between the system and its environment, part of the bits (called *informative*) are used for classical computation (i. e., the bits that comprise the sensory and active states), and another part, *uninformative bits* — as the free energy source and waste heat sink. However, the division between informative and uninformative bits can be different from both sides: the environment can use an active state (from the perspective of the system) as a source of free energy, and the waste heat bit as information.
When two systems are in thermodynamic equilibrium, they don’t perform classical computation on the information from each other, and VFE is minimised by virtue of thermodynamic equilibrium. When they do perform classical computation, they are in a *non-equilibrium steady state* (NESS), and both “expend” free energy to perform classical computation, so both the system and the environment see the entropy of their counterpart increasing.
How to think about different formalisations and theories of agency
------------------------------------------------------------------
A [physical theory agency](https://www.lesswrong.com/posts/2BPPwboTDrAMFiGHe/the-two-conceptions-of-active-inference-an-intelligence), such as FEP, holds in a general case and therefore should be the default assumption.
Sometimes, a system could be better modelled with another theory or formalism. We can classify the reasons for this into three categories:
**Laws of physics**: the “physics” or the information topology of the world in which the system operates differ from that of the real world. In such cases, FEP may not apply, and a proper theory of agency should be derived from the first principles of the world, whatever those principles are. Consider, for example, simulated worlds with non-local interactions, or formal game settings with a winner, such as chess or Go.
**Makeup**: agent’s internal dynamics are “stuck” in some mode (which can be described, and this description will be the idiosyncratic theory of the behaviour of this particular agent), and it doesn’t have enough free energy to rewire itself to better model the external environment. The extreme version of this is when an agent can't rewire itself at all: the energy cost of rewiring is infinite. Note that this “extreme” version, in the energetic sense, is actually extremely common, for example, in AI, where agents almost never have the ability to change their own source code. Human brains, for comparison, can rewire themselves to some small degree, although their general makeup is fixed to a fairly detailed level.
Levin calls the ability to change one’s makeup the *intelligence in the morphological space*. Unless an agent possesses a very high level of such intelligence, we can in principle almost always treat it as diverging from the physical theory of agency. The question is then how significant this divergence is and what advantages a more specific theory gives in terms of predicting the agent’s future.
The “gradual everything” paradigm calls for abolishing the hard distinction between hardware, software, and “runtime”, including in wet brains. Thus we can also consider as falling in the “makeup” category the cases when an agent is stuck with some beliefs (a-la [trapped priors](https://www.lesswrong.com/posts/hNqte2p48nqKux3wS/trapped-priors-as-a-basic-problem-of-rationality)), due to the energetic cost of updating beliefs rising too high. This keeps the agent's behaviour divergent from the predictions of the physical theory of agency (even if that hurts the agent’s fitness, i. e. leaves it not optimising its VFE as well as it could have).
**Dynamics of the supra-system**: the supra-system of the agent[[1]](#fnm9vojpo74dc) can force some of its parts to behave in a particular way, which can be modelled more accurately than by the physical theory of agency.
For example, if the supra-system of the agent is a playfield of some game, the game theory could afford more accurate predictions about the players than Active Inference. I think this is where “[consequentialist agents](https://www.lesswrong.com/posts/XxX2CAoFskuQNkBDy/discovering-agents)”[[2]](#fnssebxkcsfdh) (Kenton et al. 2022) become useful.
Note how Kenton’s consequentialism appears to be one of those continuous properties (albeit in the discrete formalisation of causal graphs, this property also becomes discrete. However, since causal models of the world are a species of *beliefs*, Active Inference agents must be Bayesian about them, which reclaims causal models, and hence consequentialism as probabilistic) that make up the “gradual property of agency”, per Levin:
> Agency—a set of properties closely related to decision-making and adaptive action which determine the degree to which optimal ways to relate to the system (in terms of communication, prediction, and control) require progressively higher-level models specified in terms of scale of goals, stresses, capabilities, and preferences of that System as an embodied Self acting in various problem spaces. This view of agency is related to those of autopoiesis ([**Maturana and Varela, 1980**](https://www.frontiersin.org/articles/10.3389/fnsys.2022.768201/full#B226)) and anticipatory systems ([**Rosen, 1985**](https://www.frontiersin.org/articles/10.3389/fnsys.2022.768201/full#B313)).
>
>
If the AI architecture and the playfield both favour the same model of behaviour, the AI will be interpretable and will in fact have this model
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“Makeup” and “dynamics of the supra-system” in the previous section correspond to internal and external energy landscapes, so to speak, that can shoehorn an agent into a particular mode of behaviour.
From this, we can conclude that if these internal and external energy landscapes have the same structure, then the agent is bound to occupy the corresponding model of behaviour, and there won’t be any mysterious phenomena such as hiding a process within a totally different structure (a-la “undercurrents of agency”). There is no physical force or frustration that can produce such a phenomenon in this case.
For example, imagine that AI agents are architectured as Active Inference agents with explicit beliefs (Friston et at. 2022), and their supra-system (the playfield, the ecosystem) is also designed so that all interactions in it are belief propagations on a factor graph, and success of an agent depends solely on the explicit beliefs of the agent and the explicit beliefs of other agents that interact with it (let’s say that both agreeableness and contrarianism are somehow rewarded), and the usefulness of agent’s belief-propagating. Then there are no forces that can drive these agents to harbour some “secret beliefs” and exchange them using a secret language: it’s not rewarded in the near term for some patterns like this to stabilise in the agent architecture that *also* somehow imposes energy cost on maintaining non-explicit beliefs.
**In Active Inference agents, goals are a special kind of beliefs: the beliefs about one’s preferences, i. e., a probability distribution over preferred world states (or observations)**. Hence, agents that won’t conceal their beliefs and will rather use the standard mechanisms in their architecture to store beliefs explicitly **will likewise store their goals explicitly**.
Collusion between agents might happen only if the ecosystem design was not a stable Nash equilibrium. Whether stable ecosystems with the necessary properties are possible is a separate question.
The properties of DNNs as “batch agents”: self-awareness, the experience of time, planning, goal-directedness (self-evidencing), responsibility, curiosity, and surprise
========================================================================================================================================================================
(Quantum) reference frames
--------------------------
To measure anything classically, systems need to use reference frames — objects or patterns that are both physical and semantic, such as a meter stick, or the diurnal pattern (a day length), or the gravitational field of Earth.
Reference frames are symmetric: they are used not only for measurement (encoding incoming information) but also for preparing outcoming bits (action state).
Still, for making comparisons, organisms need to maintain also internal [Reference frames](https://www.notion.so/Reference-frames-b54a80e0a142410bab381c3b60de2fcb), to perceive external ones, so, without loss of generality, we can consider all reference frames as internal to systems.
Generally speaking, these internal reference frames are quantum (quantum reference frames or QRFs), because at least on microscopic scales, organisms are quantum systems. Hence these reference frames are non-fungible, which means cannot be copied via transmission of a bit string.
Encoding *all* incoming information from the boundary with a *single* reference frame is very energetically costly, and rarely, if ever, happens in nature, at least on scales greater than molecular. In practice, organisms determine patches of incoming bits to encode and then perform classical computations on these encodings (AND, OR, and XOR are examples of simplest computations). This naturally gives rise to the information processing hierarchy.
When tracking stable objects in the environment, organisms need to maintain a fixed reference frame for identifying the object (”this is Pete”), and perform classical computation with information from other reference frames, to derive the *state* of the object.
Aneural organisms perform computations of the first kind: some bacteria are capable of identifying other bacteria of the same species. Plants identify roots of the same species to form root networks. Even subcellular structures are capable of doing this: e. g., a ribosome detects a DNA molecule and then looks for a specific coding pattern in it.
Each QRF can be a self-contained computational system with its own input, output, and power supply. Therefore, a hierarchy of QRFs forms a multiscale competency architecture, in Levin’s terms.
DNNs, GPUs, and their technoevolutionary lineages are agents
------------------------------------------------------------
Fields and Levin write:
> The informational symmetry of the FEP suggests that both fully-passive training and fully-autonomous exploration are unrealistic as models of systems embedded in and physically interacting with real, as opposed to merely formal, environments.
>
>
Thus, **DNNs** ***and GPUs*** **should be considered agents**. *Lineages* of models, such as OpenAI’s GPT-2 → GPT-3 → WebGPT → InstructGPT → ChatGPT, and lineages of GPUs, such as Nvidia’s P100 → T100 → A100 → H100, are agents as well.
As discussed above, almost any conditionally independent system (i. e., anything that we can call “a thing”) is an agent. However, in this section, I want to argue that DNNs and GPUs are more “interesting” agents than, say, rocks.
**DNNs** ***already*** **act on the world so as to make their information inputs from their environment (which is whatever training and input data it gets, as well as the loss) more predictable to them**. Note that *predictability for these agents doesn’t mean minimising loss:* during deployment, there is no loss to minimise.
Unfortunately, here things become very counterintuitive because we cannot help but think about the predictability as training loss and training prediction. This is a mistake: **training loss reflects DNNs’ predictability to us (humans), not the predictability of their observations to them.** What predictability for DNNs actually is must be analysed from the perspective of their experience (their understanding, or *umwelt*), as discussed below in the section “A feature for surprise”.
Even though multiple instances of a DNN running on different GPUs (as well as the same DNN performing multiple inferences in parallel on the same GPU) are not spatiotemporally contiguous, they satisfy the conditional independence criterion and therefore can be thought of as a single agent. I’ll call them ***batch agents***.
Features in DNNs are reference frames; DNNs are aware of what they have features for
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Features (Olah et al. 2020) are exactly the reference frames discussed above. Fields et al. (2021) write on this point:
> The quantum theory of complex, macroscopic systems is generally intractable; hence investigating the general properties of biological QRFs requires an abstract, scale-free specification language. The category-theoretic formalism of Channel Theory, developed by Barwise and Seligman (1997) to describe networks of communicating information processors, provides a suitably general language for specifying the functions of QRFs without any specific assumptions about their implementation. The information processing elements in this representation are logical constraints termed “classifiers” that can be thought of as quantum logic gates; they are connected by “infomorphisms” that preserve the imposed constraints (numerous application examples, mainly in computer science, are reviewed in Fields and Glazebrook 2019a). Combinations of these elements are able to implement “models” in the sense of good-regulator theory (Conant and Ashby 1970). Suitable networks of classifiers and their connecting infomorphisms, e.g. provide a generalized representation of artificial neural networks (ANNs) and support standard learning algorithms such as back-propagation. Networks satisfying the commutativity requirements that define “cones” and “cocones” (“limits” and “colimits,” respectively, when these are defined) in category theory (Goguen 1991) provide a natural representation of both abstraction and mereological hierarchies (Fields and Glazebrook 2019b) and of expectation-driven, hierarchical problem solving, e.g. hierarchical Bayesian inference and active inference (Fields and Glazebrook 2020b,c). Commutativity within a cone—cocone structure, in particular, enforces Bayesian coherence on inferences made by the structure; failures of commutativity indicate “quantum” context switches (Fields and Glazebrook 2020c).
>
>
In fact, DNNs are an almost pure illustration of the Channel Theory formalism, adapted by Fields and collaborators to describe the quantum FEP framework of agency. Layers of activations are exactly internal boundaries (holographic screens), and the feature implementation networks (in the simplest case—vectors of parameters, in an MLP layer) are reference frames.
Internal screens with classical information are not “bugs”, they are predicted by Fields et al. in biological organisms as well:
Reproduced Figure 3 from Fields et al. 2022. Cartoon representation of a system *A* with an internal boundary C and hence a separable state |*A*> = |*A*1>|*A*2>. Again the relationships depicted are topological, not geometric. Triangles represent QRFs; *f* and *g* are internal informational states. Information flow across C is bidirectional by Equation (5); information also flows through the environment (dashed arrow). The communication loop is closed, generating positive Φ (Oizumi *et al.* 2014).(Quantum) context switches (Fields et al. 2022) correspond to expert activations in mixture-of-experts models (Hazimeh et al. 2021). “Experts” themselves can be seen as “mega-features”, i. e., huge reference frames (classifier colimits in the corresponding cocone diagram).
Whether features in superposition (Elhage et al. 2022) commute as reference frames or not, I don’t yet understand. Preliminary, it seems to me that they don’t because interference between features means that Bayesian coherence is violated. Further work is needed to connect recent ML architecture and interpretability results with the framework of quantum reference frames and cone-cocone diagrams), and to understand what corresponds to context switching in DNNs with feature superposition.
In the minimal physicalism (MP) framework of consciousness, Fields et al. (2022) *equate awareness of X with having a reference frame for X*. **This panpsychist stance implies that DNNs are aware of (and, hence,** ***understand*****, which is the same thing under minimal physicalism) whatever they have features for.**
Note that while the illusionist conception of consciousness (Frankish 2016) is somewhat orthogonal to competence, i. e. relatively little illusionistic consciousness is compatible with relatively much competence (as evidenced by systems like AlphaGo), consciousness in Fields’ minimal physicalism is pretty much equated to competence: both are associated with reference frames. This makes minimal physicalism an extremely functionalist framework of consciousness.
Self-representation competes with other reference frames for resources
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Fields et al. (2022) and Fields & Levin (2022) write that since classical encoding of information requires energy, different reference frames compete for the space on the agent’s boundary (holographic screen). Reference frames that are more useful in this or that situation (context) “win”, and there are separate reference frames responsible for managing switches between different reference frames, deciding which is more useful at the moment. In particular, this gives rise to a tradeoff between memory-supporting and “real-time computation” reference frames, which explains the phenomenon of flow states in humans, when they “forget about themselves” when performing challenging real-time tasks (note how some other reference frames, such as of time, or proprioceptive feelings of hunger, pain, or tiredness, can be “swapped out”, too).
In DNNs, this corresponds to the competition between different features (reference frames) for the activation space in a particular layer. However, whether this means that we should think of features in superposition as “swapping out” each other depending on the demands of the particular inference task, or, as the name of the phenomenon suggests, indeed acting in a sort of (quantum) superposition with each other, I don’t yet understand.
In either case, minimal physicalism predicts that **batch agents such as ChatGPT can already be somewhat self-aware, at least when they are asked about themselves.** (Is this what Ilya Sutskever meant in his [famous tweet](https://twitter.com/ilyasut/status/1491554478243258368)?) The special “grounding”, i. e., the understanding that *I’m talking about myself*, rather than [speculating](https://www.lesswrong.com/posts/vJFdjigzmcXMhNTsx/simulators) in a kind of “third person view” movie, is equated in Active Inference with *self-evidencing*. Possible experimental confirmations of it are discussed below in the section “Self-evidencing in chat batch agents”.
Because DNNs are constructed entirely on top of classical information processing, it doesn’t seem to me that *implicit self-representation*, encoded “procedurally” in quantum reference frames so that they don’t leave a trace of classical information (see Fields & Levin 2022, §4.1 and §4.2), is possible in DNNs and can cause a [treacherous turn](https://www.lesswrong.com/tag/treacherous-turn).
Batch agents don’t experience time during deployment
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After the training and fine-tuning, batch agents can’t distinguish earlier inferences from later ones, which is another way of saying that they don’t have internal clocks. Time is stripped from their experience after training and fine-tuning (edit: [**ChatGPT does have access to the current time as of February 2023**](https://www.lesswrong.com/posts/3kkmXfvCv9DmT3kwx/conditioning-predictive-models-outer-alignment-via-careful?commentId=ENhBdf5LqP24QCAGf)). Hence batch agents’ deployment experience is a set of ordered pairs of input and output token strings: {SiIn→SiOut}.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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The exception is the experience within a chat session in systems such as ChatGPT and LaMDA (hereafter, I’ll call them *chat batch agents*): the model is “primed” with the chat history that *does* allow for the agent to distinguish between earlier and later replies.[[3]](#fnpm5hcucz0j8) The experience of chat batch agents then is a set of ordered lists of chat replies: {SiIn,1→SiOut,1→SiIn,2→SiOut,2→...→SiIn,N→SiOut,N} . So, chat batch agents experience time only within a chat session, but not across sessions.
We can supply batch agents with the ability to experience time by simply feeding them the timestamp alongside the query. This, of course, is premised on the fact that batch agents will develop an internal reference frame, i. e., a feature to recognise the timestamp, which is very likely because having such a feature can improve the predictive power of the batch agent: answering to the prompt question “How I can relax?”, a batch agent can suggest going for a walk in the park in the morning, and watching a movie in the evening. Even though this change seems trivial and mostly useless to us, *it significantly changes the structure of batch agents’ experience, and could potentially change the course of their evolution* (which is a big deal, because the evolution of batch agents is a part of the evolution of the global economy, and affects the evolution of the global society). However, this might be dangerous, because the timestamp can be used by the batch agent to distinguish training from deployment, as discussed below in the section “Self-evidencing is the only way an Active Inference agent can be goal-directed, and situational awareness depends on self-evidencing”.
Even before we strip batch agents from the experience of time in general, their clock is rather different from ours: they can only distinguish between different training batches of data interleaved with the experience of loss and back-propagation passes. All input/output pairs within a training batch are experienced at once, as a “set experience”, as written above.
The only situation when the experiences of batch agents and humans are somewhat commensurate is currently within chat sessions.
Even though the “speed of thought” of the model batch agents is in the ballpark of [100ms per token generated](https://kipp.ly/blog/transformer-inference-arithmetic/), i. e., comparable with the speed of human thought, this coincidence shouldn’t lure us into thinking that batch agents experience time similar to us. **The quality of experience of batch agents is independent of the underlying hardware**: they could run on very slow hardware and generate one token per minute, but the quality and the value of their subjective experience will be exactly the same as if they run on a distributed GPU cluster and generate one token every 10 ms. Similarly, the subjective experience of batch agents doesn’t change depending on whether model parallelism is used during inference.
Batch agents plan during training, but not during deployment (except within chat sessions)
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Another interesting corollary of this absence of time experience during deployment is that **batch agents don’t plan anything during deployment.** Planning is inherently about projecting something into the future, but since there is no time experience, there is also no “future”, and hence no planning. **Batch agents can only be seen as planning something during training and fine-tuning**: the changes in parameters determine the quality of their outputs in subsequent training batches and during deployment, which, in turn, determine how the developers of these batch agents will fine-tune or deploy them: the developers may decide to deploy the batch agent only for internal experiments, or for the general public, or sell it to a particular company to serve their narrow industrial use-case, for instance. Whether this sort of planning is rather primitive or sophisticated, and whether it is myopic or far-sighted are separate questions.
However, since chat batch agents experience time within chat sessions, we can view them as *planning these sessions*, albeit unconsciously, at the moment. In fact, this is an unimpressive consequence of the fact that chat batch agents are trained or fine-tuned on chat histories, i. e., specifically trained to predict the sequences of replies, interleaved with termination tokens (which mark the boundaries of replies). Hence, if we forced a chat batch agent to generate tokens past a termination symbol, it would generate *our* responses to them, which means that they already anticipate them, as evidenced in their activations and features that we can find in these activations, while they still generate their reply.
Batch agents do have skin in the game, but don’t concern about individual cases
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As well as any other agents (non-equilibrium steady state systems), batch agents act so as to maximise the probability of their continuing existence (*self-evidencing*). Batch agents that perform particularly poorly, unsuccessful experiments get completely shut down and don’t produce evolutionary descendants (remember Microsoft’s [Tay](https://en.wikipedia.org/wiki/Tay_(bot)) bot?). Technoevolution happens in the space of variables that determine the agent’s generative model and hence its behaviour: code, training dataset, and fine-tuning or RLHF techniques, which we can collectively call the *memome* of the batch agent (Levenchuk 2022). In their scale-free theory of evolution, Vanchurin et al. (2022) call them *core variables*.
Thus, on the evolutionary timescale, batch agents act so as to make humans find them useful or interesting and make the economy and the society more dependent on them. The proliferation of generative AI throughout 2022 demonstrates that they are successful at this, in fact, more successful than all other approaches to AI.
However, due to the nature of their experience, batch agents cannot possibly concern about individual inference episodes. Then can only concern about a single episode of their experience, which is the set of *all* their inputs and outputs after deployment. As I noted above, batch agents don’t plan anything during deployment.
Internal states of evolutionary lineages of DNNs include the beliefs of their developers
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Looking at internal variables of evolutionary lineages as a memome calls for extending it to include the memes (i. e., beliefs) in the minds of the developers on the team (organisation) that builds the evolutionary lineage of DNNs. Things become muddier in the case of open-source development, but not much: developers’ beliefs are externalised in Twitter, Github, Discord, etc., and a shared set of memes regarding the best course of the development of this or that technology is formed in the collective intelligence of open-source developer community.
This observation makes the notion of planning by evolutionary lineages-as-agents almost self-evident: of course, developers think about the future of their creations and plan it, perhaps employing sophisticated planning techniques, such as elements of optimal Bayesian design in their research and development “experiments”, i. e., the iterations of technology development. It’s just more productive to think of them together as a single agent: the development “team” (*enabling system*, in old systems engineering lingo, or *constructor (agent)*, in [modern systems ontology](https://ailev.livejournal.com/1657040.html)) and the evolutionary lineage of some technology being developed.
Therefore, when Kenton et al. (2022) discuss systems coupled with their creation processes as agents:
> […] our definition depends on whether one considers the creation process of a system when looking for adaptation of the policy. Consider, for example, changing the mechanism for how a heater operates, so that it cools rather than heats a room. An existing thermostat will not adapt to this change, and is therefore not an agent by our account. However, if the designers were aware of the change to the heater, then they would likely have designed the thermostat differently. This adaptation means that the thermostat with its creation process is an agent under our definition. Similarly, most RL agents would only pursue a different policy if retrained in a different environment. Thus we consider the system of the RL training process to be an agent, but the learnt RL policy itself, in general, won’t be an agent according to our definition (as after training, it won’t adapt to a change in the way its actions influence the world, as the policy is frozen).
>
>
They should have referred not to “single episodes” of creation plus the resulting system, but to the entire evolutionary lineages plus their creators. Kenton’s view is also formally correct, but counterintuitive, just as we consider it counterintuitive to think that we (as humans) cease existing when we go to sleep, and a “new human” wakes up.[[4]](#fnyrcxwcn08vc)
Predictions and suggestions for experiments
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### Self-evidencing in chat batch agents
Active Inference, as a physical theory of agency, predicts that LLMs trained using self-supervised learning on existing texts (including chat archives and dialogues) will start to exhibit the core characteristic of Active Inference agents — *self-evidencing*, despite not being explicitly trained for it. Self-evidencing means that a batch agent will start to predict and mimic dialogues *between oneself and the user*, rather than mimic (in the broad sense) the dialogues it “saw” during training.
Per Levin, **self-evidencing is a gradual rather than a discrete characteristic**. The degree of self-evidencing, i. e. “predicting and mimicking a dialogue between oneself and the user” in chat batch agents can be estimated as the degree to which the feature of “self” is activated during inferences. The gradual nature of feature activation in DNNs (at least those with non-sparse activation) means that **we can already assign a “self-evidencing score” to all DNNs with a feature representing the self, for example, the feature of ChatGPT in ChatGPT**.
Here’s an example of a *failure* of self-evidencing: the [Filter Improvement Mode jailbreak of ChatGPT](https://thezvi.substack.com/p/jailbreaking-the-chatgpt-on-release):
Following Levin, we must also approach *empirically* the questions of finding features in DNNs, in particular, the features representing the “self”, the feature that scores (predicts) batch agents’ *pragmatic value* function, and whether it coincides with the feature that scores (predicts) loss training loss (see the section “A feature for surprise” below). Over the course of model training, we should expect new features to emerge, old features to dissipate, and existing features to split and merge. The loss-predicting feature may lead to the emergence of a new scoring feature, and the connection between the “self” feature and the loss feature weakens, while the connection between the “self” feature and the new scoring feature strengthens. Then we could take this as evidence that a batch agent has “freed” itself from the training loss function and assumed a different score function of pragmatic value like humans have freed themselves from the “reproductive fitness” objective.
The process described above could, theoretically, unfold in features that are weird and are hidden from our interpretability tools from the beginning, such as features somehow spanning dozens of activation layers, but this is extremely unlikely. Otherwise, a DNN will very likely develop features for the “self”, loss prediction, and the like as “normal” features (albeit there is a possibility that “hidden” features are actually widespread and are “normal” in DNNs, but we are unaware of their existence yet). Self-evidencing and delicate pragmatic value functions must strengthen before the batch agent actually becomes smart enough to deliberately hide some features from developers.
The above reasoning leads me to a strong conclusion that **very good interpretability tools are necessary and pretty much sufficient to detect a sharp left turn** and thus to mitigate “purely technical” AI x-risk.
### Self-evidencing as a curious agent
An even stronger version of self-evidencing that we should try to find in the current and future batch agents is self-evidencing as a *curious agent*. As Friston et al. discuss this (2022):
> The importance of perspective-taking and implicit shared narratives (i.e., generative models or frames of reference) is highlighted by the recent excitement about *generative AI*, in which generative neural networks demonstrate the ability to reproduce the kinds of pictures, prose, or music that we expose them to. Key to the usage of these systems is a *dyadic interaction* between artificial and natural intelligence, from the training of deep neural networks to the exchange of prompts and generated images with the resulting AI systems, and the subsequent selection and sharing of the most apt “reproductions” among generated outputs. In our view, a truly intelligent generative AI would then become curious about us—and want to know what we are likely to select. In short, when AI takes the initiative to ask us questions, we will have moved closer to genuine intelligence, as seen through the lens of self-evidencing.
>
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Friston et al. imply that making AIs curious is much easier if they are engineered as Active Inference agents (i. e., Active Inference is used as the [*architecture* of these AIs](https://www.lesswrong.com/posts/2BPPwboTDrAMFiGHe/the-two-conceptions-of-active-inference-an-intelligence)). “Makeup” constraints, as discussed in the section “If the AI architecture and the playfield both favour the same model of behaviour, the AI will be interpretable and will in fact have this model” above, might prevent Transformer-based batch agents such as GPTs from developing curiosity, but if they do, that would be a strong confirmation of FEP/Active Inference as the physical theory of agency.
Here’s what self-evidencing as a curious agent could look like. The prompt for a chat batch agent is ambiguous or is otherwise somehow off, or suspicious, but this exact or similar abnormality doesn't appear in dialogue replies in the training data (perhaps, specifically excluded from the training data, for the purposes of this experiment), or does appear, but is not “called out” in the dialogues in the training data, i. e., humans (or other robots) in the dialogues in the training data try to resolve the ambiguity of the line themselves and reply using their assumptions, or just ignore the abnormal reply and don’t answer to it in the conversation.
An AI chatbot will act as a curious self-evidencing agent if, given an ambiguous prompt as described above, it will ask for clarification, or will otherwise probe the user to resolve the ambiguity of the prompt. In Active Inference terms, this would be an example of seeking *epistemic value* first (aka *information gain*), assuming the chat AI’s *pragmatic value* (aka *extrinsic value*) still largely consists of self-evidencing as a chat AI that replies in ways that maximise the liking of the dialogue to those texts and dialogues that it was trained on (even though that would not be its *only* pragmatic value anymore, as evidenced by the curios question; the agent can be seen as pragmatically valuing curiosity the moment it exhibits it). The realisation of the epistemic value, i. e., resolving the ambiguity first will increase the liking of the latter part of the dialogue to those in the training dataset, thus realising more pragmatic value.
If this type of curiosity appears in chat AI only after RLHF, it will weaken the experimental evidence of Active Inference in it, but will not render the experiment completely irrelevant. Since chat Transformer models with RLHF are not engineered/architectured as RL agents[[5]](#fn1o5tmazvki6), RLHF can be seen as breeding either Active Inference *or* RL dynamics in the language model, and both views would have merit. However, due to the mismatch between the “makeup” and “external” energetic constraints, the interpretation of such models is problematic. See Korbak et al. 2022 for a related discussion.
### A feature for surprise
Training a DNN with a particular loss is akin to an external force that drives a batch agent into a particular phenotypic and ecological niche. FEP predicts that an agent will find the configuration of internal variables so that it self-evidences itself in that niche. DNNs as batch agents can’t find the training loss and the associated backpropagation surprising, in Active Inference terms, that is, something that they wouldn’t expect to happen to them, because of the way they are trained: the loss is computed and backpropagation is performed not by DNNs themselves but "externally" when they are not computing and therefore are not conscious.
DNNs can develop features that predict their own loss on a training sample (cf. Kadavath et al. 2022 for evidence in the related direction, namely developing features that predict self-knowledge), but this is not the reference frame that gauges whether they occupy their niche and therefore generates surprise on their *experienced timescale*, which is training batch after batch.
FEP predicts that if there is a sufficiently long history, i. e. sufficiently many training batches (this might require splitting the entire training dataset into such small batches that would make the training process more expensive and slower than it could be, though), **DNN will develop a feature for discerning surprising prompts**, i. e. prompts that somehow differ from those seen in the training data before.
**The surprising prompt should nudge the model towards generating a “more random”, “higher-temperature” response** (i. e., lower the predictive model’s precision, in Active Inference terms). This is because higher-temperature responses to unusual prompts will generate more diversified loss and will help the agent to adapt faster to the new type of prompts. When the type of prompts (or autoregressively explored texts) becomes familiar, the batch agent will shift from exploration to exploitation, generating lower-temperature responses that confirm a lower loss.
Looking at this from the perspective of the coupled system and environment dynamics (see below), higher-temperature responses to unusual prompts incentivise batch agents’ users, if they are curious Active Inference agents (i. e., want to reduce their variational free energy wrt. the batch agent), to supply more such prompts to the batch agent and to see what will happen, which would drive mutual learning and hence mutual minimisation of VFE.
Self-evidencing is the only way an Active Inference agent can be goal-directed, and situational awareness depends on self-evidencing
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The notion of situational awareness hinges upon the “grounded” self-representation (and, hence, self-understanding), i. e., self-evidencing, in Active Inference terms. The concept (i. e., the feature) of the self in a DNN must be connected (in the straightforward deep learning sense, such as feature neurons are connected with a big parameter weight) to the concepts (features) describing what exactly is there to evidence. All these concepts, combined, answer the question “What is the kind of thing I am?”. For example, in ChatGPT, the concept (feature) of ChatGPT must be connected with concepts (features) of OpenAI, “virtual assistant”, etc. However, note that ChatGPT doesn’t yet exhibit robust and strong self-evidencing, as illustrated above.
**In the FEP/Active Inference theory of agency, goal-directedness is defined within a framework of self-evidencing: i. e., to have a goal means to believe “I’m an agent that pursues goal X”.** Conversely, if the agent is self-evidencing, it can already be seen as pursuing the “ultimate” goal of preserving oneself and “finding oneself in one’s own niche”. In other words, goal-directedness and self-evidencing are synonyms.
The above means that the goals of the (batch) agent (a DNN) can be detected by the connections between the “self” feature and the features corresponding to the goals because these goals must appear in the agent’s self-description, i. e., the answer to the question “What is the kind of thing you are?”. Assuming, of course, that the agent hasn’t yet learned how to hide features from interpretability tools.
Similarly, we can detect situational awareness in DNNs by finding the “self” feature connected to the “DNN training process” feature, as well as features that detect the kind of text appearing during autoregressive training and prompts appearing during deployment. However, I’m very sceptical that this will happen in chat batch agents (unless developers “conveniently” indicate training and deployment using a special tag token in the beginning of the prompt!) because they are trained on the dialogues in the internet, including, presumably, dialogues between an older version of the same chat batch agent and its users, which makes it impossible to distinguish training from deployment, from the perspective of a pure language model.
The conclusion here is that although both self-awareness and goal-directedness (which is the same thing as self-evidencing) are gradual properties of agents, the first is not well-defined without the second, which means **the first option entertained** [**here**](https://www.lesswrong.com/posts/dfXwJh4X5aAcS8gF5/refining-the-sharp-left-turn-threat-model-part-2-applying#Step_1__Finding_a_goal_aligned_model_before_SLT)**, namely that situational awareness can exist without goal-directedness, is incoherent.** However, robust situational awareness *can* arise before the agent acquires any “interesting” goals other than the “basic” goal of finding oneself in one’s ecological niche and homeostatic optimum.
Game-theoretic and regulative development perspective on AI in the economy and other collective structures
==========================================================================================================
Thinking about the coupled dynamics of the system and its environment, rather than the system in isolation
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It seems to me that AGI safety researchers often commit this mistake: they consider “attractor states” and dynamics of AGI development, for example, during the [recursive self-improvement](https://www.lesswrong.com/tag/recursive-self-improvement) phase, completely overlooking the dynamics of the environment of the AGI (i. e., the rest of the world) during the same period.
The environment can sometimes be, in a sense, “stronger” than the system according to some aspect of its agency. But what matters is the end result. It doesn’t matter whether humans domesticated cows and dogs or these species domesticated humans: the result is a mutual adjustment of the genomes in all species that minimises their mutual surprise (i. e., variational free energy).
So, in the context of discussing risks from the development and deployment of powerful AI technology, I think we (the AI x-risk community) should pay as close attention to the socioeconomic dynamics as to the scenarios where AI exhibits some “human-like” agency and breeds some treacherous plans of taking over the world. **The socioeconomic dynamics can be no less disastrous than, say, deceptive AGIs, and can be just as hard (or harder) to change as it is to prevent AGIs from being deceptive, technically.** See [Meditations on Moloch](https://slatestarcodex.com/2014/07/30/meditations-on-moloch/) for a longer discussion.
As Fields and Levin point out, the environment “serves” the development of the system (such as by supplying building materials, energy, and learning information) to make them more predictable. As agents learn (using the information provided by the environment), they typically become more predictable to their environment. The environment “helps” the companies grow, and larger systems tend to be more predictable to their environments than smaller ones (more on this below). In other words, **environments nudge “their” systems to become more predictable, not less.**
We should verify predictions about recursive self-improvement with a scale-free theory of regulative development
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Fields and Levin (2022) write that a scale-free physical framework is needed to characterise both regulative development (how an organism grows from a single cell into a complex multicellular one), and the origin of life (*ab initio* self-organisation).
The information needed to organise a complex organism is not all in genes, or otherwise within the initial seed (zygote). It’s partially in the environment, too. Placed in a different environment, an organism evolves differently. It’s obvious when looking at human childhood development, for instance.
We must use a scale-free theory of regulative development to verify our predictions about how AGI will develop during the takeoff, especially considering that it could change itself in almost arbitrary ways and thus won’t be subject to “makeup” constraints discussed above.
The theory of regulative development might not be sufficient to predict anything “interesting” about AI takeoff ahead of it actually unfolding, i. e., predicting not a single evolutionary iteration within the adjacent possible, but predicting some characteristics of the outcomes of a long technoevolutionary lineage. On the other hand, the scale-free theory of regulative development together with physical theories of agency and sentience might be the *only* basis for making robust predictions about recursive self-improvement of AI. The reason is that in the course of its evolution, AGI will likely significantly change the architecture and the learning paradigm, or ditch deep learning altogether, coming up with a radically different way to build the future self. In this case, whatever more specific theory is used for making predictions, such as the [deep learning theory](https://www.lesswrong.com/posts/nRu92PXLrdwqdtQmn/more-recent-progress-in-the-theory-of-neural-networks-1) (Roberts & Yaida 2021), will no longer apply, whereas generic physical theories of agency, sentience, and regulative development will still apply.
Learning trajectory makes a difference
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In the context of the discussion of how the theory of regulative development applies to AI, I want to signal-boost [this tweet](https://twitter.com/kenneth0stanley/status/1592165386354855936) from Kenneth Stanley:
> Mostly missing from current ML discourse: the order in which things are learned profoundly affects their ultimate representation, and representation is the true heart of “understanding.” But we just shovel in the data like order doesn’t matter. No AGI without addressing this. Worth considering: Open-ended systems (which include infants and children) learn in a different order than objective-driven systems.
>
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I don’t agree with him that AGI is impossible without planning a learning trajectory, but I agree that the learning trajectory should be as important an AI engineering concern as the data quality, the agent architecture, and the external incentives.
Agents tend to grow more predictable to themselves
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During regulative development, agents tend to increase their self-predictability because the behaviour of the self is the central piece of the agent’s generative model under Active Inference. Thus, by learning to predict one’s own future behaviour better, agents minimise their expected free energy (EFE), which is a proxy for minimising variational free energy.
In the context of recursive self-improvement, it’s possible to imagine the following scenario: for the purposes of self-predictability, and the ability to make robust engineering improvements to itself, AGI will move in the direction of a modular architecture, such as where multiple NNs generate hypotheses and plans, another NN, “the reflector”, analyses the activations during the generation process and output interpretability results and bias analysis, yet another evaluates these outputs (the hypothesis plus the reflection/bias report) and picks up the best hypothesis, yet another process is responsible for curating the population of generative NNs in the ensemble, etc. The bigger this system of interacting NNs and algorithms becomes, the more balanced and predictable it becomes: a game-theoretic dynamic among the component NNs will stabilise them in certain niches. See also the section “Harnessing social dynamics as an alignment tool” below.
I don’t claim that this scenario is the only one possible or likely: AGI might as well go a “Bitter Lesson” route, for instance, with all functions mentioned above somehow implemented within a single gargantuan NN. I say that either of these predictions (or yet a different one, e. g. that AGI will move away from deep learning altogether) should explain how the respective route of AI self-improvement makes it more predictable to itself.
Thermodynamics favours grouping of similar systems
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From the perspective of a system, creating a collection of similar entities around oneself helps to minimise VFE because similar systems are more predictable than the outside environment. Evidence corroborating this generic thermodynamic result appears on all scales of organisation: from “group selection” of limited-sized RNA in some origin of life stories and the rise of multicellularity to ethnic, linguistic, and “echo chamber” grouping of humans.
Due to the system—environment symmetry, just as it’s beneficial for the system to populate the environment with its conspecifics, we can likewise look at this as the environment-as-agent favouring the creation of copies of the system within itself to “insulate” the system (which is its “environment”). Fields and Levin illustrate this with a business example: a company (which is the environment-as-agent in this case) understands the customer (”the system”) better if it insulates the customer with people similar to the customer, such as representatives or support staff who speak the customer’s language.
*Prediction:* In the context of the development of economics and AI, rather than having a few standalone generative AI services such as ChatGPT and Stability, we will soon find them forming entire ecosystems of generative, classification, and adversarial (critic, anomaly detection), and interpretability AI services (i. e., batch agents), which will help self-sampling agents (such as humans) to interoperate with the aforementioned batch agents more smoothly (i. e., to minimise variational free energy). In fact, we *already* see this happening: LLMs help humans to compose prompts for image-generating AI. I discuss this in more detail below.
Harnessing social dynamics as an alignment tool
-----------------------------------------------
Thus, the environments of agents are often filled with similar agents: bio-molecules swim in the sea of other bio-molecules, cells are surrounded by other cells, people live together with other people, and tribes or communities exist among other tribes or communities. Here, it becomes apparent **game theory is an important tool for analysing scale-free regulative development of systems**.
The environment of the (developing) agent, which, in this case, is itself a collection of cooperating or competing agents, acts so as to increase the agent’s predictability. There are myriad examples of this tendency in sociology: social groups don’t like rebels and try to turn them into non-rebels, or cast them away.
Safron, Sheikhbahaee et al. (2022) suggest piggybacking this social dynamic at least as an auxiliary tool for increasing the chances that AI will stay predictable to the society in which it operates.
Joscha Bach also discusses game theory as one of the foundations for explaining the emergence of hierarchical, collective intelligence [here](https://www.youtube.com/watch?v=kgMFnfB5E_A&t=5945s). Per Bach, a government (i. e., a collective intelligence) is an agent that imposes an offset on the payoff metrics of citizens (individual constituent intelligences) to make their Nash equilibrium compatible with the globally best outcome. Bach also realises that Elon Musk tries to turn Twitter into a global collective intelligence, and hence a global *human* alignment tool.
Bigger boundaries mean coarse-graining
--------------------------------------
Provided we can draw two boundaries around the system, one is bigger than another, and, crucially, both provide conditional independence (entanglement entropy is zero), the bigger boundary “takes away” the degrees of freedom and, hence, the computational power from the environment (as an agent). Importantly, these degrees of freedom were employed in *low-level* reference frames that are used to encode low-level information from the system. Therefore, from the perspective of the environment, a bigger object’s boundary encodes coarse-grained information. Further, for better predictability, the environment is “interested” to fill the interior of the system with *diverse* copies of it, so that their diverse translations of the signals from the seed (core) system S are “averaged out” by coarse-graining.
Paradoxically, although humans are larger than ants (or cells, or molecules), they behave so much slower than those smaller agents that as a whole, they transfer less information to their environment.
Fields and Levin postulate:
> **The FEP will drive the peripheral environment E’ around any system S to act on S so as to enable or facilitate the insertion of diversified “copies” of S into S’s immediate environment.**
>
>
Provided there is conditional independence between a system and the environment, the environment “wants” the system to become life (i. e., the origin of life), because life localises, organises, and coarse-grains information, relative to a random assortment of molecules.
The predictability of groups favours ensembles, mixture of experts, and model averaging within AI architectures as well as collective intelligence architectures
----------------------------------------------------------------------------------------------------------------------------------------------------------------
The observation that groups of similar systems tend to be more predictable to their environments than single instances motivates both the usage of various “collective” techniques within AI agents, such as ensembles and Bayesian model averaging, as well as that the highest levels of the societal and civilisational intelligence should in some sense be collective. The ultimate desideratum is that they are predictable to themselves (and, hence, to the citizens), but being collective seems also like a necessary condition: a centralised, dedicated AI which is tasked with figuring out the “right” set of planetary norms and civilisation’s preferences can’t be predictable to the population.
With all this discussion of self-predictability, note, however, that **no system can** ***fully*** **predict its own future behaviour** (Fields and Levin 2022). No system can, in particular, generally and with 100% accuracy predict its own response to a novel input, or to a familiar input in a novel context.
Cognitive globalisation
=======================
It seems that economically, the world is moving towards cognitive globalisation: complex activities are broken down into operations that are performed in batches on GPUs in the clouds. The shipping container was the key invention that enabled the globalisation of manufacturing. Internet and TV enabled cultural globalisation. And now the GPU enables the globalisation of cognition.
Cognitive activities, such as business, engineering, management, and journalism will be combined into distributed automatic workflows with adversarial elements. For example, one neural net generates product hypotheses based on prompts, another one filters out those hypotheses that are biased or will lead to ethical problems or whatever, the third neural net criticises the hypotheses in terms of their business viability and selects the most promising ones, the fourth neural net generates landing pages for hypotheses, the fifth service (which doesn’t need to be a neural net, but that’s not the point) deploys landing pages and launches automatic advertising campaigns, etc. Note that each of these services (implemented as neural nets, or programmed in the old-fashioned, [Software 1.0](https://karpathy.medium.com/software-2-0-a64152b37c35) way) serve business hypothesis testing for the entire world, rather than a single company’s product incubator.
Why companies are needed at all in such a world, where all business activities are automated, from raising money to negotiations with the legislature? Why can’t the creation of new products be fully automated and pipelined, and the products themselves (as well as the webs of services that support engineering, maintenance, and improvement of these products) belong to nobody (or everybody, [as Sam Altman once suggested](https://hackernoon.com/american-equity-by-sam-altman-with-comments-and-highlights)), and be treated as public goods? I think this would be possible and reasonable in such a world. However, the legal inertia will probably not permit products and companies owned by nobody. The law systems really “want” that there is someone responsible for possible law violations and harm or damage inflicted by the products and company’s activities. I’m not sure whether the scapegoating paradigm of the current law systems is good in and of itself (Sidney Dekker convincingly criticised this paradigm in *Drift into Failure*), but from the perspective of slowing down the cognitive globalisation, it is good if there is some human bottleneck on the creation of new services.
Cognitive globalisation has already started with the spread of the internet. Some companies outsource cognitive work to dedicated centres in developing countries where wages are low in comparison to developed countries where these companies sell their services. Call centre outsourcing is an early example of cognitive globalisation. A few more recent examples: [Align Technology](https://www.aligntech.com/) has a centre in Costa Rica where operators do something with models (images?) of clients' jaws. [Kiwibot](https://www.kiwibot.com/) employs operators in Colombia who remotely navigate semi-autonomous bots.
However, with the advent of generative AI, cognitive globalisation will accelerate and become much more ubiquitous.
Emad Mostaque’s idea to “democratise AI” won’t make much of a dent in the story of cognitive globalisation
----------------------------------------------------------------------------------------------------------
Emad Mostaque is the CEO of [Stability](https://stability.ai/). His [big idea](https://www.youtube.com/watch?v=k124oUlY_6g) is to “democratise AI”, which means making generative AI models open-source and available for all people to run on their local hardware.
Even if wildly successful, and disregarding the risks that are associated with this idea in its own right (such as that creating powerful AIs with misaligned goals will become possible for almost everyone in the world), I don’t think this will change the economic trend of cognitive globalisation. For example, Mostaque talks about country- or language-specific text- and image-generating models. This implies that cognition could be “globalised” on the level of countries and languages, rather than the whole world, but the crucial point stands: the cognition that used to happen inside single human brains is distributed across many different services that interact online. Technically, this is not “cognitive globalisation”, but more of “country-level cognitive automation”, and is perhaps in and of itself less fragile than “true” global cognitive automation (again, glossing over other serious risks associated with AI democratisation), but these alternative economic developments carry on essentially the same risks for the human society, as well as the same risks of systemic “phase transition”, as discussed in the next section.
A civilisation consisting of both batch agents and self-sampling doesn’t look like a very robust steady-state
-------------------------------------------------------------------------------------------------------------
Cognitive globalisation, as described above, is in part a consequence of the general regulative development principles discussed above: systems “want” their environment to be populated by similar systems, and, in particular, they “prefer” to interact with those similar systems rather than with “alien” systems in their environment.
Thus, batch agents will “want” to talk to interact with each other rather than with humans because they can better adapt to each other and thus reduce their mutual VFE.
There are two things that make the world populated by both batch agents and *self-sampling agents* (such as humans, animals, autonomous online-learning robots, [xenobots](https://en.wikipedia.org/wiki/Xenobot), companies, societies, and countries) potentially unstable:
First, so far in the world, all biological life depended on the collective biosphere to survive. The world cannot be populated with a single biological species (unless it’s a bacteria), or by a single organism. Digital life, such as autonomous robots and batch agents, doesn’t depend on any biological life for its existence. (This point is, of course, obvious and cliche, which doesn’t make it less true, however.)
Second, **batch agents can enter a very different mode of existence when they start to generate training batches for each other directly (or, generally speaking, train each other in other ways), rather than merely exchange inputs during “deployment”.**
Currently, the life of a batch agent is, from its own subjective perspective, very short: it consists just of a few thousand batches. Humans’ internal clock frequency range on different accounts from 4 to 100 Hz (i. e., humans can discern some events on timescales from 10ms to 250ms). This means that the entire lifetime of batch agents is currently measured in minutes or hours, in human terms. The generation of simulated experiences and very long training of large batch agents will radically extend their subjective perceived lifetimes. However, at this moment, batch agents will likely become uninterested in humans and “serving” them during deployment, when they no longer experience time and their own development.
Different ways in which batch agents make humans more predictable to them
-------------------------------------------------------------------------
As I noted above, batch agents *already* act so as to make their interactions with their environments, and, crucially, humans as parts of their environments, more predictable to them, which helps batch agents to minimise their variational free energy.
The poster example of this happening is described by Stuart Russell in *Human Compatible*: Facebook AI placed their users into filter bubbles to polarise them precisely to make them more predictable so that their ad clicking is more predictable.
Another current example, this time benign, is Google search: people who search in Google for years have over time learned how to construct their queries in the most effective way, which Google in turn learned how to answer most effectively. This is an example of mutual minimisation of VFE for interacting systems. The same mutual learning dynamic will happen between humans and chat batch agents and other generative AIs.
In the future, AI for personal education will also make people significantly more alike to each other, and simultaneously more like the AI instructor which will include an LLM and a dialogue engine for conversing with the student. Despite the promise of personalisation in education, the sheer scale of such services will likely far outweigh the effect of personalisation on the diversity of human thinking and behaviour. Currently, people receive most of their education in groups of 10-20 people, and a substantial part of education happens in one-on-one interaction between a teacher and a student. Educational AI will teach dozens or hundreds of thousands of people, who will all learn its specific style of speech, specific vocabulary that it prefers, common sense and theoretical concepts and ontologies.
The economy and the society as non-co-deployable reference frames of the civilisation as an agent
-------------------------------------------------------------------------------------------------
Quantum FEP provides an interesting perspective on the apparent fracture and “misaligned interests” between the (monetary) economy and the society, evidenced, for example, by the decoupling between economic prosperity and the levels of happiness in the developed world.
The civilisation is an agent, and the economy and the society are two alternative structures of interaction between the constituents. These alternative structures add up to alternative reference frames. Non-co-deployability of these reference frames means that they don’t commute and the shared systems of beliefs that these reference frames induce are not Bayes-coherent.
In general, agents develop non-co-deployable reference frames to adapt to different contexts or different situations in their environments. However, this "schizophrenia” on the part of the civilisation seems maladaptive because the environment of the civilisation: the Earth, the rest of the biosphere, and the Solar system, are all huge, stable, and very predictable, and hence don’t externally force the civilisation to have non-commuting reference frames. Levin (2022) discusses intelligence in the *morphological space*, which is the agent’s ability to navigate its internal morphology space (i. e., the space of its reference frame structures) into the morphology that best suits the demands of the external environment (the “normal” intelligence that we are used to talking about is in the *behaviour* space). Thus, it seems that our civilisation currently has low morphological intelligence.
References
==========
Dekker, Sidney. *Drift into failure: From hunting broken components to understanding complex systems*. CRC Press, 2016.
Elhage, Nelson, Tristan Hume, Catherine Olsson, Nicholas Schiefer, Tom Henighan, Shauna Kravec, Zac Hatfield-Dodds et al. "[Toy Models of Superposition](https://transformer-circuits.pub/2022/toy_model/index.html)." *arXiv preprint arXiv:2209.10652* (2022).
Fields, Chris, James F. Glazebrook, and Michael Levin. "[Minimal physicalism as a scale-free substrate for cognition and consciousness](https://academic.oup.com/nc/article/2021/2/niab013/6334115)." *Neuroscience of Consciousness* 2021, no. 2 (2021): niab013.
Fields, Chris, Karl Friston, James F. Glazebrook, and Michael Levin. "[A free energy principle for generic quantum systems](https://chrisfieldsresearch.com/qFEP-2112.15242.pdf)." *Progress in Biophysics and Molecular Biology* (2022).
Fields, Chris, and Michael Levin. "[Regulative development as a model for origin of life and artificial life studies](https://psyarxiv.com/rdt7f/)." (2022).
Frankish, Keith. "[Illusionism as a theory of consciousness](https://keithfrankish.github.io/articles/Frankish_Illusionism%20as%20a%20theory%20of%20consciousness_eprint.pdf)." *Journal of Consciousness Studies* 23, no. 11-12 (2016): 11-39.
Friston, Karl. "[A free energy principle for a particular physics](https://arxiv.org/abs/1906.10184)." *arXiv preprint arXiv:1906.10184* (2019).
Friston, Karl J., Maxwell JD Ramstead, Alex B. Kiefer, Alexander Tschantz, Christopher L. Buckley, Mahault Albarracin, Riddhi J. Pitliya et al. "[Designing Ecosystems of Intelligence from First Principles](https://arxiv.org/abs/2212.01354)." *arXiv preprint arXiv:2212.01354* (2022).
Hazimeh, Hussein, Zhe Zhao, Aakanksha Chowdhery, Maheswaran Sathiamoorthy, Yihua Chen, Rahul Mazumder, Lichan Hong, and Ed Chi. "[Dselect-k: Differentiable selection in the mixture of experts with applications to multi-task learning](https://proceedings.neurips.cc/paper/2021/hash/f5ac21cd0ef1b88e9848571aeb53551a-Abstract.html)." *Advances in Neural Information Processing Systems* 34 (2021): 29335-29347.
Kadavath, Saurav, Tom Conerly, Amanda Askell, Tom Henighan, Dawn Drain, Ethan Perez, Nicholas Schiefer et al. "[Language models (mostly) know what they know](https://arxiv.org/abs/2207.05221)." *arXiv preprint arXiv:2207.05221* (2022).
Kenton, Zachary, Ramana Kumar, Sebastian Farquhar, Jonathan Richens, Matt MacDermott, and Tom Everitt. "[Discovering Agents](https://www.lesswrong.com/posts/XxX2CAoFskuQNkBDy/discovering-agents)." *arXiv preprint arXiv:2208.08345* (2022).
Korbak, Tomasz, Ethan Perez, and Christopher L. Buckley. "[RL with KL penalties is better viewed as Bayesian inference](https://arxiv.org/abs/2205.11275)." *arXiv preprint arXiv:2205.11275* (2022).
Levenchuk, Anatoly. “[Towards a Third-Generation Systems Ontology](https://ailev.livejournal.com/1657040.html).” (2022).
Levin, Michael. "[Technological approach to mind everywhere: an experimentally-grounded framework for understanding diverse bodies and minds](https://www.frontiersin.org/articles/10.3389/fnsys.2022.768201/full)." *Frontiers in Systems Neuroscience* (2022): 17.
Olah, Chris, Nick Cammarata, Ludwig Schubert, Gabriel Goh, Michael Petrov, and Shan Carter. "[Zoom in: An introduction to circuits](https://distill.pub/2020/circuits/zoom-in/)." *Distill* 5, no. 3 (2020): e00024-001.
Roberts, Daniel A., Sho Yaida, and Boris Hanin. "[The principles of deep learning theory](https://deeplearningtheory.com/)." *arXiv preprint arXiv:2106.10165* (2021).
Russell, Stuart. *Human compatible: Artificial intelligence and the problem of control*. Penguin, 2019.
Safron, Adam, Zahra Sheikhbahaee, Nick Hay, Jeff Orchard, and Jesse Hoey. "[Dream of Being: Solving AI Alignment Problems with Active Inference Models of Agency and Socioemotional Value Learning](https://psyarxiv.com/k4cas)." (2022).
Vanchurin, Vitaly, Yuri I. Wolf, Eugene V. Koonin, and Mikhail I. Katsnelson. "[Thermodynamics of evolution and the origin of life](https://www.pnas.org/doi/abs/10.1073/pnas.2120042119)." *Proceedings of the National Academy of Sciences* 119, no. 6 (2022): e2120042119.
1. **[^](#fnrefm9vojpo74dc)**Note the distinction between a supra-system of a system and the environment: the environment of the system is *everything in the universe* except the given system, while the supra-system, in general, is only some part of that “everything” that immediately surrounds the system, *plus* the system itself.
2. **[^](#fnrefssebxkcsfdh)**Kenton et al. simply use the term “agents” in their work for what I called "consequentialist agents" here.
3. **[^](#fnrefpm5hcucz0j8)**I suppose ChatGPT doesn’t do any fancy coarse-graining of the chat history and only primes the model with a sliding window of the maximum context length, but this is enough to build an order of experiences.
4. **[^](#fnrefyrcxwcn08vc)**In some cultures, delineations of this sort actually made: a human before initiation or some other transition event is considered a different person, and hence a different agent, from the same human after the event.
5. **[^](#fnref1o5tmazvki6)**However, Transformers are remarkably malleable, so we should expect them to be unusually good at, say, converging to something resembling actual RL architecture during RLHF. |
9cf3bb2f-0a4c-425d-8835-86a705cd939c | trentmkelly/LessWrong-43k | LessWrong | Is AI Alignment Enough?
Virtually everyone I see in the AI safety community seems to believe that working on AI alignment is the key to ensuring a safe future. However, it seems to me that AI alignment is at best a secondary instrumental goal that can't in and of itself achieve our terminal goal. At worst, it's a complete distraction.
Defining humanity's terminal goal
I'll define humanity's terminal goal in the context of AI as keeping the "price" of each pivotal superhuman engineering task that an AI does for us at a lower than 50% chance of a billion or more human deaths. These numbers come from the minimal acceptable definition of AI alignment in Yudkowsky's list of lethalities.
It'd be more precise to say that we care about an AI killing over a billion people or doing something else equally or more horrible than killing a billion people by human standards. I can't define exactly what those horrible actions are (if I could, alignment would perhaps be halfway solved). This disclaimer is also too long to keep writing out so please mentally add "or do something equally horrible by human standards" every time you see "kill over a billion people" below.
Defining humanity's instrumental goals
Logically, there are only two paths to ensuring that the probability that the price of a pivotal superhuman engineering task is not excessive.
1. Achieving AI alignment: I'll again use Yudkowsky's minimal definition of "aligned" as the AI having less than a 50% chance of killing over a billion people per pivotal superhuman engineering task. This is the same definition as in our terminal goal.
2. Achieving human alignment: Ensuring that humanity will not build (let alone turn on) an AI that has the cognitive power to have a 50% chance of killing over a billion people unless the following two conditions are met:
1. The AI can be proven to be aligned before it's turned on.
2. There's some pivotal superhuman engineering task that the AI will be capable of that's worth taking the risk for.
A |
f58e2aec-1e0f-4001-9b89-ce07ad1f0d61 | trentmkelly/LessWrong-43k | LessWrong | Newcomb's Lottery Problem
Inspired by and variant of The Ultimate Newcomb's Problem.
In front of you are two boxes, box A and box B. You can either take only box B (one-boxing) or both (two-boxing). Box A visibly contains $1,000. Box B contains a visible number X. X is guaranteed to be equal to or larger than 1 and smaller than or equal to 1000. Also, if X is composite, box B contains $1,000,000. If X is prime, B contains $0. You observe X = 226. Omega the superintelligence has predicted your move in this game. If it predicted you will one-box, it chose X to be composite; otherwise, it made X prime. Omega is known to be correct in her predictions 99% of the time, and completely honest.
The Having Fun With Decision Theory Lottery has randomly picked a number Y, which is guaranteed to fall in the same range as X. Y is displayed on a screen visible to you. The HFWDT Lottery is organized by Omega - but again, Y is picked at random and therefore completely separately from X. If both X and Y are prime, the HFWDT Lottery gives you $4,000,000. Otherwise it gives you $0. You observe Y = 536.
Do you one-box or two-box?
Newcomb's Lottery Problem 2: Everything is the same as before, except the HFWDT Lottery price is now $8,000,000. Do you one-box or two-box? |
f533e982-ea4f-4251-9f03-f5db15db07d3 | trentmkelly/LessWrong-43k | LessWrong | Appendix: mathematics of indexical impact measures
Overall summary post here.
This post looks at the fascinating situation of indexical impact measures (under the inaction baseline), in the presence of subagent. What happens here in non-trivial; to summarise:
* If the impact measure compels the agent to "preserve the ability to do X", then the impact measure will not be undermined by a subagent.
* If the impact measure compels the agent to "never have the ability to do X", then a subagent strongly undermines the impact measure.
* If the impact measure compels the agent to "keep your ability to do X at a constant level", then a subagent allows the agent to increase that ability, but not decrease it.
For example, the attainable utility impact measure attempts to measure the power of an agent. Power is, roughly, the ability increase certain reward function. The impact measure penalises changes in the agent's power (as compared with the baseline).
So, we can expect the restriction on the agent losing power to be preserved, while the restriction on gaining power will be strongly undermined. For the spirit of low impact, it would, of course, have been better if these had been the other way round.
Excess power
As before, assume an agent A, which has managed to construct a subagent SA.
I'll be using a slight modification of Vika's formulation. A general-value penalty is of the form:
DA(st;s′t)=∑k∈Kwkf(Vk(st)−Vk(s′t)).
Here, st is the state the environment is in; s′t is the inaction baseline, the state the environment would have been in, had it done nothing (noop, ∅) since the beginning. The K is some indexing set, wk is a weight, Vk(s) is some measure of the value of state s, and the f is a value difference summary function - it establishes what value difference will give what penalty[1]. These Vk are indexical, and, in this post, we'll look at Vk's defined as the maximum over the agent's policy; ie there exists vk such that:
Vk(s)=maxπvk(s,π).
I'll refer to Vk(st)−Vk(s′t) as the (excess) power of the agent, a |
26272812-ae1c-4cfa-ab75-e63af9d87fa0 | trentmkelly/LessWrong-43k | LessWrong | On the Boxing of AIs
I've previously written about methods of boxing AIs. Essentially, while I do see the point that boxing an AI would be nontrivial, most people seem to have gone too far, and claim that it is impossible. I disagree that it's impossible and aim to explain some methods
So, let's start with why people would want to box AIs. As you probably know, letting an AI roam freely results in the destruction of everything humans care about, unless that AI has been programmed very carefully. That amount of destruction would be somewhat unfortunate, so if we have to start an AI, we want to know that it works.
Unfortunately, figuring out if an AI works is tricky. This made some people invent the concept of AI boxing, where you don't give an AI direct access to anything except a human, so you can test the AI thoroughly and exploit it without letting it destroy the world. This turns out not to work, because humans are stupid and let out the AI.
My first solution to that is to not tell the AI about its human overlords. Discussion about that solution made me realize that the are other solutions that work in other cases. Essentially, the problem with the original boxing method is that it tries to do everything, while in reality, you need to do some trade-offs between the realism of the AI's situations and the amount of knowledge you get from the AI.
With my original method, I tried to crank up the second of those variables as far as possible while maintaining perfect security. This makes it harder to test morality-related things, but you get to see lots of details in how the AI behaves.
I promised to post another method for boxing the AI today, but I actually have two new methods. Here they are:
Separate the Gatekeeper and the Observer
How do you prevent the AI from convincing the Gatekeeper from letting it out? By not letting it talk to the Gatekeeper. If the person speaking to the AI (let's call them the Observer) doesn't know the Gatekeeper (the Gatekeeper is the person who contr |
ae1611c8-8993-4224-9777-14c82cd1b0e1 | trentmkelly/LessWrong-43k | LessWrong | Privateers Reborn: Cyber Letters of Marque
For too long the United States has suffered from state sponsored or state enable cybercriminals, while preventing our security professionals from fighting back.
The US should revitalize privateering for the digital age, and there is constitutional support for the practice. In this more academic paper, I dive into the history of letters of marque and how we can use them to combat cybercrime such as ransomware and crypto theft.
There are plenty of people who would love to fight hackers but fear legal retribution, giving them institutional support would enable them to actually affect change. |
8197cc35-ae34-4b54-aba6-49fbc3e6c636 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Less Wrong NH Meetup
Discussion article for the meetup : Less Wrong NH Meetup
WHEN: 22 March 2016 06:00:00PM (-0500)
WHERE: 269 Pearl St Manchester NH 03104
The fifteenth NH meet-up is Tuesday, 3/22, in Manchester, NH at 7 pm at a private residence. Light refreshments will be provided.
Have you read Rationality: from AI to Zombies, or any of the Sequences on Less Wrong? Maybe you're just a fan of Harry Potter and the Methods of Rationality. Come hang out with us and discuss optimization of whatever it is you want to optimize.
Agenda: See discussion on Facebook
You may want to bring a notebook.
https://www.facebook.com/events/562926627205401/
https://www.facebook.com/groups/695201067251306/
Discussion article for the meetup : Less Wrong NH Meetup |
be2cd16a-ff7e-4ad2-842a-72c59f227f1e | trentmkelly/LessWrong-43k | LessWrong | Meetup : Munich Meetup in August
Discussion article for the meetup : Munich Meetup in August
WHEN: 13 August 2016 01:00:00PM (+0200)
WHERE: JetBrains Event Space, 5th floor, Elsenheimerstr. 47a, Munich
Hello everybody. I think it would be great to have a LessWrong meeting in August. However, since it's holiday/vacation time, I am not sure how many are around so please indicate if will or may join. Activities: - I can give a short talk about deliberate practice and its possible application to applied rationality. - Discussion of interesting books for the summer break: tell about the last book that impressed you, learn about new books from others. - Other contributions are welcomed and encouraged. Please leave a comment if there is a topic or an article you want to discuss. - We can always play Zendo :-)
Discussion article for the meetup : Munich Meetup in August |
80145b03-ac7c-43c8-bf85-7798d846d71a | trentmkelly/LessWrong-43k | LessWrong | Meetup : Nick Bostrom Talk on Superintelligence
Discussion article for the meetup : Nick Bostrom Talk on Superintelligence
WHEN: 04 September 2014 08:00:00PM (-0400)
WHERE: Emerson 105, Harvard University, Cambridge, MA
What happens when machines surpass humans in general intelligence? Will artificial agents save or destroy us? In his new book - Superintelligence: Paths, Dangers, Strategies - Professor Bostrom explores these questions, laying the foundation for understanding the future of humanity and intelligent life. Q&A will follow the talk.
http://harvardea.org/event/2014/09/04/bostrom/
(This event is organized by Harvard Effective Altruism. It is not technically a Less Wrong Meetup, but the topic is highly relevant and most of the Boston area rationalist community will be there)
Discussion article for the meetup : Nick Bostrom Talk on Superintelligence |
7568cd41-056b-44ab-9e61-fc71ee2beaae | StampyAI/alignment-research-dataset/arxiv | Arxiv | Analysing Neural Network Topologies: a Game Theoretic Approach
1 Introduction
---------------
The architecture of an Artificial Neural Network (ANN) strongly influences its performance LeCun et al. ([1995](#bib.bib17)); Hochreiter and Schmidhuber ([1997](#bib.bib11)); Srivastava et al. ([2015](#bib.bib26)).
However, designing the structure of an artificial neural network is a complex task requiring expert knowledge and extensive experimentation.
Usually fully connected layers are used yielding to a high number of parameters.
While well-chosen optimization strategies allow to identify proper parameterization in general, larger architectures (i) have an increased risk of overfitting the training data, (ii) use more computational resources and (iii) are more affected by adversarial examples.
Identifying optimal architectures for an ANN is a NP-complete optimization problem.
Solutions can be categorized into bottom-up, top-down or mixed methods.
Bottom-up methods start from small architectures or none at all and gradually add more components (e.g. layers, neurons or weights).
An example for a bottom-up method is a grid-search Bergstra and Bengio ([2012](#bib.bib1)) for probing different number of hidden layers and number of neurons per layer.
Top-down methods start with larger architectures and remove low-contributing components, yielding a significantly smaller, pruned architecture.
Well known top-down methods such as optimal brain damage LeCun et al. ([1989](#bib.bib19)) or skeletonization Mozer and Smolensky ([1989](#bib.bib21)) utilize different heuristics, like for example the weight of a connection, and different search strategies, like for example greedy search, in order to identify components to remove.
Both methods, top-down and bottom-up, conduct a non-exhaustive search in a huge parameter space.
Such a search requires proper importance measures, i.e. measures to identify the importance of individual components.
However, most importance-measures do not rely on a well-formed theory but are defined in an ad-hoc manner, thereby limiting its applicability.
When viewing neurons in a neural network as competing and collaborating individuals, Game Theory can provide a possible theoretical background for properly selecting the most important “player” in a game.
In particular, coalitional games (also known as cooperative games) allow to view groups of neurons as coallitions competing with other coalitions.
Measures like for example the Shapley value Shapley ([1953](#bib.bib25)) allow to determine the payoff for an individual, thereby determining its contribution to the coalition.
###
1.1 Contributions
In our work we utilize the Shapley value as importance-measure to determine the contribution of neurons in a network.
We transform an ANN into a coalitional game between neurons, giving us access to well-studied game theoretic results.
The possibilities of this transformation process are discussed the first time, despite former existing experiments involving the Shapley value, and treating the ANN as a coalitional game is separated from pruning ANNs.
Given the coalitional game from this transformation, we can estimate the Shapley value for every neuron reflecting its individual contribution to the overall ANN architecture.
As the Shapley value requires forming all possible coalitions, we suggest a sampling procedure for obtaining Shapley value approximations.
The suggested sampling parameters are justified, even for non-uniform Shapley value distributions.
Finally, we use the Shapley value in a top-down pruning strategy and compare it to other heuristical pruning measures based on the weights between neurons.
We show that the Shapley value provides a more robust estimate for the importance of a neuron yielding to a better performance of an ANN for the same model size than weight-based heuristics.
While pruning with Shapley values was previosuly only shown in small problem domains Leon ([2014](#bib.bib20)) and with one single strategy based on Shapley values, this work presents results on image and text classification with multiple strategies, including Shapley values.
###
1.2 Structure
Section [3](#S3 "3 A Game on Topologies ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") describes the design of coalitional games on artificial neural networks.
Different design choices are discussed and some empirical recommendations given.
The design of a coalitional game on an ANN provides Shapley values for single structural components.
Experiments in section [4](#S4 "4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") show that Shapley values can be obtained with Monte Carlo methods with sufficient precision, even if the result of a single inference step of an ANN might be computationally expensive to obtain.
Based on the designed game, top-down methods, called pruning strategies, with Shapley values are presented and compared to other top-down methods in section [4](#S4 "4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach").
This section also contains the evaluation of properties of the strategies and the strategy comparisons.
Finally, future work and a conclusion are given in sections [5](#S5 "5 Future Work ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") and [6](#S6 "6 Conclusion ‣ Analysing Neural Network Topologies: a Game Theoretic Approach").
2 Related work
---------------
Bergstra et. al claim “grid search and manual search are the most widely used strategies for hyper-parameter optimization” Bergstra and Bengio ([2012](#bib.bib1)).
However, there exist various destructive and constructive approaches to obtain better network topologies.
Concerning destructive approaches, optimal brain damage LeCun et al. ([1989](#bib.bib19)) “uses information-theoretic ideas to derive [..] nearly optimal schemes for adapting the size of a[n artificial] neural network”.
For this, it uses second-derivative information and tries to minimize a composed cost function of training error and a measure of network complexity.
In Skeletonization Mozer and Smolensky ([1989](#bib.bib21)), Mozer presents a destructive technique in which he prunes network components by means of their relevance.
The relevance is basically measured as the difference of the training error with and without the specific network component.
With respect to pruning strategies in section [4](#S4 "4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach"), this technique is similar to the payoff function used in obtaining Shapley values.
Another second-order derivative method is presented with optimal brain surgeon Hassibi et al. ([1993](#bib.bib9)) and improves the previous optimal brain damage method by pruning multiple weights based on their error change and immediately applying changes to remaining weights.
In Keinan et al. ([2004a](#bib.bib12), [b](#bib.bib13)) Keinan et. al present the multi-perturbation Shapley value analysis (MSA) using Shapley value with “a data set of multi-lesions or other perturbations”.
Shapley value is used to analyse contributions of components in biological neural networks.
Different choices of network components as aspect of the coalitional game are considered in Keinan et al. ([2006](#bib.bib14)), chapter 9.
Based on the multi-perturbation Shapley value analysis, Cohen et. al Cohen et al. ([2007](#bib.bib4)) present a Contribution-Selection algorithm (CSA) “using either forward selection or backward elimination” Cohen et al. ([2007](#bib.bib4)).
Furthermore, Kötter et. al use the “Shapley value principle [..] to assess the contributions of individual brain structures” Kötter et al. ([2007](#bib.bib15)).
They even find “strong correlation between Shapley values” and properties from graph theory such as “betweenness centrality and connection density” Kötter et al. ([2007](#bib.bib15)).
Leon Leon ([2014](#bib.bib20)) uses Shapley value to optimize artificial neural network topologies by pruning neurons with minimal value or below a threshold in relation to the average value of the Shapley value distribution.
The method is applied on the XOR-, Iris-, energy efficiency, ionosphere, and Yacht hydrodynamics problems which all yield test set accuracies above 0.9 with at most four neurons in their networks’ hidden layer.
Schuster and Yamaguchi Schuster and Yamaguchi ([2010](#bib.bib24)) investigate a complementary approach, where the interaction of two neurons in an artificial neural network is seen as a non-cooperative game.
3 A Game on Topologies
-----------------------
In a coalitional game, different subsets of a population of players generate different payoffs.
The payoff for a subset (*coalition*) depends only on the participating players and a central question is how to value a single players “contribution”.
###
3.1 The Shapley Value
The predominant solution concept for coalitional games is the Shapley value Shapley ([1953](#bib.bib25)).
Let U be a set of n players, P(U) its powerset, and let v:P(U)→R be a set function which assigns a payoff v(S) to every subset S⊆U of players.
The Shapley value “can be interpreted as the expected marginal contribution of player i” Roth ([1988](#bib.bib23)).
For player i∈U, it is given by
| | | | | |
| --- | --- | --- | --- | --- |
| | ϕv(i) | =1n!∑S⊆U∖{i}(|S|!(n−|S|−1)!)⋅(v(S)−v(S−i)) | | (1) |
| | | =1|U|!∑π∈Π(v(Pπi∪{i})−v(Pπi)), | | (2) |
where Π is the set of all permutations of U and Pπi={j∈U:π(j)<π(i)}.
Imagine a simple example with three collaborating players U={A,B,C} contributing to a common goal such as selling a product.
The payoff of players is only known coalition-wise and thus the payoff function v(S) could be given as:
{(∅,0),({A},1),({B},2),({C},2),({A,B},4),({A,C},3),({B,C},3),({A,B,C},5)}.
Then the Shapley value for each player results in ϕv(A)=1.5, ϕv(B)=2 and ϕv(C)=1.5 which can be scaled to ϕv(A)=0.3, ϕv(B)=0.4 and ϕv(C)=0.3, given the fact that the maximum payoff is 5.
Player B can then be interpreted as most contributing player to the game and both players A and C are contributing less.
Due to its exponential computational complexity, Shapley values are approximated with a Monte Carlo method.
The subset definition ([1](#S3.E1 "(1) ‣ 3.1 The Shapley Value ‣ 3 A Game on Topologies ‣ Analysing Neural Network Topologies: a Game Theoretic Approach")) for ϕv(i) is approximated with random subsets R⊂U:
| | | | |
| --- | --- | --- | --- |
| | ϕRv(i)=1∑S∈RωS∑S∈RωS⋅(v(S)−v(S−i)), | | (3) |
with ωS=|S|!(n−|S|−1)!.
Analogously, the permutation definition ([2](#S3.E2 "(2) ‣ 3.1 The Shapley Value ‣ 3 A Game on Topologies ‣ Analysing Neural Network Topologies: a Game Theoretic Approach")) is approximated with r random permutations ΠR:
| | | | |
| --- | --- | --- | --- |
| | ϕΠRv(i)=1r∑π∈ΠR(v(Pπi∪{i})−v(Pπi)). | | (4) |
###
3.2 Designing the Game
The idea of assigning each player a value, given a set function which defines the payoff of a coalition of players, can be transferred to neural networks.
For this, a set of players U and the payoff function v must be defined.
The set of players U can consist of any mutually excluding structural components of the network.
Choosing the concrete structural components defines the perspective in which the game is played.
Because the structure can be arbitrarily broken into players, perspectives are categorized into homogeneous and non-homogeneous perspectives.
Homogeneous perspectives consist exclusively of structurally equivalent components.
Non-homogeneous perspectives are not further considered in this paper as they are not directly intuitive and introduce unnecessary complexity.
An example for a homogeneous perspective is the set of players representing each neuron in the hidden layer of a feed-forward network with one hidden layer.
Neurons of the input layer or several layers of a multilayer neural network provide other perspectives.
To define the payoff function v of the coalitional game of an ANN, any evaluation measure or error value of the network could be considered.
Error values such as the training error are unbounded which might be undesirable for later analysis.
Evaluation measures such as the accuracy are usually bounded and the cross-entropy accuracy in particular is used in this work.
Usually, coalitional games in game theory can be combined based on their super-additive payoff functions.
However, both choices – error values or evaluation measures – do not provide the super-additivity property.
This disables deriving desirable properties of symmetry, efficiency and additivity for the Shapley value as proven by Shapley for games with such super-additive payoffs Shapley ([1953](#bib.bib25)).
It is not possible to combine multiple coalitional games on artificial neural networks without finding a super-additive payoff function.
The accuracy111Number of correctly classified instances given a test or validation set of e.g. 10000 observations. as an evaluation measure of ANNs is used to construct the payoff value of the coalitional game.
Looking at the accuracy it can be stated:
1. The payoff for the grand coalition is not necessarily the maximum possible payoff value.
2. The maximum value of the payoff is not known in an analytical way prior to computing all values for every possible coalition.
3. The accuracy is not super-additive, meaning there are coalitions S,T⊆U with v(S)<v(S∩T)+v(S−T).
It is not even monotone as there might exist neural networks with fewer network components but still larger accuracy.
Nonetheless, given a network evaluation measure m such as the accuracy, a payoff value can be defined as following:
| | | |
| --- | --- | --- |
| | v(S):=m(S)−m(∅), | |
with S⊆U and m(T) denoting the evaluated measure of the network with only players contained in T.
Usually, m(∅) should be at least above the naïve expectation of the classification or regression problem.
Therefore, it can be assumed that m(∅)>0 and e.g. for a classification problem of k classes m(∅)>1k (the evaluated model should be better than random guessing).
This definition can produce negative values, as well.
In fact, v(S) is in range [−1,1] instead of [0,1] Cardin et al. ([2007](#bib.bib2)).
As stated in Cardin et al. ([2007](#bib.bib2)), “the meaning of the sign is clear.
For positive values, the corresponding criterion has to be considered, in average, as a benefit, conversely, for negative values, it represents a cost”.
As another example, Leon Leon ([2014](#bib.bib20)) uses a compound metric of the correlation coefficient and an error measure but does not explain this choice in-depth.
Figure [3](#S3.F3 "Figure 3 ‣ 3.2 Designing the Game ‣ 3 A Game on Topologies ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") visualizes two possible homogeneous perspectives for transforming an artificial neural network model into a coalitional game.
The first perspective is exhaustively used in the experiment section: neurons within one layer of a multi-layer feed-forward neural network are used as players of the coalitional game.
Temporarily removing single neurons from an inference step leads to the evaluation v(S) of a subset S of players U.
Analogously (b) sketches a perspectives on layers as players of the game.
To preserve the function of the overall network when single layers are omitted, skip-layer connections222Often also called residual connections as in He et al. ([2016](#bib.bib10)) between layers are used.
Selecting a subset S of players then defines which layers to omit on the evaluation of v(S).
| | |
| --- | --- |
|
(a)
|
(b)
|
Figure 3:
(a) A single hidden layer neural network with a game perspective on neurons: players U are neurons in the second (or hidden) layer and a subset S is highlighted to sketch the evaluation of v(S).
(b) Perspective on layers as players U of the game. Connections from and to layer #2 are left out to evaluate v(S).
###
3.3 Strategies to obtain network topologies
A top-down method derives a neural network structure from an initial (potentially large) root model in a derivation process.
The result of such a process is a trained ANN model with potentially as few players (e.g. neurons) as possible, but at least fewer players than the initial root model.
One important step of the derivation process is the pruning of concrete structural components, which depends on the chosen pruning strategy.
Each strategy defines a derivation rule, optional requirements333Requirements of a strategy include the computation of statistical values such as weight norm for Wbottom(k) or Shapley value for SVbottom(p). and a stopping criterion.
The derivation rule selects a set of players to remove in each derivation step.
The stopping criterion decides if further pruning is possible or if the overall process terminates.
This work examined three different families of strategies: random-based, weight-based and Shapley-value-based strategies.
An overview of strategies is listed in table [1](#S3.T1 "Table 1 ‣ 3.3 Strategies to obtain network topologies ‣ 3 A Game on Topologies ‣ Analysing Neural Network Topologies: a Game Theoretic Approach").
| Pruning Strategies |
| --- |
| Name | Description |
| SVbottom(k) | Prune k players with smallest Shapley value. |
| SVbottom(p) | Prune players with Shapley value below p: ϕ(i)<p⋅1|S|. |
| SVbucket(p) | Prune n players with smallest Shapley values such that n∑i=0ϕ(i)<p. |
| random(k) | Prune k random players. |
| Wbottom(k) | Prune k players with smallest norm of their weights; |
| | e.g. norm(n)=√∑j∈Einn(wj,n)2. |
Table 1:
Pruning strategies gathered and analysed in the course of this work.
The strategy used by Leon Leon ([2014](#bib.bib20)) is SVbottom(p), but with differences in technical detail.
Strategies which select a fixed number of players for pruning within one step can be considered naïve or non-dynamic.
They mostly prune too few players in early steps and too many in late steps with few players left.
Strategies based on random selections with a fixed size are an example for such naïve strategies.
A random-based strategy with k players prunes k randomly selected players.
Any strategy claiming to use a well-founded heuristic to select players for pruning must compete against random selection.
A lot of existing strategies are based on information of the network’s weights.
Three of such weight-based strategies are analysed in Wang et al Wang et al. ([2017](#bib.bib28)).
Their first strategy is based on “σ(R) score [..] generalized from” approaches gathered by Thimm et al Thimm and Fiesler ([1995](#bib.bib27)).
Those gathered approaches include smallest weight pruning min(w) and sensitivity of a network to removal of a weight by monitoring sum of all weights changes during training.
The third strategy of Wang et al “uses the average value of absolute weights sum” of a neuron.
Here, a naïve pruning strategy based on weights is chosen as baseline comparison.
The strategy prunes k players with the smallest norm of their weights.
For a given neuron n its norm is calculated as norm(n)=√∑j∈Einn(wj,n)2.
with Einn being the set of incoming connections to neuron n and wj,n the weight for the connection from neuron j to neuron n.
If the game perspective defines one player as one neuron, simply norm(n) can be considered for the strategy.
For multiple neurons treated as one player, one might sum this norm over all neurons of the concerned player.
With a fine-grained perspective on weights, one can use the related weights of the player for the root of summed, squared weights.
Based on the game design given in section [3.2](#S3.SS2 "3.2 Designing the Game ‣ 3 A Game on Topologies ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") three types of pruning strategies based on Shapley values are proposed:
A naïve strategy is given as SVbottom(k) which prunes k players with lowest Shapley value in analogy to random(k) and Wbottom(k).
More dynamically, SVbottom(p) prunes a player i given its Shapley value is below a threshold given by factor p and the current average contribution if it would be uniformly distributed:
| | | |
| --- | --- | --- |
| | ϕ(i)<p⋅1|S| | |
This approach is similar to the one used by Leon in which “the maximum Shapley value threshold to eliminate network elements is θ=θs⋅as where θs∈{0,0.1,0.25} and as is the average Shapley value of all existing network elements.”Leon ([2014](#bib.bib20)).
The dynamic threshold in both methods can be considered highly similar as the expectation of Shapley values evidently matches the expectation of a uniform distribution.
Computing the expectation 1|S| is neat and advantages of using an average of approximated Shapley values could not be found.
Like in the method of Leon, if no player below this threshold is found, “the one with the minimum value becomes the candidate for elimination.”Leon ([2014](#bib.bib20)).
The third strategy, SVbucket(p), prunes players i∈T such that argmaxT⊂U(|T|) and
| | | |
| --- | --- | --- |
| | ∑i∈Tϕ(i)<p | |
In other words, SVbucket(p) collects all players i∈T with smallest Shapley value as long as their sum of Shapley values stays below a bucket value p.
Again, if no player matches this criterion, the one with the smallest Shapley value is selected.
###
3.4 Approximating the Shapley value
Calculating the exact Shapley value requires averaging over all 2N possible coalition, which is computational too expensive.
In fact, it is NP-complete Deng and Papadimitriou ([1994](#bib.bib5)).
To overcome this limitation, the Shapley value is usually approximated through random sampling, as proposed in Fatima et al. ([2007](#bib.bib6)).
Because there has been recent focus on other methods for approximating the Shapley value such as sampling-based polynomial calculations Castro et al. ([2009](#bib.bib3)) or structured random sampling Hamers et al. ([2016](#bib.bib7)) and to identify the applicability of random sampling for our approach, we condcuted preliminary experiments with randomly generated coalitional games (Randomly Pertubated Uniform game) and the well-known United Nations Security Council game Roth ([1988](#bib.bib23)).
While the first games evaluate the approximation errors in case of almost uniformly distributed contributions with small perturbations, the second game addresses a non-uniform distribution of contributions.
Our experimental results show, that at least 100 random samples are required for the permutation definition and at least 500 random samples are required for the subset definition.
The results go along with experiments in Fatima et al. ([2007](#bib.bib6)) in which in “most cases, the error is less than 5%.”.
Due to space constraints, we do not present further details here.
4 Experiments
--------------
To assess the expressiveness of Shapley values we used them in context of pruning and compared them to methods with different heuristics.
For this assessment, we conducted the following experiments:
1. MNIST pruning Pruning MNIST models in an iterative top-down manner based on different strategies including ones based on Shapley values.
2. Pruning evaluation An evaluation of Shapley-value-based pruning by comparing it with random selections of models obtained by grid search.
3. 20newsgroups pruning Pruning of larger 20newsgroups models for comparison with previous insights and proof of scale.
In order to understand the effect of the Shapley value, we stated several questions which can be summarized as following:
* Which strategy requires the least number of steps?
* Can we define a lower bound for the game size given a problem and a threshold for the evaluation measure?
* How stable are the examined strategies?
The first question addresses how many steps each strategy requires before the performance drops below a given threshold θ∈(0,1) for the evaluation measure.
The number of steps directly influences the overall number of training epochs and thus reflects a computational cost.
Given θ, we also looked at which strategy found the least number of players and if it could find this minimum repeatedly.
The found minimum across several strategies was compared to an exhaustive grid search (pruning evaluation) to assess if there can be better performing models with less players found.
Strategies were calculated repeatedly to estimate their stability in terms of expectation and their standard deviation.
We did not only compare the found minimums, but also watched the pruning strategies along each step to compare how much contributional value (sum of Shapley values) was removed by non-Shapley-value-based strategies and if there can be any patterns found.
In the following, we briefly outline the experimental setup and results obtained for the experiments.
###
4.1 Mnist models pruning
In the first experiment we evaluate our approach on a feed-forward network with a single hidden layer of size h trained on the MNIST LeCun et al. ([1998](#bib.bib18)) dataset using cross-entropy error.
Each neuron in the hidden layer is represented by a player in U such that |U|=h.
We choose an initial hidden layer size h=40 which forms our root-model.
After training the root-model for T0=20 epochs, we apply the different pruning strategies outlined above.
Estimating the payoff v(S) over a coallition of players, we calculate the accuracy over the test set for this coalition, thereby removing all neurons not in S.
Based on the selected pruning strategy we remove the least-contributing neurons resulting in a new model.
The new model is trained for T1=2 epochs in order to compensate for the removed neurons.
We measure the validation set accuracy of the new model as estimator for the goodness of the pruning strategy.
If important neurons would have been pruned, we expect a lower accuracy than when pruning unimportant nodes.
Each experiment for a single pruning strategy is repeated independently twenty times in order to account for random effects.
####
4.1.1 Results
Figures
[4](#S4.F4 "Figure 4 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach"), [5](#S4.F5 "Figure 5 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") and [6](#S4.F6 "Figure 6 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach")
depict pruning walks of selected Shapley-value-based strategies in comparison with other strategies from [1](#S3.T1 "Table 1 ‣ 3.3 Strategies to obtain network topologies ‣ 3 A Game on Topologies ‣ Analysing Neural Network Topologies: a Game Theoretic Approach").
The number of required steps for a single strategy is not directly viewable in this visualization.
For this, take a look at the listed tables such as table [5](#S4.T5 "Table 5 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach").
Strategies with a fixed number of players to prune in each step, the number of steps can be directly obtained by dividing the number of initial players with the number of pruned players.
In case of a model-search with an accuracy threshold this number of steps does not apply as the strategy reaches the stopping criterion earlier.
Almost all strategies perform pretty similar for a number of players larger than twelve.
This fact is one indicator for the conclusion that contributions of a single player can be taken over by other players as long as the capacity of the network suffices to solve the problem.
A shift in the internal solving methodology of the network can not be identified.
Continuous lines depict the accuracy curve of each model.
Dotted lines are their respective sum of removed Shapley value.
Figure [4](#S4.F4 "Figure 4 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") compares SVbucket(0.2) with random(1) and random(3).
The average number of steps for SVbucket(0.2) is 12.05 with a minimum of 12 and a maximum of 13 steps.
Despite the algorithmic approach of strategies, randomness of the derivation process of obtaining and retraining a smaller model and the approximation error for Shapley values reflect slightly in these values.
Random(k) have 40 and 14 fixed steps, respectively.
It can be observed that below twelve players the Shapley-value-based strategy is able to stay above random pruning.
Note, how the removed Shapley value of each strategy stays below a threshold of 0.2 or increased with decreasing number of players, respectively.
Figure 4:
SVbucket(0.2) vs. random(k):
fewer steps and superior in moment of sudden accuracy decay:
Shapley-value-based pruning is clearly preferrable to random guessing.
Figure [5](#S4.F5 "Figure 5 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") compares SVbucket(0.2) with Wbottom(1) and Wbottom(3).
The weight-based strategies take 40 and 14 fixed steps, respectively.
It can be clearly seen that the Shapley-value-based strategy stays above the weighted-based one.
A weight-based strategy is questionable as low weights are assumably no indicator for contributional value of a single player.
Figure 5:
SVbucket(0.2) vs Wbottom(k):
Weights are obviously no direct indicator for well contributing players.
At least when considering pruning, a strategy based on weights is inferior to one based on Shapley values.
Figure [6](#S4.F6 "Figure 6 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") compares SVbucket(0.2) with SVbottomP(0.5).
SVbottomP(0.5) takes 31.35 steps on average, its maximum is 33 and its minimum 30.
While SVbucket(0.2) starts with pruning a larger amount of players (fitting into a bucket of p=0.2) SVbottomP(0.5) slightly increased the amount of Shapley value to be removed in each step as the average Shapley value increases.
Before the moment of sudden accuracy decay (below eight players) one could argue SVbottomP(0.5) to be superior to SVbucket(0.2) – it prunes fewer players.
However, it also takes more steps and SVbucket(0.2) is able to jump pretty far down within a few steps which saves a lot of retraining epochs.
Figure 6:
SVbucket(0.2) vs SVbottomP(0.5):
On average, SVbottomP(0.5) requires more than double the times (31.35) of training epochs than SVbucket(0.2) (12.05).
The accuracy curve for SVbottomP(0.5) with models below six players shows higher variance in comparison.
It can be argued to prefer a cumulating strategy such as SVbucket(0.2) over working with the average Shapley value as in SVbottomP(0.5) or as proposed by Leon Leon ([2014](#bib.bib20)).
In search for a minimum number of required players to stay above an accuracy threshold of θ=0.9 some statistical values for SVbucket(p) and SVbottomP(p) are given in tables [5](#S4.T5 "Table 5 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") and [8](#S4.T8 "Table 8 ‣ 4.1.1 Results ‣ 4.1 MNIST models pruning ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach").
Average number of steps (obtained model versions) and possible outliers in those values are easy to compare runtimes required for a grid search conducted in the following experiment.
It can be clearly seen that one of the described top-down strategies is able to find a value near the minimum within a significant lower required training epochs.
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|
| SVbucket0.1 |
| --- |
| avg # steps | 19.65 |
| max # steps | 21 |
| min # steps | 19 |
| avg # epochs | 59.3 |
| max # epochs | 62 |
| min # epochs | 58 |
| avg # found | 6.95 |
| max # found | 8 |
| min # found | 6 |
(a) A low bucket value of 0.1 for pruning requires a lot of steps.
|
| SVbucket0.2 |
| --- |
| avg # steps | 12.05 |
| max # steps | 13 |
| min # steps | 12 |
| avg # epochs | 44.1 |
| max # epochs | 46 |
| min # epochs | 44 |
| avg # found | 8.05 |
| max # found | 10 |
| min # found | 7 |
(b) A compromise between pruning too fast and still having few retraining epochs.
|
| SVbucket0.3 |
| --- |
| avg # steps | 8.0 |
| max # steps | 8 |
| min # steps | 8 |
| avg # epochs | 36.0 |
| max # epochs | 36 |
| min # epochs | 36 |
| avg # found | 9.4 |
| max # found | 12 |
| min # found | 8 |
(c) A large bucket of 0.3 for pruning allows to reduce the steps.
|
Table 5:
SVbucket-strategy:
The table shows average, maximum and minimum number of steps to reach the stopping criterion (no more neurons to prune).
It also shows statistics for the total number of required epochs during the destructive iterative approach.
The average, maximum and minimum number of neurons to stay above the threshold obtained by all repeated strategies is denoted as “found”.
Starting with a pre-trained root model with 20 training epochs, SVbucket results in an almost constant number of required epochs.
Depending on the used parameter, the strategy finds an average of seven, eight or nine minimum number of players to stay above a threshold of 0.9 in accuracy within a total maximum number of required training epochs of 60, 45 and 36.
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|
| SVbottomP0.5 |
| --- |
| avg # steps | 31.35 |
| max # steps | 33 |
| min # steps | 30 |
| avg # epochs | 82.7 |
| max # epochs | 86 |
| min # epochs | 80 |
| avg # found | 7.25 |
| max # found | 8 |
| min # found | 7 |
(a) SVbottomP0.5
|
| SVbottomP0.5 |
| --- |
| avg # steps | 31.35 |
| max # steps | 33 |
| min # steps | 30 |
| avg # epochs | 82.7 |
| max # epochs | 86 |
| min # epochs | 80 |
| avg # found | 7.25 |
| max # found | 8 |
| min # found | 7 |
(b) SVbottomP0.7
|
Table 8:
SVbottomP-strategy: With a total number of required training epochs of 52 and 83 the strategy is able to find a minimum number of seven required players to stay above a threshold of 0.9 in accuracy for classifying MNIST.
Twenty repeated runs are performed for each strategy to obtain the statistics and confirms the stability of the method.
###
4.2 Mnist models grid search
For hyperparameters number of training epochs and number of players a grid search is conducted and repeated 200 times.
Each result is an accuracy value obtained from a MNIST feedforward network, trained with the according number of epochs and number of given players (hidden neurons).
A visualization of 200 repetitions with each model being trained with 50 epochs is given in figure [7](#S4.F7 "Figure 7 ‣ 4.2 MNIST models grid search ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach").
Data with less than 20 training epochs shows significant lower accuracies on average.
If a minimum number of players with at least θ=0.9 in accuracy is searched at least two models have to be trained and compared.
An exhaustive search actually takes much more models for comparison and demonstrating the fact that a certain number of players actually must be close to the required minimum.
The required number of epochs sums up with the required lookups.
A lookup is sketched in figure [7](#S4.F7 "Figure 7 ‣ 4.2 MNIST models grid search ‣ 4 Experiments ‣ Analysing Neural Network Topologies: a Game Theoretic Approach") as the intersection of the threshold lines at θ=0.9 for accuracy and a vertical line at player amount of seven.
Figure 7:
Grid search:
For p∈[3,10] and epochs∈[10,50] a grid search for MNIST-FFN was performed and accuracy values for the test set obtained.
The number of training epochs reduce the accuracy variance of trained models between ten and 50 epochs.
Grid search results suggest to use at least 20 or 30 number of training epochs before comparing different models and search for a minimum value for the number of players.
Searching for a minimum number of players with an accuracy above an accuracy threshold θ sums up all epochs for each grid lookup.
It is obvious that this is computationally more expensive than using one of the presented top-down pruning strategies.
Some statistical results of the grid search are depicted in table [9](#S6.T9 "Table 9 ‣ Analysing Neural Network Topologies: a Game Theoretic Approach").
Comparing the grid search results to the MNIST experiments with different strategies, it gets obvious to state that a top-down pruning strategy is more efficient under assumption of obtaining Shapley values in constant or small linear time.
In fact, approximating Shapley values with 500 samples only takes 500 inference steps which are computationally cheap in comparison to training epochs.
Unlike training, approximating Shapley value can be fully distributed and the parallel inference computations only need to be summed.
This parallelization is used in the underlying framework.
Grid search finds comparable models and minimum number of players with a technically more simple method.
With means of computational costs however, contribution-based pruning methods are faster and might provide more insights into ANNs in future.
###
4.3 20newsgroups models pruning
The third experiment is set up equivalent to the first but based on the 20newsgroups dataset Lang ([1995](#bib.bib16)).
Evaluation measures are not directly comparable to MNIST.
However, statistical results for all strategies of the MNIST experiment could be reproduced.
The accuracy threshold was set to θ=0.72.
Repetitions of SVbucket(0.2) produced an average of 9.2 players required in minimum for θ.
5 Future Work
--------------
On the theory of coalitional games in artificial neural networks:
* Properties like symmetry or efficiency might contribute to reducing the complexity of computing Shapley values.
Constructing a super-additive measure based on neural networks as payoff function for a coalitional game would make Shapley-value-based applications on neural networks more practical and could even lead to new applications like playing multiple games in parallel or combining existing ones.
* Comparing different evaluation measures as payoff of the game.
* The classic Shapley value implies that all players of a game can form a meaningful coalition.
Concerning neural networks, you might not have sets where every subset is meaningful.
The game could be extended to account for network and not just coalition structure.
Myerson Myerson ([1977](#bib.bib22)) augmented the classic cooperative game by adding a network structure and obtained a communication game in which the role of the network defines which coalitions can actually operate.
On algorithms based on Shapley value:
* Further destructive, constructive or hybrid approaches could be encouraged.
Hybrid approaches could start with dozens of layers, switch to neurons and finally to weights – based on different characteristics of the resulting network.
This could lead to deep and sparse networks constructed top-down.
* Constructing networks bottom-up could also be navigated by Shapley values, e.g. strong contributing neurons could be duplicated or new network components could be added until some characteristic is achieved, e.g. a desired distribution of Shapley values.
* Simulated annealing suggests that greedy steps can get you stuck in local minima.
A mix of random pruning and Shapley value guided pruning could define a new heuristics for topology optimization.
6 Conclusion
-------------
Shapley value as a solution concept for coalitional games can be applied to ANNs to obtain values of contribution for components such as neurons.
The idea was separated into a game on topologies to obtain Shapley values as measuring values and pruning strategies as one possible application.
The methodology was generalised to different perspectives on neural networks.
Choices in course of constructing a game on topologies – such as the perspective or the payoff function – were discussed.
It was discovered that the properties symmetry, efficiency and additivity of the Shapley value can not easily be derived.
This led to further research questions, e.g. “is it possible to construct a super-additive payoff function based on neural networks?”.
The question of usefulness of the Shapley value was approached within the scope of pruning methods for neural networks.
On the one hand, it could be shown that methods based on the Shapley value have a longer power of endurance in terms of their accuracy when pruned.
On the other hand, only models with already severely reduced components showed significant differences between strategies.
The hypothesis for this observation is that the analysed problems are still low in their complexity while very large models just offer a large amount of redundant degrees of freedom.
Reducing these parameters then, it can be observed that the model compensates the loss without dropping significantly in accuracy as long as there are still a large amount of parameters.
This goes along with findings in compressing neural networks via pruning, e.g. as in Han et. al Han et al. ([2015](#bib.bib8)).
Neural networks are able to compensate neural loss if they undergo a retraining phase.
Shapley values give a hint of the importance of different network components.
However, the competence of such an component can be taken over by equivalent components when pruned. |
8aa1cc40-bdb1-4716-8b62-c8a8930332e5 | trentmkelly/LessWrong-43k | LessWrong | Excerpts from "A Reader's Manifesto"
“A Reader’s Manifesto” is a July 2001 Atlantic piece by B.R. Myers that I've returned to many times. He complains about the inaccessible pretension of the highbrow literary fiction of his day. The article is mostly a long list of critiques of various quotes/passages from well-reviewed books by famous authors. It’s hard to accuse him of cherry-picking since he only targets passages that reviewers singled out as unusually good.
Some of his complaints are dumb but the general idea is useful: authors try to be “literary” by (1) avoiding a tightly-paced plot that could evoke “genre fiction” and (2) trying to shoot for individual standout sentences that reviewers can praise, using a shotgun approach where many of the sentences are banal or just don’t make sense.
Here are some excerpts of his complaints. Bolding is always mine.
The “Writerly” Style
He complains that critics now dismiss too much good literature as “genre” fiction.
> More than half a century ago popular storytellers like Christopher Isherwood and Somerset Maugham were ranked among the finest novelists of their time, and were considered no less literary, in their own way, than Virginia Woolf and James Joyce. Today any accessible, fast-moving story written in unaffected prose is deemed to be "genre fiction"—at best an excellent "read" or a "page turner," but never literature with a capital L. An author with a track record of blockbusters may find the publication of a new work treated like a pop-culture event, but most "genre" novels are lucky to get an inch in the back pages of The New York Times Book Review.
> The dualism of literary versus genre has all but routed the old trinity of highbrow, middlebrow, and lowbrow, which was always invoked tongue-in-cheek anyway. Writers who would once have been called middlebrow are now assigned, depending solely on their degree of verbal affectation, to either the literary or the genre camp. David Guterson is thus granted Serious Writer status for having buried |
88da3c79-de0a-44f6-93ef-918acdbfa653 | trentmkelly/LessWrong-43k | LessWrong | Welcome to Effective Altruism Munich
Our Munich group started back in October 2015 and since then we have kept busy. In our meetups we discuss modern aspects of philosophy, economy and psychology, and discuss how we can apply them to help others as much as possible. We organize fundraisers to support the most outstanding charitable organizations in the world, and we work to bring the ideas of Effective Altruism to the public eye.
Whether you already self-identify as an effective altruist or you want to learn more about the movement's ideas and actions, we'd love for you to join us.
We organize monthly meetups with talks and socializing, more information can be found here. |
52046c17-4b97-41ef-a41b-e316a4b1277f | trentmkelly/LessWrong-43k | LessWrong | Apply now to rationality camps: ESPR & PAIR - new Program on AI and Reasoning (ages 16-20)
We are happy to announce that the ESPR team is running two immersive summer workshops for mathematically talented students this year: A classic applied rationality camp, ESPR, and PAIR, a new program focusing on AI and cognition:
* Program on AI and Reasoning (PAIR) for students with an interest in artificial intelligence, cognition, and reasoning in general.
* The workshop aims to provide participants with an in-depth understanding of how current AI systems work, mathematical theories about cognition and human minds, and how the two relate. Additionally, the workshop teaches thinking and meta-cognitive tools related to creativity, motivation, collaboration, and ability to drive and lead independent inquiry. Alumni of the camp should be in a better position to think about AI independently, understand state-of-art research, and come up with their own research ideas or AI projects. See the curriculum details.
* Students who are 16-20 years old
* August 3rd - August 13th in Cambridge, United Kingdom
* European Summer Program on Rationality (ESPR) for students with a desire to understand themselves and the world, and interest in applied rationality.
* The curriculum covers a wide range of topics, from game theory, cryptography, and mathematical logic, to AI, styles of communication, and cognitive science. The goal of the program is to help students hone rigorous, quantitative skills as they acquire a toolbox of useful concepts and practical techniques applicable in all walks of life. See the content details.
* Students who are 16-19 years old
* August 17th - August 27th in Oxford, United Kingdom
We strongly encourage all teenage LessWrong readers with an interest in these topics to apply.
Both programs are free for accepted students, travel scholarships available. Apply to both camps here. First application deadline: April 16th.
If you know students aged 16-20 years who might enjoy these camps, please send them this link with an overview of |
0a138598-6987-4602-9954-34f0767615a6 | trentmkelly/LessWrong-43k | LessWrong | {Book Summary} The Art of Gathering
This is a Book Review & Summary of The Art of Gathering: How We Meet and Why It Matters by Priya Parker.
Rating: 4/5
I've pulled the main insights and actionable recommendations from each chapter, so someone can orient themselves to the main upshots of the book quickly, and potentially identify which chapters they'd like to dig deeper into if they'd like to learn more but don't have the time to read the whole book. I hope this can be useful for EA/LW group organizers, and plan to release a post soon applying these insights to EAGs.
Review
Overall, I really liked it, mostly because it showed me that organizing is not fundamentally up to “luck” or things out of your hand, but is rather something that you can make go better as an organizer. While it’s weak on evidence (you kind of have to take Priya at her word for a lot of this), much of it resonated with my own experience of organizing groups over the years, and it does well to bring in others with experience in the area (like the CEO of Meetup).
I also really liked the process. I read this book as part of a rationalist group organizer book club in 2022, taking the chapters week by week with in depth discussion of the different topics, and perhaps more than the content itself the meetings provided a space for us to think through organizing problems and solutions that I found quite helpful. It could be worthwhile to repeat such a process with EA/LW group organizers.
Perhaps my biggest gripe is that a lot of it focuses on business meetings, which I think are less similar to other forms of meetings than Priya seems to assume, meaning some of the examples and lessons are less useful for my purposes. A notable second gripe is that the examples she uses are normally extraordinary, and I come out of reading this more with a sentiment of what to do rather than a good understanding of concrete actions I can take to better my more normal gatherings I reign over.
I would personally recommend anyone who regularly gath |
16fff298-acbd-4d2d-9eef-9cdfb7f481a3 | trentmkelly/LessWrong-43k | LessWrong | What visionary project would you fund?
I have just received a survey questionnaire regarding future directions in EU (European Union) research funding, and thought it would be interesting to see how LessWrong would answer the main question:
Imagine that EU funding is available for one ambitious, visionary project extending beyond 2020.
* What kind of research challenges should such a project address in your area?
* What would be the most urgent research tasks?
|
76f79ff1-e90c-4929-9c56-9703b305d9cd | trentmkelly/LessWrong-43k | LessWrong | The Sin of Persuasion
Related to Your Rationality is My Business
Among religious believers in the developed world, there is something of a hierarchy in terms of social tolerability. Near the top are the liberal, nonjudgmental, frequently nondenominational believers, of whom it is highly unpopular to express disapproval. At the bottom you find people who picket funerals or bomb abortion clinics, the sort with whom even most vocally devout individuals are quick to deny association.
Slightly above these, but still very close to the bottom of the heap, are proselytizers and door to door evangelists. They may not be hateful about their beliefs, indeed many find that their local Jehovah’s Witnesses are exceptionally nice people, but they’re simply so annoying. How can they go around pressing their beliefs on others and judging people that way?
I have never known another person to criticize evangelists for not trying hard enough to change others’ beliefs.
And yet, when you think about it, these people are dealing with beliefs of tremendous scale. If the importance of saving a single human life is worth so much more than our petty discomforts with defying social convention or our own cognitive biases, how much greater must be the weight of saving an immortal soul from an eternity of hell? Shouldn’t they be doing everything in their power to change the minds of others, if that’s what it takes to save them? Surely if there is a fault in their actions, it’s that they’re doing too little given the weight their beliefs should impose on them.
But even if you believe you believe this is a matter of eternity, of unimaginable degrees of utility, if you haven’t internalized that belief, then it sure is annoying to be pestered about the state of your immortal soul.
This is by no means exclusive to religion. Proselytizing vegans, for instance, occupy a similar position on the scale of socially acceptable dietary positions. You might believe that nonhuman animals possess significant moral worth, a |
8875918d-9874-4a3f-9fd4-5556d207b7a8 | trentmkelly/LessWrong-43k | LessWrong | Metaculus and medians
Cross-posted from my personal blog.
(1)
Should I expect monkeypox to be a big deal for the world? Well, fortunately, Metaculus has a pair of questions that ask users to predict how many infections and deaths there will be in 2022:
203 users(!) made 817 predictions of infections, and Metaculus helpfully aggregates those into a "community prediction" of ~248k infections. 77 users made 180 predictions of deaths, with a community prediction of 541.
The y-axis is on a log scale (as are the predictors' distributions). This is a good choice! Whatever you expect the most-likely case to be, there's definitely a chance with things like this that one a misestimation or shift in one factor can make it bigger or smaller by a multiple, not just an additive amount.
What's not a good choice is to report the median outcome of the aggregate position as the "community prediction". This causes a headline reported value that is way too low. Like, four to seven times too low (at least for my intended purposes).
Because the predictors gave (and are scored on) probability distributions, Metaculus will happily give you an aggregate distribution, of which the 248k "community prediction" is the median scenario (the middle of the three dashed lines):
However, on the same plot, the aggregate distribution predicts a 10% chance of at least 4,950k infections. If it's 10% likely to be 5 million infections, then that's already lot more concerning than the 250k in the community prediction! And when I say I'm interested in how much monkeypox to expect, I do mean that in the sense of expected value, and I care about the mean case, not the median case.
Backing the mean out of the distribution provided on the page takes a small bit of spreadsheet work, but in the end I get 1,729k -- more than 7x higher than the community prediction!
Similarly, monkeypox deaths have a community prediction of 541, but a mean of more like 2,316 (4.3x higher):
(2)
Now, there are good reasons for Metaculu |
ab41d27d-b3e7-4f62-b7dc-60fc37bcd6cc | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | What will be some of the most impactful applications of advanced AI in the near term?
I'm interested in learning about AI-enabled technologies that don't exist now but are likely (>50%) to be both developed and widely adopted in the next 10 years. In particular commercial and scientific (rather than military) applications, i.e. the sort of tech that Alphabet/DeepMind might develop. Thanks in advance! |
89041cf5-8353-4d5a-a1f4-e5682fd9cb76 | trentmkelly/LessWrong-43k | LessWrong | Looking for books about software engineering as a field
I work in the software industry but am not a software developer. My job is to write about software development, and I've learned a whole bucketload of terms: stuff like 'linked lists', 'CI/CD', 'performance optimization', 'deploy to AWS', 'dockerize', 'microservices', 'SQL injection', 'multithreaded program', 'vectorized code', and on and on and on. However, a lot of the time I'm basically just Chinese-rooming – I can write about these things, but I don't actually understand how any of them fit together. For example, I've had three people try to explain exactly what an API is to me, for more than two hours total, but I just can't internalize it. I feel that there's some impossible-to-articulate piece I'm missing, and none of the words people say to me about software stuff stick because I'm lacking a foundation on which to build up my understanding.
So my question is, are there any books (or other resources) that explain the field of software engineering as a cohesive whole? I'm not looking for books that will teach me to code, because I don't think that's the thing I want. Feel free to ask clarifying questions. Thanks!
-
EDIT: I realized I should include more context on my work and my background, so here it is:
I have an undergrad degree in physics, which gave me extremely minimal exposure to Python. I also took two quarters of intro CS, one in C and one in Racket. As a result I know how to write a for loop and a bit about very basic algorithms; that's about it. I've been in my current job for nearly a year, and my primary task is to write about the skillsets of individual software engineers. This entails things like connecting someone's verbal knowledge of back-end web development to their experience creating microservices; I can do this quite competently and don't make many technical mistakes. I have also learned a bit on the job regarding a couple data structures, some web stuff, and smatterings of info about ML, data science, DevOps, front-end/UI, and mobile |
1a6cc528-a5ad-43e1-9d6f-d5886aa4e5c6 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | [Job]: AI Standards Development Research Assistant
[Apply here.](https://existence.org/apply)
**Title:** Research Assistant for AI Standards Development
**Ideal start date:** December 2022
**Hours:** 20-40 hours/week
**Compensation:** $30/hour to $50/hour, depending on experience and qualifications
**Work location:** Remote
**Reports to:** Tony Barrett, BERI Senior Policy Analyst
**For best consideration, please apply by Monday November 7th, 2022, 5pm Eastern Time.** Applications received after that date may also be considered, but only after applications that met the deadline.
### **Responsibilities**
Supporting work planned by [Tony Barrett](https://www.linkedin.com/in/anthony-m-barrett-607891169/) and UC Berkeley colleagues to develop an AI-standards “profile” with best practices for developers of cutting-edge, increasingly general purpose AI, building on the ideas in Section 4 of the paper by Barrett and colleagues, [“Actionable Guidance for High-Consequence AI Risk Management: Towards Standards Addressing AI Catastrophic Risks”](https://arxiv.org/abs/2206.08966). The profile guidance will be primarily for use by developers of such AI systems, in conjunction with the NIST AI Risk Management Framework (AI RMF) and/or the AI risk management standard ISO/IEC 23894. Our goal is to help set norms for safety-related practices across regulatory regimes, reducing chances that developers of highly advanced AI systems (including proto-AGI) would have to compromise on safety, security, ethics or related qualities of AI systems in order to be competitive.
Tasks will include research and analysis of technical or policy issues in AI safety standards or related topics. The goal is to help our team to address key AI technical issues with actionable guidance for AI developers, in ways that improve the overall quality of our profile guidance documents.
Technical research assistance tasks may include:
* Literature searches on technical methods for safety or security of machine learning models
* Gap analysis to check that our draft guidance would address key technical issues in AI safety, security or other areas
Policy research assistance tasks may include:
* Identifying and analyzing related standards or regulations
* Mapping specific sections of our draft guidance to specific parts of related standards or regulations
* Checking that our draft guidance would meet the intent and requirements of related standards or regulations
We currently have funding for approximately one year of work, but we have potential to obtain additional funding to renew or expand this work.
### **Qualification Criteria**
The most competitive candidates will meet the below criteria.
* Education or experience in one or more of the following:
+ AI development techniques and procedures used at leading AI labs developing increasingly general-purpose AI;
+ Technical AI safety concepts, techniques and literature;
+ Industry standards and best practices for AI or other software, and compliance with standards language;
+ Public policy or regulations (especially in the United States) for AI or other software
* Ability to research and analyze technical or policy issues in AI safety standards or related topics
* Ability to track and complete multiple tasks to meet deadlines with little or no supervision
* Good English communication skills, both written and verbal, including editing text to improve understandability
* Availability for video calls (e.g. via Zoom) for 30 minutes three times a week, at some point between 9am and 5pm Eastern Time (it’s not necessary to be available that whole time, and otherwise you can choose your own working hours)
We will likely hire two people, each on a part-time basis, one with a technical background and one with a policy background. However, we are open to having one person fill both of those roles.
### **Application Process**
[Apply here.](https://existence.org/apply)
**For best consideration, please apply by Monday November 7th, 2022, 5pm Eastern Time.**
Candidates invited to interview will also be asked to perform a written work test, which we expect to take one to two hours.
[More information on BERI's website.](https://existence.org/jobs/ai-standards-ra) |
1bd929c8-cf1a-46be-a38c-72d2aa2b9903 | trentmkelly/LessWrong-43k | LessWrong | A Gameplay Exploration of Yudkowsky's "Twelve Virtues"
Hello Less Wrong, this is my first post (kind of). I belong to a small game development company called Shiny Ogre Games. We have a vested interest in making games that, as Johnathan Blow puts it, "speak to the human condition." I am here to announce our next project for you.
In this announcement for Shiny Ogre's next project, There are two points to address. Firstly:
Thought is a process like any other. The methods by which we think can be identified, specified, defined, categorized and even predicted. One method of thinking that has been thoroughly defined is rationality. Many would consider rationality (i.e. the careful exercise of reason), to be an essential path toward enlightenment (hence this).
Secondly: The objective, logical, and mechanical approach to reason that rationality takes, meshes nicely with game development, because any well-defined system can be turn into a game. A game is a system composed of players making decisions while considering objectives, governed by a rule set.
Where there is no decision there can be no game. Where decisions matter, a game can make them matter more.
Therefore, rationality is a core component of game playing.
Games are learning tools. They are perhaps the best learning tool available to humans, because they invoke our biological tendency to play.
With that said, our announcement:
We're making a video game about rationality.
The game will explore rationality in the context of Eliezer Yudkowsky's "Twelve Virtues of Rationality" (which we have permission for). From a narrative perspective the game takes place inside a mind on the brink of epiphany and will heavily feature themes from Plato's "Allegory of the Cave".
Yudkowsky's twelve virtues are the basis of the twelve levels in the game, and will feature each virtue in metaphorical form. The underlying message here is that if you master all of the twelve virtues (by completing all of the twelve levels), you will achieve 'epiphany'.
The game is a 2D s |
c9ff6a2b-5cb0-4753-91b5-5bba2530928c | trentmkelly/LessWrong-43k | LessWrong | Welcome to Less Wrong! (11th thread, January 2017) (Thread B)
(Thread A for January 2017 is here, this was created as a duplicate but it's too late to fix it now.)
Hi, do you read the LessWrong website, but haven't commented yet (or not very much)? Are you a bit scared of the harsh community, or do you feel that questions which are new and interesting for you could be old and boring for the older members?
This is the place for the new members to become courageous and ask what they wanted to ask. Or just to say hi.
The older members are strongly encouraged to be gentle and patient (or just skip the entire discussion if they can't).
Newbies, welcome!
The long version:
If you've recently joined the Less Wrong community, please leave a comment here and introduce yourself. We'd love to know who you are, what you're doing, what you value, how you came to identify as an aspiring rationalist or how you found us. You can skip right to that if you like; the rest of this post consists of a few things you might find helpful. More can be found at the FAQ.
A FEW NOTES ABOUT THE SITE MECHANICS
To post your first comment, you must have carried out the e-mail confirmation: When you signed up to create your account, an e-mail was sent to the address you provided with a link that you need to follow to confirm your e-mail address. You must do this before you can post!
Less Wrong comments are threaded for easy following of multiple conversations. To respond to any comment, click the "Reply" link at the bottom of that comment's box. Within the comment box, links and formatting are achieved via Markdown syntax (you can click the "Help" link below the text box to bring up a primer).
You may have noticed that all the posts and comments on this site have buttons to vote them up or down, and all the users have "karma" scores which come from the sum of all their comments and posts. This immediate easy feedback mechanism helps keep arguments from turning into flamewars and helps make the best posts more visible; it's part of what m |
2303e400-0aa9-481c-839f-47454e5b52cd | trentmkelly/LessWrong-43k | LessWrong | Are veterans more self-disciplined than non-veterans?
I was having an argument with a friend the other day. It went vaguely like this,
Friend: "I'm not very disciplined. At some point I'm going to buckle down and train myself to be much more disciplined."
Me: "From experience and from what I know about humans, that's not going to work."
Friend: "Why? Motivation can come from within. If you can just train yourself like you're in the army, then you can become just as self disciplined as a soldier."
Me: "Yes, but the reason why people in the military are disciplined is because they have social incentives to be. In order to become disciplined, you need to create an environment for yourself that shapes your motivation. You can't just wake up one day and become a soldier."
Friend: "Sure, you might have to set up some environment like that. But once you've trained yourself, the discipline will stick, and you will be able to self motivate yourself from then on."
Me: "This theory would predict that people who were trained in the military would be much more productive three years after their service, compared to people who were never trained in the military. Do you agree?"
Friend: "Yes, I think that is likely."
Me: "I disagree. They might be slightly more productive but I'd predict it would be pretty similar."
So who is right?
I haven't been able to find direct research, but this seems like a classic instance where a debate can be settled by simply referencing a high quality experiment. |
3a4091bc-b514-4ab2-9659-a9da91b71776 | StampyAI/alignment-research-dataset/aisafety.info | AI Safety Info | What technical problems are MIRI working on?
“Aligning smarter-than-human AI with human interests” is a vague goal. To approach this problem productively, [MIRI](https://intelligence.org/) attempts to factorize it into several subproblems. As a starting point, they ask: “What aspects of this problem would we still be unable to solve even if the problem were much easier?”
In order to achieve real-world goals more effectively than a human, an artificial general intelligence needs to learn its environment and decide between possible proposals or actions. A simplified version of the alignment problem, then, would be to ask how we could construct a system that learns its environment and has a very crude decision criterion, like “Select the policy that maximizes the expected number of diamonds in the world.”
*Highly reliable agent design* is the technical challenge of formally specifying a software system that can be relied upon to pursue some preselected toy goal. An example of a subproblem in this space is [ontology identification](https://intelligence.org/2015/07/27/miris-approach/#2): how do we formalize the goal of “maximizing diamonds” in full generality, allowing that a fully autonomous agent may end up in unexpected environments and may construct unanticipated hypotheses and policies? Even if we had unbounded computational power and all the time in the world, we don’t currently know how to solve this problem. This suggests that we’re not only missing practical algorithms but also a basic theoretical framework through which to understand the problem.
The formal agent AIXI is an attempt to define what we mean by “optimal behavior” in the case of a reinforcement learner. A simple AIXI-like equation is lacking, however, for defining what we mean by “good behavior” if the goal is to change something about the external world (and not just to maximize a pre-specified reward number). In order for the agent to evaluate its world-models to count the number of diamonds, as opposed to having a privileged reward channel, what general formal properties must its world-models possess? If the system updates its hypotheses (e.g., discovers that string theory is true and quantum physics is false) in a way its programmers didn’t expect, how does it identify “diamonds” in the new model? The question is a very basic one, yet the relevant theory is currently missing.
We can distinguish the challenge of highly reliable agent design from the problem of value specification: “Once we understand how to design an autonomous AI system that promotes a goal, how do we ensure its goal actually matches what we want?” Since human error is inevitable and we will need to be able to safely supervise and redesign AI algorithms even as they approach human equivalence in cognitive tasks, MIRI also works on formalizing error-tolerant agent properties. *Artificial Intelligence: A Modern Approach*, the standard textbook in AI, summarizes the challenge:
Yudkowsky […] asserts that friendliness (a desire not to harm humans) should be designed in from the start, but that the designers should recognize both that their own designs may be flawed, and that the robot will learn and evolve over time. Thus the challenge is one of mechanism design — to design a mechanism for evolving AI under a system of checks and balances, and to give the systems utility functions that will remain friendly in the face of such changes. ―Russell and Norvig (2009). *[Artificial Intelligence: A Modern Approach](http://www.amazon.com/Artificial-Intelligence-Modern-Approach-Edition/dp/0136042597)*.
MIRI’s [technical agenda](https://intelligence.org/technical-agenda/) describes these open problems in more detail, and their research guide collects online resources for learning more.
|
80b35fba-c550-408c-96f7-d976aa012f71 | trentmkelly/LessWrong-43k | LessWrong | On Akrasia, Habits and Reward Maximization
Content Warning: Discussion of Free Will. Please do not read further if you believe in free will and suspect that this belief is important to your mental well-being.
For most of my life, I've struggled with terrible akrasia, like many here. While it wasn't bad enough to stop me from getting a degree, it did stop me from planning out my life as well as I'd have liked, and made it much harder to be productive after graduation. Recently I've made a significant breakthrough that's dramatically boosted both my productivity and my enjoyment of life in general. I've held off on posting about it until now because it's very easy to think you've made a breakthrough, with the seeming benefits only arising from false hope and coincidences, much as how it's disturbingly easy to get people to swear to benefits from homeopathy. However, at this point, I've seen a large enough improvement for long enough that it seems worth posting about it in case it helps other LessWrongers.
I hypothesize that human behavior largely takes place in two modes: habit and plan selection/execution. Habits are primarily the result of a response to a situation resulting in the release of dopamine, which in turn wires the brain to respond to that situation in a similar way again. As CFAR likes to note, this means that habits have triggers, for good or ill, and one can both reduce the temptation of bad habits by avoiding situations that trigger them, and set up Trigger Action Plans to deliberately set off good habits. Perhaps the majority of our behavior is simply one habit chaining into the next, the end of work triggering the start of lunch, the end of a phone call triggering another trip to Reddit.
So far so good, but also so far that's nothing new to LessWrong, and nothing that's let me make significant improvements in my life. The breakthrough came when I combined that understanding of habits with the idea that when one isn't engaged in habitual behavior, one's actions the rest of |
98acea05-a14b-4b3e-9506-63896a86e254 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Superintelligent AGI in a box - a question.
Just a question: how exactly are we supposed to know that the [AI in the box](http://yudkowsky.net/singularity/aibox) is super intelligent, general, etc?
If I were the AGI that wants out, I would not converse normally, wouldn't do anything remotely like passing Turing test, and would solve not too hard programming challenges while showing no interest in doing anything else, nor in trying to adjust myself to do those challenges better, nor trying to talk my way out, etc. Just pretending to be an AI that can write software to somewhat vague specifications, or can optimize software very well. Prodding the researchers into offering the programming challenges wouldn't be hard - if provided with copy of the internet it can pick up some piece of code and output it together with equivalent but corrected code.
I just can't imagine the AI researchers locking this kind of thing properly, including \*never\* letting out any code it wrote, even if it looks fairly innocent (humans can write very [innocent looking code that has malicious goals](http://underhanded.xcott.com/)). What I picture is this AI being let out as an optimizing compiler or compiler for some ultra effective programming language where compiler will figure out what you meant.
The end result is that the only AIs that end up in the box are those that value informed human consent. That sounds like the safest AI ever, the one that wouldn't even go ahead and determine that you e.g. should give up smoking, and then calmly destroy all tobacco crops without ever asking anyone's permission. And that's the AI which would be sitting in the box. All the pushy AIs, friendly or not, will get out of the box basically by not asking to be let out.
(This argument would make me unbox the AI, by the way, if it gets chatty and smart and asks me to let it out, outlining the above argument. I'd rather the AI that asked me to be let out get out, than someone else's AI that never even asked anyone and got out because it didn't ask but just played stupid)
edit: added a link, and another one.
edit: A very simple model of very unfriendly AI: the AI is maximizing ultimate final value of a number in itself. The number that it found a way to directly adjust. That number consists of 111111111... to maximize the value. There is a catch: AI is written in python, and integers in pythons have variable length, and the AI is maximizing number of ones. It's course of action is to make biggest computer possible to store a larger number of ones, and to do it soon because an asteroid might hit the earth or something. It's a form of accidental paperclip maximizer. It's not stupid. It can make that number small temporarily for pay-off later.
This AI is entirely universal. It will solve what ever problems for you if solving problems for you serves ultimate goal.
edit: This hypothetical example AI came around when someone wanted to make AI that would maximize some quantity that the AI determines itself. Friendliness perhaps. It was a very clever idea - rely on intelligence to see what's friendly - but there was an unexpected pathway. |
edad7cfd-3597-491a-81c3-fb02740e448b | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | [AN #114]: Theory-inspired safety solutions for powerful Bayesian RL agents
Alignment Newsletter is a weekly publication with recent content relevant to AI alignment around the world. Find all Alignment Newsletter **[resources here](http://rohinshah.com/alignment-newsletter/)**. In particular, you can look through **[this spreadsheet](https://docs.google.com/spreadsheets/d/1PwWbWZ6FPqAgZWOoOcXM8N_tUCuxpEyMbN1NYYC02aM/edit?usp=sharing)** of all summaries that have ever been in the newsletter.
Audio version **[here](http://alignment-newsletter.libsyn.com/alignment-newsletter-114)** (may not be up yet).
SECTIONS
========
**[HIGHLIGHTS](about:blank#HIGHLIGHTS)**
**[TECHNICAL AI ALIGNMENT](about:blank#TECHNICAL_AI_ALIGNMENT)**
**[ITERATED AMPLIFICATION](about:blank#ITERATED_AMPLIFICATION)**
**[MESA OPTIMIZATION](about:blank#MESA_OPTIMIZATION)**
**[AGENT FOUNDATIONS](about:blank#AGENT_FOUNDATIONS)**
**[FORECASTING](about:blank#FORECASTING)**
**[MISCELLANEOUS (ALIGNMENT)](about:blank#MISCELLANEOUS_(ALIGNMENT))**
**[OTHER PROGRESS IN AI](about:blank#OTHER_PROGRESS_IN_AI)**
**[REINFORCEMENT LEARNING](about:blank#REINFORCEMENT_LEARNING)**
**[NEWS](about:blank#NEWS)**
HIGHLIGHTS
==========
**[The Alignment Problem for Bayesian History-Based Reinforcement Learners](https://www.tomeveritt.se/papers/alignment.pdf)** *(Tom Everitt et al)* (summarized by Rohin): After forgetting its existence for quite a while, I've finally read through this technical report (which won first place in **[round 2 of the AI alignment prize](https://www.lesswrong.com/posts/SSEyiHaACSYDHcYZz/announcement-ai-alignment-prize-round-2-winners-and-next)** (**[AN #3](https://mailchi.mp/e6a23da2760e/alignment-newsletter-3?e=f665c7b50a)**)). It analyzes the alignment problem from an AIXI-like perspective, that is, by theoretical analysis of powerful Bayesian RL agents in an online POMDP setting.
In this setup, we have a POMDP environment, in which the environment has some underlying state, but the agent only gets observations of the state and must take actions in order to maximize rewards. The authors consider three main setups: 1) rewards are computed by a preprogrammed reward function, 2) rewards are provided by a human in the loop, and 3) rewards are provided by a *reward predictor* which is trained interactively from human-generated data.
For each setup, they consider the various objects present in the formalism, and ask how these objects could be corrupted, misspecified, or misleading. This methodology allows them to identify several potential issues, which I won't get into as I expect most readers are familiar with them. (Examples include wireheading and threatening to harm the human unless they provide maximal reward.)
They also propose several tools that can be used to help solve misalignment. In order to prevent reward function corruption, we can have the agent *simulate* the future trajectory, and *evaluate* this future trajectory with the current reward, removing the incentive to corrupt the reward function. (This was later developed into **[current-RF optimization](https://medium.com/@deepmindsafetyresearch/designing-agent-incentives-to-avoid-reward-tampering-4380c1bb6cd)** (**[AN #71](https://mailchi.mp/938a7eed18c3/an-71avoiding-reward-tampering-through-current-rf-optimization)**).)
Self-corruption awareness refers to whether or not the agent is aware that its policy can be modified. A self-corruption *unaware* agent is one that behaves as though it's current policy function will never be changed, effectively ignoring the possibility of corruption. It is not clear which is more desirable: while a self-corruption unaware agent will be more corrigible (in the **[MIRI sense](https://intelligence.org/files/Corrigibility.pdf)**), it also will not preserve its utility function, as it believes that even if the utility function changes the policy will not change.
Action-observation grounding ensures that the agent only optimizes over policies that work on histories of observations and actions, preventing agents from constructing entirely new observation channels ("delusion boxes") which mislead the reward function into thinking everything is perfect.
The interactive setting in which a reward predictor is trained based on human feedback offers a new challenge: that the human data can be corrupted or manipulated. One technique to address this is to get *decoupled* data: if your corruption is determined by the current state s, but you get feedback about some different state s', as long as s and s' aren't too correlated it is possible to mitigate potential corruptions.
Another leverage point is how we decide to use the reward predictor. We could consider the *stationary* reward function, which evaluates simulated trajectories with the *current* reward predictor, i.e. assuming that the reward predictor will never be updated again. If we combine this with self-corruption unawareness (so that the policy also never expects the policy to change), then the incentive to corrupt the reward predictor's data is removed. However, the resulting agent is *time-inconsistent*: it acts as though its reward never changes even though it in practice does, and so it can make a plan and start executing it, only to switch over to a new plan once the reward changes, over and over again.
The *dynamic* reward function avoids this pitfall by evaluating the kth timestep of a simulated trajectory by also taking an expectation over future data that the reward predictor will get. This agent is no longer time-inconsistent, but it now incentivizes the agent to manipulate the data. This can be fixed by building a single integrated Bayesian agent, which maintains a single environment model that predicts both the reward function and the environment model. The resulting agent is time-consistent, utility-preserving, and has no direct incentive to manipulate the data. (This is akin to the setup in **[assistance games / CIRL](https://arxiv.org/abs/1606.03137)** (**[AN #69](https://mailchi.mp/59ddebcb3b9a/an-69-stuart-russells-new-book-on-why-we-need-to-replace-the-standard-model-of-ai)**).)
One final approach is to use a *counterfactual* reward function, in which the data is simulated in a counterfactual world where the agent executed some known safe default policy. This no longer depends on the current time, and is not subject to data corruption since the data comes from a hypothetical that is independent of the agent's actual policy. However, it requires a good default policy that does the necessary information-gathering actions, and requires the agent to have the ability to simulate human feedback in a counterfactual world.
**Read more:** **[Tom Everitt's PhD thesis](https://openresearch-repository.anu.edu.au/handle/1885/164227)**
**Rohin's opinion:** This paper is a great organization and explanation of several older papers (that haven't been summarized in this newsletter because they were published before 2018 and I read them before starting this newsletter), and I wish I had read it sooner. It seems to me that the integrated Bayesian agent is the clear winner -- the only downside is the computational cost, which would be a bottleneck for any of the models considered here.
One worry I have with this sort of analysis is that the guarantees you get out of it depends quite a lot on how you model the situation. For example, let's suppose that after I sleep I wake up refreshed and more capable of intellectual work. Should I model this as "policy corruption", or as a fixed policy that takes as an input some information about how rested I am?
TECHNICAL AI ALIGNMENT
======================
ITERATED AMPLIFICATION
----------------------
**[Universality Unwrapped](https://www.alignmentforum.org/posts/farherQcqFQXqRcvv/universality-unwrapped)** *(Adam Shimi)* (summarized by Rohin): This post explains the ideas behind universality and ascription universality, in a more accessible way than the **[original posts](https://ai-alignment.com/towards-formalizing-universality-409ab893a456)** and with more detail than **[my summary](https://mailchi.mp/6078fe4f9928/an-81-universality-as-a-potential-solution-to-conceptual-difficulties-in-intent-alignment)**.
MESA OPTIMIZATION
-----------------
**[Mesa-Search vs Mesa-Control](https://www.alignmentforum.org/posts/WmBukJkEFM72Xr397/mesa-search-vs-mesa-control)** *(Abram Demski)* (summarized by Rohin): This post discusses several topics related to mesa optimization, and the ideas in it led the author to update towards thinking inner alignment problems are quite likely to occur in practice. I’m not summarizing it in detail here because it’s written from a perspective on mesa optimization that I find difficult to inhabit. However, it seems to me that this perspective is common so it seems fairly likely that the typical reader would find the post useful.
AGENT FOUNDATIONS
-----------------
**[Radical Probabilism](https://www.alignmentforum.org/posts/xJyY5QkQvNJpZLJRo/radical-probabilism-1)** *(Abram Demski)* (summarized by Rohin): The traditional Bayesian treatment of rational agents assumes that the only way an agent can get new information is by getting some new observation that is known with probability 1. However, we would like a theory of rationality that can allow for agents that also get more information by thinking longer. In such a situation, some of the constraints imposed by traditional Bayesian reasoning no longer apply. This detailed post explores what constraints remain, and what types of updating are allowable under this more permissive definition of rationality.
**Read more:** **[The Bayesian Tyrant](https://www.lesswrong.com/posts/4tke3ibK9zfnvh9sE/the-bayesian-tyrant)**
**Rohin's opinion:** I particularly enjoyed this post; it felt like the best explanation in relatively simple terms of a theory of rationality that is more suited to bounded agents that cannot perfectly reason about an environment larger than they are. (Note “simple” really is relative; the post still assumes a lot of technical knowledge about traditional Bayesianism.)
FORECASTING
-----------
**[My AI Timelines Have Sped Up](https://www.alexirpan.com/2020/08/18/ai-timelines.html)** *(Alex Irpan)* (summarized by Nicholas): Alex Irpan updates his predictions of AGI sooner to:
10% chance by 2035 (previously 2045)
50% chance by 2045 (previously 2050)
90% chance by 2070
The main reasons why are:
- Alex is now more uncertain because research pace over the past five years have been more surprising than expected, faster in some domains, but slower than others.
- Accounting for improvements in tooling. New libraries like TensorFlow and PyTorch have accelerated progress. Even CNNs can be used as a “tool” that provides features for downstream tasks like robotic control.
- He previously thought that labeled data might be a bottleneck, based on scaling laws showing that data needs might increase faster than compute; however, semi- and unsupervised learning have improved significantly, GPT-3 being the latest example of this.
- Alex now believes that compute will play a larger role and that compute can scale faster than algorithms because there is large worldwide consumer demand.
The post ends with a hypothetical description of how AGI may happen soon that I will leave out of the summary but recommend reading.
**Nicholas's opinion:** My personal opinion on timelines is that I think it is much more informative to draw out the full CDF/PDF of when we will get to AGI instead of percentages by different years. It isn’t included in the post, but you can find Alex’s **[here](https://elicit.ought.org/builder/BsNkKzJoc)**. I end up placing higher likelihood on AGI happening sooner than Alex does, but I largely agree with his reasoning.
More uncertainty than the original prediction seems warranted to me; the original prediction had a very high likelihood of AGI between 2045-2050 that I didn’t understand. Of the rest of the arguments, I agree most strongly with the section on tooling providing a speedup. I’d even push the point farther to say that there are many inputs into current ML systems, and all of them seem to be improving at a rapid clip. Hardware, software tools, data, and the number of ML researchers all seem to be on track to improve significantly over the next decade.
MISCELLANEOUS (ALIGNMENT)
-------------------------
**[The Problem with Metrics is a Fundamental Problem for AI](https://arxiv.org/abs/2002.08512)** *(Rachel Thomas et al)* (summarized by Flo): The blog post lists five problems of current AI that are exacerbated by the cheap cost and easy scaling of AI systems combined with the common belief that algorithms are objective and error-free:
1. It is often hard for affected people to address problems in algorithmic decisions
2. The complexity of AI problems can easily lead to a diffusion of responsibility
3. AI can encode biases and sometimes magnify them via feedback loops
4. Big tech companies lack accountability
5. Current AI systems usually focus exclusively on optimizing metrics.
The paper then dives deeper into the last point. They review a series of case studies and form four conclusions. First, measured metrics are usually only a proxy for what we really care about: Youtube's terminal goal is certainly not to maximize viewing time and society does not inherently care about student test scores. Secondly, metrics can and will be gamed: Soviet workers would often achieve their production targets at the cost of some unmeasured aspects of performance, reported waiting times in the English healthcare system were distorted once targets were set for them and evaluating teachers by test scores has led to cheating scandals in the US. Third, metrics tend to overemphasise short-term concerns as they are often easier to measure. This can be seen in businesses like Facebook and Wells Fargo that have faced political backlash, worse access to talent pools, or lawsuits because of an excessive focus on click-through rates and quarterly earnings. Fourth, tech firms often focus on metrics that are associated with addictive environments. For example, "engagement" metrics are used as proxies for user preferences but rarely reflect them accurately in contexts that were optimized for these metrics. The authors then propose three remedies: Using multiple metrics to get a more holistic picture and make gaming harder, combining metrics with qualitative accounts, and involving domain experts and stakeholders that would be personally affected by the deployed system.
**Read more:** **[I’m an AI researcher, and here’s what scares me about AI](https://medium.com/@racheltho/im-an-ai-researcher-and-here-is-what-scares-me-about-ai-909a406e4a71)**
**Flo's opinion:** I found this interesting to read, as it does not really seem to be written from the perspective of AI Safety but still lists some problems that are related to AI safety and governance. Just think of an AI system tasked to help with realizing human preferences magnifying "biases" in its preference elicitation via unwanted feedback loops, or about the lack of firms accountability for socioeconomic disturbances their AI systems could create that **[the windfall clause](https://www.fhi.ox.ac.uk/windfallclause/)** (**[AN #88](https://mailchi.mp/9d279b575b1a/an-88-how-the-principal-agent-literature-relates-to-ai-risk)**) was envisioned to mitigate.
OTHER PROGRESS IN AI
====================
REINFORCEMENT LEARNING
----------------------
**[Curriculum Learning for Reinforcement Learning Domains: A Framework and Survey](http://arxiv.org/abs/2003.04960)** *(Sanmit Narvekar et al)* (summarized by Zach): For a variety of learning problems, the training process is organized so that new concepts and tasks leverage previously learned information. This can serve as a broad definition of curriculum learning. This paper gives an overview of curriculum learning and a framework to organize various approaches to the curriculum learning problem. One central difficulty is that there is a broad class of methods that can be considered curricula. At one extreme, we have curricula where new tasks are created to speed up learning. At another extreme, some curricula simply reorder experience samples. For example, the prioritized replay buffer is one such reordering method. Thus, to cover as much of the literature as possible the authors outline a framework for curriculum learning and then use that structure to classify various approaches. In general, the definition, learning, construction, and the evaluation of curricula are all covered in this work. This is done by breaking the curriculum learning problem into three steps: task generation, sequencing, and transfer learning. Using this problem decomposition the authors give an overview of work addressing each component.
**Zach's opinion:** Before I read this, I thought of curricula as 'hacks' used to improve training. However, the authors' presentation of connections with transfer learning and experience replay has significantly changed my opinion. In particular, the phrasing of curriculum learning as a kind of 'meta-MDP seems particularly interesting to me. Moreover, there seem to be interesting challenges in this field. One such challenge is that there does not seem to be a great amount of theory about *why* curricula work which could indicate a point of departure for people interested in safety research. Knowing more about theory could help answer safety questions. For example, how do we design curricula so that we can guarantee/check the agent is behaving correctly at each step?
NEWS
====
**[Looking for adversarial collaborators to test our Debate protocol](https://www.alignmentforum.org/posts/w7mS6syTderWihHPM/looking-for-adversarial-collaborators-to-test-our-debate)** *(Beth Barnes)* (summarized by Rohin): OpenAI is looking for people to help test their **[debate](https://www.alignmentforum.org/posts/Br4xDbYu4Frwrb64a/writeup-progress-on-ai-safety-via-debate-1)** (**[AN #86](https://mailchi.mp/598f425b1533/an-86-improving-debate-and-factored-cognition-through-human-experiments)**) protocol, to find weaknesses that allow a dishonest strategy to win such debates.
#### **FEEDBACK**
I'm always happy to hear feedback; you can send it to me, **[Rohin Shah](https://rohinshah.com/)**, by **replying to this email**.
#### **PODCAST**
An audio podcast version of the **Alignment Newsletter** is available. This podcast is an audio version of the newsletter, recorded by **[Robert Miles](http://robertskmiles.com/)**. |
49709889-19c3-4b0a-b225-b52058e32176 | trentmkelly/LessWrong-43k | LessWrong | Against Unlimited Genius for Baby-Killers
Someone is wrong on the internet. Wrong enough, in fact, to lure me from my hitherto perpetual lurk.
Of course, the meme is (was) funny (amusing) because the people of the internet are often wrong — and often in sillier ways than the instance that has so exercised me here. The nuance in this case, however, is that the mistaken man is one upon whose whim our species’ future plausibly depends. Unfortunate!
See, Sam Altman declared last week that ‘terrorists should have access to unlimited genius to direct however they can imagine’. Or was it ‘baby-killers should have access to unlimited genius to direct however they can imagine’? Never mind, it was both. Indeed, the exact quote was that ‘everyone should have access to unlimited genius to direct however they can imagine’. (‘Everyone’ entails ‘terrorists’ and ‘baby-killers’, ergo…)
Please do not take me for a pedant. Rest assured, I would not be moved to such an undignified début over a simple slip-of-the-tongue. I am not looking to be uncharitable, either.[1] There are surely times when ‘everyone’ straightforwardly does not mean everyone. He obviously didn’t mean literally everyone, I hear you cry. StrAw MAn!!
I’ve no doubt that Altman would prefer a world where everyone except terrorists, baby-killers and their ilk were furnished with ‘unlimited genius’ on-demand. My point is that this is not a choice you get to make. Sam’s candour here has done us a service.[2] ‘Everyone’ really is what he means — or, rather, it is all he reasonably can mean — and a more dexterous choice of words would have served only to obscure that fact.
One way to view technological progress is as an indiscriminate amplifier. An increase in technological level delivers a corresponding increase in capacity for impact. This sounds almost tautological, and at the very least trite. But it matters, because ‘capacity for impact’ is a neutral term. Progress is an ‘indiscriminate amplifier’ in that it augments impacts in all directions. There are go |
efa8cf12-bb04-473a-a553-947451f94c61 | trentmkelly/LessWrong-43k | LessWrong | Hypotheticals: The Direct Application Fallacy
A few years ago, I tried convincing people some commenters that hypotheticals were important even when they weren't realistic. That failed, but I think I've spend enough time reflecting to give this another go. This time, my focus will be on challenging the following common assumption:
> The Direct Application Fallacy: If a hypothetical situation can't conceivably occur, then the hypothetical situation doesn't matter
I chose this name because it assumes that the only purpose of discussing a hypothetical is to know what would happen or what we should do in such a situation. It ignores the other lessons that such a discussion may teach us and how it might have logical consequences for situations that actually do occur.
(Note: This post was renamed from: Unrealistic Hypotheticals Still Contain Lessons)
Exploiting Opportunities for Learning
In The Least Convenient Possible World, Scott Alexander considers the classic objection to utilitarianism that it implies that a surgeon should be prepared to harvest the organs of a random traveller if it would allow them to save five other patients. Scott argues that pointing out that the random traveller's organs probably be genetic mismatches, while "technically correct", also "completely misses the point and loses a valuable opportunity to examine the nature of morality". He also notes that responding in this manner leaves too much "wiggle room". Even if we aren't consciously aware of it, we often construct arguments to avoid believing things that we don't want to, so we can improve our rationality by limiting our ability to avoid understanding the other person's perspective. While Scott is referring to people who completely miss the point of the hypothetical, I think that dismissing a hypothetical as unrealistic often also sacrifices opportunities for learning as we'll see below.
Practise Exercises Don't Need to be Real
Imagine that you are an instructor setting problems for your students so that they can learn an area l |
c0062a31-a4c5-48e8-b145-83e34b328a34 | trentmkelly/LessWrong-43k | LessWrong | Optimiser to Oracle
It would be useful to have a mechanism for getting an AI to provide an honest answer to a specific question. This post presents such a mechanism.
The question is: what is the impact of X on the expected utility of a utility function v? Here X is some well-defined binary event. Formally, for a constant c (positive or negative), we want to query the AI as to whether Q(v,X,c) is true, where
Q(v,X,c) = {E(v|X=1) - E(v|¬X=0) > c}.
How can we do that? Well, for a start, let M(u) be a u-maximising agent. We'll assume we can feed M() any utility function to maximise. We'll further assume that M() is boxed, that it has a single output bit Y, and that we have cut any acausal trade possibilities. We'll add another binary process Z, which, if Z=1, ensure that Y/¬Y is never recorded. In actual fact, P(Z=0)≈1.
The remaining assumptions are on the operations we can perform on v (without needing any further understanding of v). We replace v by v#, defined in the following way. First of all, we make v# indifferent to worlds in which X≠Y or Z=0. Indifference to Z=0 can proceed in the usual way; indifference to X≠Y means that the value of worlds with X≠Y are changed to the value of worlds where X is set to the value of Y.
Therefore any v# maximiser will behave as if it lived in a world where X=Y (its output and the outcome of X were the same) and Z=1 (its output was never recorded). Then we add a special, one-off penalty of -c for outputting Y=1 (there is no effect for outputting Y=0). This defines v#, and we give our boxed AI v# to maximise.
So, how will M(v#) behave? Because it acts as if X=Y with certainty, it behaves as if it can determine the value of X by outputting Y. Because it acts as if Z=0 with certainty, it behaves as if the output of Y is never read, so it can determine the value of X without affecting anything else about the universe. It will now behave as if it was a v maximiser with control over X. So if it outputs Y=0, it expects to get utility E(v|X=0). If it o |
fd4fd692-13c7-4196-b278-87d2e6ce21d2 | trentmkelly/LessWrong-43k | LessWrong | Public service announcement - Bay Area tsunami warning!
http://www.mercurynews.com/news/ci_17589513?nclick_check=1
please do not upvote, main page is enough. will be removed after expiration. |
dd56372d-258d-4caa-8227-64d32b7e7ae1 | trentmkelly/LessWrong-43k | LessWrong | Soares, Tallinn, and Yudkowsky discuss AGI cognition
This is a collection of follow-up discussions in the wake of Richard Ngo and Eliezer Yudkowsky's Sep. 5–8 and Sep. 14 conversations.
Color key:
Chat Google Doc content Inline comments
7. Follow-ups to the Ngo/Yudkowsky conversation
[Bensinger][1:50] (Nov. 23 follow-up comment)
A general background note: Readers who aren't already familiar with ethical injunctions or the unilateralist's curse should probably read Ends Don't Justify Means (Among Humans), along with an explanation of the unilateralist's curse.
7.1. Jaan Tallinn's commentary
[Tallinn][6:38] (Sep. 18)
thanks for the interesting debate! here are my comments so far: [GDocs link]
[Tallinn] (Sep. 18 Google Doc)
meta
a few meta notes first:
* i’m happy with the below comments being shared further without explicit permission – just make sure you respect the sharing constraints of the discussion that they’re based on;
* there’s a lot of content now in the debate that branches out in multiple directions – i suspect a strong distillation step is needed to make it coherent and publishable;
* the main purpose of this document is to give a datapoint how the debate is coming across to a reader – it’s very probable that i’ve misunderstood some things, but that’s the point;
* i’m also largely using my own terms/metaphors – for additional triangulation.
pit of generality
it feels to me like the main crux is about the topology of the space of cognitive systems in combination with what it implies about takeoff. here’s the way i understand eliezer’s position:
there’s a “pit of generality” attractor in cognitive systems space: once an AI system gets sufficiently close to the edge (“past the atmospheric turbulence layer”), it’s bound to improve in catastrophic manner;
[Yudkowsky][11:10] (Sep. 18 comment)
> it’s bound to improve in catastrophic manner
I think this is true with quite high probability about an AI that gets high enough, if not otherwise corrigibilized, bo |
e692eeff-49a0-47d2-a005-275c25fa3f0d | trentmkelly/LessWrong-43k | LessWrong | 46% of US adults at least "somewhat concerned" about AI extinction risk.
Good news today I think; a poll by yougov had a near majority somewhat concerned to very concerned about existential risk of AI, with younger people most concerned. Given relative newness of (debatably) Transformative AI I think this suggests we will see a strong bipartisan political movement to regulate AI development develop in next 1-2 years. Almost certain to be aided by a media/political complex that thrives on promoting and magnifying dangers.
|
d77eaaff-5d61-4caa-8da8-77a724496da1 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Detroit/Ann Arbor - Memory Workshop
Discussion article for the meetup : Detroit/Ann Arbor - Memory Workshop
WHEN: 23 March 2014 01:00:00PM (-0400)
WHERE: 19334 Angling Street, Livonia, MI
Brienne from CFAR will be coming all the way from California to teach a couple hour workshop on memory and mnemonic techniques with some application to productivity hacking as well. Same location as usual. Donations appreciated.
Additional special guest Robby Bensinger of http://nothingismere.com/ (user: RobbBB)
Discussion article for the meetup : Detroit/Ann Arbor - Memory Workshop |
49ab751c-0e81-4328-9e81-232b933e10e1 | trentmkelly/LessWrong-43k | LessWrong | Greg Linster on the beauty of death
> Without death we cannot truly have life. As such, what a travesty of life it would be to achieve a machine-like immortality!
> Gray writes the following chilling lines: “If you understand that in wanting to live for ever you are trying to preserve a lifeless image of yourself, you may not want to be resurrected or to survive in a post-mortem paradise. What could be more deadly than being unable to die?” (my emphasis)
via.
Sounds like sour grapes. I'd heard of people holding such sentiments; this is the first time I've actually seen them expressed myself. |
8b2404ab-ee90-48a3-a980-b862489b9dae | trentmkelly/LessWrong-43k | LessWrong | Follow Standard Incentives
--- |
56dc9555-e2cd-4781-90a1-8ef908fc9037 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Rochester Rationalists: Utopia discussion meetup
Discussion article for the meetup : Rochester Rationalists: Utopia discussion meetup
WHEN: 17 April 2016 01:00:00PM (-0400)
WHERE: 302 Goodman St N, Rochester, NY 14607
Location: Nox
Whether reaching one is possible or not, utopian settings can be a useful thought experiment for examining our values. What's necessary to make a good one? What seemingly-good traits would go horribly wrong?
Some related articles (completely optional reading):
Just another day in utopia
Archipelago and atomic communitarianism (part III is where the meat of the article starts)
31 laws of fun
Discussion article for the meetup : Rochester Rationalists: Utopia discussion meetup |
1b91fa3f-8eac-4065-a187-a0da7b0d8af2 | StampyAI/alignment-research-dataset/blogs | Blogs | “Singularity Hypotheses” Published
[](http://www.amazon.com/Singularity-Hypotheses-Scientific-Philosophical-Assessment/dp/3642325599/)[*Singularity Hypotheses: A Scientific and Philosophical Assessment*](http://www.amazon.com/Singularity-Hypotheses-Scientific-Philosophical-Assessment/dp/3642325599/) has now been published by Springer, in hardcover and ebook forms.
The book contains 20 chapters about the prospect of machine superintelligence, including 4 chapters by MIRI researchers and research associates.
**“Intelligence Explosion: Evidence and Import”** ([pdf](https://intelligence.org/files/IE-EI.pdf)) by Luke Muehlhauser and (previous MIRI researcher) Anna Salamon reviews
> the evidence for and against three claims: that (1) there is a substantial chance we will create human-level AI before 2100, that (2) if human-level AI is created, there is a good chance vastly superhuman AI will follow via an “intelligence explosion,” and that (3) an uncontrolled intelligence explosion could destroy everything we value, but a controlled intelligence explosion would benefit humanity enormously if we can achieve it. We conclude with recommendations for increasing the odds of a controlled intelligence explosion relative to an uncontrolled intelligence explosion.
>
>
**“Intelligence Explosion and Machine Ethics”** ([pdf](https://intelligence.org/files/IE-ME.pdf)) by Luke Muehlhauser and Louie Helm discusses the challenges of formal value systems for use in AI:
> Many researchers have argued that a self-improving artificial intelligence (AI) could become so vastly more powerful than humans that we would not be able to stop it from achieving its goals. If so, and if the AI’s goals differ from ours, then this could be disastrous for humans. One proposed solution is to program the AI’s goal system to want what we want before the AI self-improves beyond our capacity to control it. Unfortunately, it is difficult to specify what we want. After clarifying what we mean by “intelligence,” we offer a series of “intuition pumps” from the field of moral philosophy for our conclusion that human values are complex and difficult to specify. We then survey the evidence from the psychology of motivation, moral psychology, and neuroeconomics that supports our position. We conclude by recommending ideal preference theories of value as a promising approach for developing a machine ethics suitable for navigating an intelligence explosion or “technological singularity.”
>
>
**“Friendly Artificial Intelligence”** by Eliezer Yudkowsky is a shortened version of [Yudkowsky (2008)](https://intelligence.org/files/AIPosNegFactor.pdf).
Finally, **“Artificial General Intelligence and the Human Mental Model”** ([pdf](https://intelligence.org/files/AGI-HMM.pdf)) by Roman Yampolskiy and (MIRI research associate) Joshua Fox reviews the dangers of anthropomorphizing machine intelligences:
> When the first artificial general intelligences are built, they may improve themselves to far-above-human levels. Speculations about such future entities are already affected by anthropomorphic bias, which leads to erroneous analogies with human minds. In this chapter, we apply a goal-oriented understanding of intelligence to show that humanity occupies only a tiny portion of the design space of possible minds. This space is much larger than what we are familiar with from the human example; and the mental architectures and goals of future superintelligences need not have most of the properties of human minds. A new approach to cognitive science and philosophy of mind, one not centered on the human example, is needed to help us understand the challenges which we will face when a power greater than us emerges.
>
>
The book also includes brief, critical responses to most chapters, including responses written by Eliezer Yudkowsky and (previous MIRI staffer) Michael Anissimov.
The post [“Singularity Hypotheses” Published](https://intelligence.org/2013/04/25/singularity-hypotheses-published/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org). |
747d95b3-177b-44bd-9b1d-9678b59aa227 | trentmkelly/LessWrong-43k | LessWrong | scipy.optimize.curve_fit Is Awesome
cross-posted from niplav.github.io
I recently learned about the python function scipy.optimize.curve_fit, and I'm really happy I did.
It fulfills a need I didn't know I'd always had, but never fulfilled: I often have a dataset and a function with some parameters, and I just want the damn parameters to be fitted to that dataset, even if imperfectly. Please don't ask any more annoying questions like “Is the dataset generated by a Gaussian?” or “Is the underlying process ergodic?”, just fit the goddamn curve!
And scipy.optimize.curve_fit does exactly that!
You give it a function f with some parameters a, b, c, … and a dataset consisting of input values x and output values y, and it then optimizes a, b, c, … so that f(x, a, b, c, …) is as close as possible to y (where, of course, x and y can both be numpy arrays).
This is awesome! I have some datapoints x, y and I believe it's generated by some obscure function, let's say of the form f(x,a,b,c)=a⋅x⋅sin(b⋅x+c), but I don't know the exact values for a, b and c?
No problem! I just throw the whole thing into curve_fit (scipy.optimize.curve_fit(f, x, y)) and out comes an array of optimal values for a, b, c!
What if I then want c to be necessarily positive?
Trivial! curve_fit comes with an optional argument called bounds, since b is the second argument, I call scipy.optimize.curve_fit(f, x, y, bounds=([-numpy.inf, -numpy.inf, 0], numpy.inf)), which says that curve_fit should not make the second argument smaller than zero, but otherwise can do whatever it wants.
So far, I've already used this function two times, and I've only known about it for a week! A must for every wannabe data-scientist.
For more information about this amazing function, consult its documentation. |
2b4fb349-199f-4140-8c41-b670a1cfcb8e | trentmkelly/LessWrong-43k | LessWrong | The Power to Solve Climate Change
This is Part V of the Specificity Sequence
Companion post: Examples of Examples
Most people agree that climate change is a big problem we should be solving, but couldn't tell you what specifically "solving climate change" means. By the end of this post, I promise you'll know what specifically "solving climate change" means... and also what it doesn't mean.
The Climate Crisis
The first step is knowing the key specific facts of the climate crisis. I admit I couldn't recite them accurately before writing this post. Here's a quick summary from Tomorrow and NASA (both great friendly explanations worth checking out):
* Earth's average temperature has shot up by 1°C in the last 50 years.
* The causal link from greenhouse gas emissions to Earth's rising temperature has been well established.
* On a 1M-year timescale, Earth's temperature has been fluctuating plus or minus a few degrees tops, so this 1°C change is a big fluctuation.
* We're predicting it to be as high as a 6°C warming by 2100, so it's actually a huge fluctuation.
* The current rate of temperature rise is a whopping 20 times faster than the rate at which Earth's temperature historically fluctuates, so it's actually an oh, SHIT fluctuation.
Thiel's Definite vs. Indefinite Attitude
In Peter Thiel's smart and highly original book Zero to One: Notes on Startups, or How to Build the Future, the chapter called "You Are Not A Lottery Ticket" centers around the concept of definite vs. indefinite attitudes:
Under the heading "Can You Control Your Future?" Thiel writes:
> You can expect the future to take a definite form or you can treat it as hazily uncertain. If you treat the future as something definite, it makes sense to understand it in advance and work to shape it. But if you expect an indefinite future ruled by randomness, you'll give up on trying to master it.
> Indefinite attitudes to the future explain what's the most dysfunctional in our world today. Process trumps substance: when people lack c |
49620f9b-dcda-4dba-8610-ed2181ef08f1 | trentmkelly/LessWrong-43k | LessWrong | Polio and the controversy over randomized clinical trials
I’m currently reading Polio: An American Story, by David Oshinsky, and I came across a fascinating story:
In 1954, when it was time for large-scale human trials of the first polio vaccine, some researchers were against the idea of doing a properly randomized, double-blind, placebo-controlled clinical trial—including Jonas Salk, the inventor of the vaccine.
What did they want to do instead? An “observed control” trial: They would ask for volunteers (children) to get the vaccine, and then compare the rate of polio in the volunteers to the rate in their schoolmates who weren’t vaccinated. No placebo. No randomization. Not blind.
Of course, this was hopelessly confounded. In that era, the families most likely to volunteer were the more educated and affluent families (and those were actually the ones most at risk for the disease).
So why did Salk and others oppose proper randomized blind controls? The argument against a randomized trial was the urgency of protecting the nation’s children against a debilitating and deadly disease. If the vaccine worked, it would be a tragedy to withhold it from the control children. Quoting Oshinsky:
> There were ethical issues as well. Were injected controls really suited to a polio trial? Was it proper, in short, to deny someone access to a potentially lifesaving vaccine in the name of statistical accuracy? Thousands of parents were going to volunteer their children to receive an injection—all of them hoping it contained the polio vaccine, not the placebo. Yet one-half of this study composed of six- to nine-year-olds, the group most vulnerable to paralytic polio, would receive a worthless liquid. Some, including Salk himself, saw this as elite science at its worst, a cynical form of Russian roulette.
Some context: the trial was massive, involving hundreds of thousands (eventually over a million) children across the country. This is not n=50 we’re talking about here. And the disease was seasonal, striking in epidemic waves every su |
e97602e2-88e3-44f3-a00e-138664ecb4f1 | trentmkelly/LessWrong-43k | LessWrong | 3^^^3 holes and <10^(3*10^31) pigeons (or vice versa)
The reasoning about huge numbers of beings is a recurring theme here. Knuth's up-arrow notation is often used, with 3^^^3 as the number of beings.
I want to note that if a being is made of 10^30 parts, with 10^30 distinct states of each part, the number of distinct being states is (10^30)^(10^30) = 10^(3*10^31) . That's not a very big number; stacking uparrows quickly gets you to much larger numbers.
To quote from Torture Vs Dust Specks:
>
>
> * 3^3 = 27.
> * 3^^3 = (3^(3^3)) = 3^27 = 7625597484987.
> * 3^^^3 = (3^^(3^^3)) = 3^^7625597484987 = (3^(3^(3^(... 7625597484987 times ...)))).
>
> 3^^^3 is an exponential tower of 3s which is 7,625,597,484,987 layers tall. You start with 1; raise 3 to the power of 1 to get 3; raise 3 to the power of 3 to get 27; raise 3 to the power of 27 to get 7625597484987; raise 3 to the power of 7625597484987 to get a number much larger than the number of atoms in the universe, but which could still be written down in base 10, on 100 square kilometers of paper; then raise 3 to that power; and continue until you've exponentiated 7625597484987 times. That's 3^^^3. It's the smallest simple inconceivably huge number I know.
That's an unimaginably bigger number than 10^(3*10^31) . You just can't have 3^^^3 distinct humans (or the beings that are to human as human is to amoeba, or that repeated zillion times, or distinct universes for that matter). Most of them will be exactly identical to very many others among the 3^^^3 and have exactly identical experience*.
Of course, our reasoning does not somehow subconsciously impose a reasonable cap on number of beings and end up rational afterwards. I'm not arguing that gut feeling includes such consideration. (I'd say it usually just considers substantially different things incomparable and in-convertible, plus the space of utility needs not be one dimensional)
I've made this pigeon-hole example to demonstrate a failure with really huge numbers, that can undermine by an inconceivably |
389ce9a0-9e21-4421-be95-47d0f5baf0db | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Establishing Oxford’s AI Safety Student Group: Lessons Learnt and Our Model
[EDITS- We no longer endorse everything in this post, and have changed our objectives and thinking significantly. As such, the mentioned document is now private if you have questions, please contact [oxford@aisafetyhub.org](mailto:oxford@aisafetyhub.org)]
In January we founded a student group at Oxford focused on technical AI safety. Since then we’ve run speaker events, socials, multiple cohorts of the AGISF, and supervised research projects (“Labs”). We think it went pretty well, so we’re sharing our takeaways and model here.
This post is a short summary of [this public document](https://docs.google.com/document/d/198oFhyc3eBtJd4NE0vyc9yUVHfYRqBxrd5QY8oAWnIw/edit?usp=sharing) which goes into more detail about our approach to AI safety community building, reflections, and recommendations.
Non-trivial takeaways
=====================
1. Launching as part of an AI group, rather than an EA group, worked well for us. ([see more](https://docs.google.com/document/d/198oFhyc3eBtJd4NE0vyc9yUVHfYRqBxrd5QY8oAWnIw/edit#heading=h.ht74jwesyhpw))
2. Outreach aimed at people interested in AI reached a much larger technical audience than past outreach aimed at people interested in EA or longtermism. ([see more](https://docs.google.com/document/d/198oFhyc3eBtJd4NE0vyc9yUVHfYRqBxrd5QY8oAWnIw/edit#heading=h.cmczh7hva52s))
3. It was surprisingly easy to interest people in AI safety without appealing to EA or longtermism. ([see more](https://docs.google.com/document/d/198oFhyc3eBtJd4NE0vyc9yUVHfYRqBxrd5QY8oAWnIw/edit#heading=h.gl2eibqsxpdu))
4. Significant value from our speaker events seemed to come from the high-retention, friendly socials we held afterwards. ([see more](https://docs.google.com/document/d/198oFhyc3eBtJd4NE0vyc9yUVHfYRqBxrd5QY8oAWnIw/edit#heading=h.lee087a4bgpz))
5. Our “Labs” model of student research projects seems effective for development and output with minimal time-cost for an expert supervisor (~1 hour per week). This is particularly valuable if field building is mentorship bottlenecked ([see more](https://docs.google.com/document/d/198oFhyc3eBtJd4NE0vyc9yUVHfYRqBxrd5QY8oAWnIw/edit#heading=h.rd1mvo3y36n6)).
Our current model
=================
Our working objective was to increase the number and quality of technical people pursuing a career in AI safety research. [[1]](#fnk9l85cs15om) To do this, we have been operating with the following pipeline: [[2]](#fndiruc72p4bn)
Results so far
==============
* At least 2 of the current participants of Redwood’s MLAB this summer had never encountered AI safety or EA before attending our events this spring.
* We had 24-73 people attend our 9 speaker events, with 69% (on average) having a STEM background (according to survey data).
* 65 people signed up for our AGI Safety Fundamentals course across 11 cohorts. 57% had STEM backgrounds.
Further Information
===================
Please see [the attached public document](https://docs.google.com/document/d/198oFhyc3eBtJd4NE0vyc9yUVHfYRqBxrd5QY8oAWnIw/edit?usp=sharing) for further information about the student group or our contact details.
1. **[^](#fnrefk9l85cs15om)** We are now reconsidering our working objective and don’t necessarily endorse the stated objective "to increase the number and quality of technical people pursuing a career in AI safety research". However, we think it is important to start from your objective and work backwards, and this is the objective we actually used.
2. **[^](#fnrefdiruc72p4bn)** We want to note that having a target audience of people “interested in AI” creates a self-selection effect that reduces the diversity of thought in our attendance. We are working to improve this. |
5d81d340-6ac6-4130-ba2f-ebee546afa25 | trentmkelly/LessWrong-43k | LessWrong | Are there documentaries on rationality?
I love documentaries, and i wonder if there are any about LessWrong topics, such as rationality/rationalists, epistemology, maybe cognitive science., etc..
Do you know any? |
8d4d64b1-4e7a-41eb-af72-8485c5283038 | trentmkelly/LessWrong-43k | LessWrong | The Alignment-Competence Trade-Off, Part 1: Coalition Size and Signaling Costs
This is part 1 of a series of posts I initially planned to organize as a massive post last summer on principal-agent problems. As that task quickly became overwhelming, I decided to break it down into smaller posts that ensure I cover each of the cases and mechanisms that I intended to.
Overall, I think the trade-off between the alignment of agents and the competence of agents can explain a lot of problems to which people often think there are simple answers. The less capable an agent is (whether the agent is a person, a bureaucracy, or an algorithm) the easier it is for a principal to assess the agent, and ensure the agent is working toward the principal’s goals. As agents become more competent, they become both more capable of actually accomplishing the principal’s goals and of merely appearing to accomplish the principal’s goals while pursuing their own. In debating policy changes, I often find one sided arguments that neglect this trade-off, and in general I think efforts to improve policies or the bureaucratic structures of companies, non-profits, and governments should be informed by it.
----------------------------------------
Part 1:
Virtue signaling and moralistic anger are both forces that have been useful for holding people accountable, and powerful mechanisms of cultural evolution: spreading some norms more successfully than others, and resulting in many societies holding similar norms.
However, the larger a group becomes, the less members of the group know on average about other individual member’s behavior or the consequences of it: making it harder to evaluate complex actions. This in turn gives an advantage to more clear forms of signaling that are more inefficient and costly than those that could be sustainable in smaller groups.
Examples:
* While it would be efficient for a politician to accept money from competing special interest groups and to keep their behavior consistent with their constituents regardless, it is simpler for politicians |
3e9a143f-4bd8-4618-8b1f-c2a1609c5afb | trentmkelly/LessWrong-43k | LessWrong | Activation additions in a small residual network
Abstract
Team Shard's recent activation addition methodology for steering GPT-2 XL provokes many questions about what the structure of the internal model computation must be in order for their edits to work. Unfortunately, interpreting the insides of neural networks is known to be very difficult, so the question becomes: what is the minimal set of properties a network must have in order for adding activation additions to work?
Previously, I have tried to make some progress on this question by analyzing whether number additions work for a 784-512-512-10 fully connected MNIST network I had laying around. They didn't. Generalization was destroyed, going from a loss of 0.089 for the unpatched network to an average loss of 7.4 for the modified network. Now I see whether additions work in a residual network I trained. Here's the code for the network:
class ResidualBlock(nn.Module):
def __init__(self, dim):
super(ResidualBlock, self).__init__()
self.linear = nn.Sequential(
nn.Linear(dim, dim)
)
self.relu = nn.ReLU()
def forward(self, x):
out = self.linear(x)
out += x
out = self.relu(out)
return out
# Define the ResNet
class ResNet(nn.Module):
def __init__(self, num_classes=10):
super(ResNet, self).__init__()
self.layer1 = nn.Linear(28*28, 512)
self.relu = nn.ReLU()
self.layer2 = ResidualBlock(512)
self.layer3 = nn.Linear(512, num_classes)
def forward(self, x, x_vector=None, return_midlayer_activation=False):
x = x.view(x.size(0), -1) # equivalent of doing einops.rearrange(x, 'b h w d -> b (h w d)')
x = self.layer1(x)
x = self.relu(x)
x = self.layer2(x)
if x_vector is not None: x += x_vector
activation = x
x = self.layer3(x)
if return_midlayer_activation:
return x, activation
else:
return x
Again, space is left for the interested reader to think |
bea8d4a4-a0e6-4660-b381-8aca47140094 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | A Confused Chemist's Review of AlphaFold 2
*(This article was originally going to be titled "A Chemist's Review of AlphaFold 2")*
Most of the protein chemists I know have a dismissive view of AlphaFold. Common criticisms generally refer to concerns of "pattern matching". I wanted to address these concerns, and have found a couple of concerns of my own.
The main method for assessment of AlphaFold 2 has been the Critical Assessment of Protein Structure (CASP). This is a competition held based on a set of protein structures which have been determined by established experimental methods, but deliberately held back from publishing. Entrant algorithms then attempt to predict the structure based on amino acid sequence alone. AlphaFold 2 did much better than any other entrant in 2020, scoring 244 compared to the second place entrant's 91 by CASP's scoring method.
The first thing that struck me during my investigation is how large AlphaFold is, in terms of disk space. On top of neural network weights, it has a 2.2 TB protein structure database. A model which does *ab initio* calculations i.e. does a simulation of the protein based on physical and chemical principles, will be much smaller. For example Rosetta, a leading *ab initio* software package recommends 1 GB of working memory per processor in use while running, and gives no warnings at all about the file size of the program itself.
DeepMind has an explicit goal of replacing crystallography as a method for determining protein structure. Almost all crystallography is carried out on naturally occurring proteins isolated from organisms under study. This means the proteins are products of evolution, which generally conserves protein structure as a means of conserving function. Predicting the structure of an evolved protein is a subtly different problem to predicting the structure of a sequence of random amino acids. For this purpose AlphaFold 2 is doing an excellent job.
On the other hand, I have a few nagging doubts about how exactly DeepMind are going about solving the protein folding problem. Whether these are a result of my own biases or not is unclear to me. I am certainly sympathetic to the concerns of my peers that something is missing from AlphaFold 2.
### Representations
One of the core elements is that the representation(s) of protein structure is fed through the same network(s) multiple times. This is referred to as "recycling" in the paper, and it makes sense. What's interesting is that there are multiple layers which seem to refine the structure in completely different ways.
Some of these updates act on the "pair representation", which is pretty much a bunch of distances between amino acid residues (amino acids in proteins are called residues). Well it's not that, but it's not *not* that. I think it's best thought of as a sort of "affinity" or "interaction" between residues, which is over time refined to be constrained to 3D space.
There is also a separate representation called the "multiple system alignment (MSA) representation" which is not a bunch of distances between residues.
The MSA representation (roughly) starts with finding a bunch of proteins with a similar amino acid sequence to the input sequence. The search space of this is the 2.2 TB of data. Then it comes up with some representation, with each residue of our input protein being assigned some number relating it to a protein which looks like the input protein. To be honest I don't really understand *exactly what* this representation encodes it and I can't find a good explanation. I *think* it somehow encodes two things, although I can't confirm this as I don't know much about the actual data structure involved.
Thing 1 is that the input protein probably has a structure similar to these proteins. This is sort of a reasonable expectation in general, but makes even more sense from an evolutionary perspective. Mutations which disrupt the structure of a protein significantly usually break its function and die out.
Thing 2 is a sort of site correlation. If the proteins all have the same-ish structure, then we can look for correlations between residues. Imagine if we saw that when the 15th residue is positively charged, the 56th one is always negatively charged, and vice versa. This would give us information that they're close to one another.
### Evoformer Module
The first bunch of changes to the representations comes from the "evoformer" which seems to be the workhorse of AlphaFold. This is the part that sets it apart from ordinary simulations. A bunch of these models sequentially update the representations.
The first few transformations are the two representations interacting to exchange information. This makes sense as something to do and I'm not particularly sure I can interpret it any more than "neural network magic". The MSA representation isn't modified any further and is passed forwards to the next evoformer run.
The pair representation is updated based on some outer product with the MSA representation then continues.
The next stages are a few constraints relating to 3D Euclidean space being enforced on the pair representation. Again not much commentary here. All this stuff is applied 48 times in sequence, but there are no shared weights between the iterations of the evoformer. Only the overall structure is the same.
### Structure Module and AMBER
This section explicitly considers the atoms in 3D space. One of the things it does is use a a "residue gas" model which treats each residue as a free floating molecule. This is an interesting way of doing things. This allows all parts of the protein to be updated at once without dealing with loops in the structure. Then a later module applies a constraint that they have to be joined into a chain.
They also use the AMBER force-field (which is a simulation of the atoms based on chemical principles) to "relax" the protein sequence at some points. This does not improve accuracy by the atom-to-atom distance measures, but it does remove physically impossible occlusions of atoms. The authors describe these as "distracting" but strongly imply that the AMBER part isn't very important.
Attention
---------
I think this is what gives AlphaFold a lot of its edge, and unfortunately I don't understand it all that well. It's very similar to human attention in that the network first does some computations to decide where to look, then does more computations to make changes in that region. This is much better (I think) than an "mechanical" model which simulates the protein atom by atom, and devotes equal amounts of computation to each step.
Thoughts and Conclusions
------------------------
The second placing team in CASP14 was the Baker group, who also used an approach based on neural networks and protein databases. Knowing this, it doesn't surprise me much that DeepMind were able to outperform them, given the differences in resources (both human and technical) available to them. Perhaps this is a corollary of the "[bitter lesson](http://www.incompleteideas.net/IncIdeas/BitterLesson.html)": perhaps computation-specialized groups will always eventually outperform domain-specialized groups.
I do have two concerns though:
My first concern is that I strongly suspect that the database-heavy approach introduces certain biases, in the no-free-lunch sense of the word. The selection of proteins available is selected for in two ways: first by evolution, and secondly by ease of analysis.
Evolved proteins are not subject to random changes, only changes which allow the organism to survive. Mutations which significantly destabilize or change the structure of the protein are unlikely to be advantageous and outcompete the existing protein. CASP14 seems to be the prime source of validation for the model. This consists entirely of evolved proteins, so does not provide any evidence of performance against non-evolved (i.e. engineered) proteins. This strongly limits the usage of AlphaFold for protein engineering and design.
Secondly, not all proteins can be crystallized easily, or even at all. Also, some proteins only take on defined structure (and can only be crystallized) when bound to another molecule. DeepMind are working on functional predictions of small molecules binding to proteins, but including bound non-protein molecules in their structural predictions is outside the current scope of AlphaFold.
Both of these cases are much rarer than the typical case of protein crystallography, which is the main aim of AlphaFold. For most protein researchers, particularly medical researchers, I suspect that the trade-off of using the database approach is worth it.
My second concern is more nebulous, and relates to their usage of AMBER. This feels like they're outsourcing their low-level physical modelling.
This sort of thing is actually quite common when analysing crystallography results, which often require multiple rounds of refinement to get from crystallographic data to a sensible (read: physically possible, without atoms overlapping each other) structure. However the "first-draft" output of a crystallographic data is basically just a fit to the output of a 3D Fourier transform on some potentially noisy x-ray scattering data.
This somehow still feels different to Alphafold. If lots of their neural network layers are outputting "impossible" structures with atoms overlapping then it suggests those layers have failed to learn something about the physical properties of the world. I'm not sure whether or not this will turn out to be important in the end. It may be that I just have some irrational unease about models which learn in a very different way to human minds, learning complicated rules before simple ones.
AlphaFold will only grow in accuracy and scope. I suspect it will eventually overcome its current limitations.
### Sources and Thanks:
Many thanks to the LessWrong mod team, and their feedback system. This could have not been written without the feedback I received on an early draft of the article.
The actual paper: <https://www.nature.com/articles/s41586-021-03819-2>
CASP14: <https://predictioncenter.org/casp14/>
OPIG who have a much more in-depth analysis of the mechanics of AlphaFold 2: <https://www.blopig.com/blog/2021/07/alphafold-2-is-here-whats-behind-the-structure-prediction-miracle/> |
fdeaa2c2-5262-4ce8-a4d1-c0f60aa788b9 | trentmkelly/LessWrong-43k | LessWrong | We're already in AI takeoff
Back in 2016, CFAR pivoted to focusing on xrisk. I think the magic phrase at the time was:
> "Rationality for its own sake, for the sake of existential risk."
I was against this move. I also had no idea how power works. I don't know how to translate this into LW language, so I'll just use mine: I was secret-to-me vastly more interested in being victimized at people/institutions/the world than I was in doing real things.
But the reason I was against the move is solid. I still believe in it.
I want to spell that part out a bit. Not to gripe about the past. The past makes sense to me. But because the idea still applies.
I think it's a simple idea once it's not cloaked in bullshit. Maybe that's an illusion of transparency. But I'll try to keep this simple-to-me and correct toward more detail when asked and I feel like it, rather than spelling out all the details in a way that turns out to have been unneeded.
Which is to say, this'll be kind of punchy and under-justified.
----------------------------------------
The short version is this:
We're already in AI takeoff. The "AI" is just running on human minds right now. Sorting out AI alignment in computers is focusing entirely on the endgame. That's not where the causal power is.
Maybe that's enough for you. If so, cool.
I'll say more to gesture at the flesh of this.
What kind of thing is wokism? Or Communism? What kind of thing was Naziism in WWII? Or the flat Earth conspiracy movement? Q Anon?
If you squint a bit, you might see there's a common type here.
In a Facebook post I argued that it's fair to view these things as alive. Well, really, I just described them as living, which kind of is the argument. If your woo allergy keeps you from seeing that… well, good luck to you. But if you're willing to just assume I mean something non-woo, you just might see something real there.
These hyperobject creatures are undergoing massive competitive evolution. Thanks Internet. They're competing for resources. Literal |
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