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Section: In scientific research > Benefits. Scientists have also found that hallucinations can serve as a valuable tool for scientific discovery, particularly in fields requiring innovative approaches to complex problems. At the University of Washington, David Baker's lab has used AI hallucinations to design "ten milli... | Wikipedia - Hallucination (artificial intelligence) - In scientific research > Benefits | 347 | 1,986 | null |
Section: Mitigation methods. The hallucination phenomenon is still not completely understood. Researchers have also proposed that hallucinations are inevitable and are an innate limitation of large language models. Therefore, there is still ongoing research to try to mitigate its occurrence. Particularly, it was shown ... | Wikipedia - Hallucination (artificial intelligence) - Mitigation methods | 340 | 1,831 | null |
An extra layer of logic-based rules was proposed for the web search mitigation method, by using different ranks of web pages as a knowledge base, which differ in hierarchy. When there are no external data sources available to validate LLM-generated responses (or the responses are already based on external data as in RA... | Wikipedia - Hallucination (artificial intelligence) - Mitigation methods | 277 | 1,429 | null |
Article: Hierarchical navigable small world. The Hierarchical navigable small world (HNSW) algorithm is a graph-based approximate nearest neighbor search technique used in many vector databases. Nearest neighbor search without an index involves computing the distance from the query to each point in the database, which ... | Wikipedia - Hierarchical navigable small world - Summary | 239 | 1,216 | null |
Section: Use in vector databases. HNSW is a key method for approximate nearest neighbor search in high-dimensional vector databases, for example in the context of embeddings from neural networks in large language models. Databases that use HNSW as search index include: Apache Lucene Vector Search Chroma Qdrant Vespa Ve... | Wikipedia - Hierarchical navigable small world - Use in vector databases | 155 | 702 | null |
Article: Hierarchical Risk Parity. Hierarchical Risk Parity (HRP) is an advanced investment portfolio optimization framework developed in 2016 by Marcos LΓ³pez de Prado at Guggenheim Partners and Cornell University. HRP is a probabilistic graph-based alternative to the prevailing mean-variance optimization (MVO) framewo... | Wikipedia - Hierarchical Risk Parity - Summary | 155 | 857 | null |
Section: Key Features. Algorithms within the HRP framework are characterized by the following features: Machine Learning Approach: HRP employs hierarchical clustering, a machine learning technique, to group similar assets based on their correlations. This allows the algorithm to identify the underlying hierarchical str... | Wikipedia - Hierarchical Risk Parity - Key Features | 179 | 1,007 | null |
Article: Highway network. In machine learning, the Highway Network was the first working very deep feedforward neural network with hundreds of layers, much deeper than previous neural networks. It uses skip connections modulated by learned gating mechanisms to regulate information flow, inspired by long short-term memo... | Wikipedia - Highway network - Summary | 220 | 1,139 | null |
Section: Model. The model has two gates in addition to the H ( W H , x ) {\displaystyle H(W_{H},x)} gate: the transform gate T ( W T , x ) {\displaystyle T(W_{T},x)} and the carry gate C ( W C , x ) {\displaystyle C(W_{C},x)} . The latter two gates are non-linear transfer functions (specifically sigmoid by convention).... | Wikipedia - Highway network - Model | 192 | 569 | null |
Section: Structure. The structure of a hidden layer in the Highway Network follows the equation: y = H ( x , W H ) β
T ( x , W T ) + x β
C ( x , W C ) = H ( x , W H ) β
T ( x , W T ) + x β
( 1 β T ( x , W T ) ) {\displaystyle {\begin{aligned}y=H(x,W_{H})\cdot T(x,W_{T})+x\cdot C(x,W_{C})\\=H(x,W_{H})\cdot T(x,W_{T})+x\... | Wikipedia - Highway network - Structure | 177 | 354 | null |
Section: Related work. Sepp Hochreiter analyzed the vanishing gradient problem in 1991 and attributed to it the reason why deep learning did not work well. To overcome this problem, Long Short-Term Memory (LSTM) recurrent neural networks have residual connections with a weight of 1.0 in every LSTM cell (called the cons... | Wikipedia - Highway network - Related work | 339 | 1,392 | null |
It is like a 2000 LSTM with forget gates unfolded in time, while the later Residual Nets have no equivalent of forget gates and are like the unfolded original 1997 LSTM. If the skip connections in Highway Networks are "without gates," or if their gates are kept open (activation 1.0), they become Residual Networks. The ... | Wikipedia - Highway network - Related work | 336 | 1,607 | null |
Section: History. The company was founded in 2016 by French entrepreneurs ClΓ©ment Delangue, Julien Chaumond, and Thomas Wolf in New York City, originally as a company that developed a chatbot app targeted at teenagers. The company was named after the U+1F917 π€ HUGGING FACE emoji. After open sourcing the model behind t... | Wikipedia - Hugging Face - History | 322 | 1,546 | null |
The company also said the next generation of BLOOM will be run on Trainium, a proprietary machine learning chip created by AWS. In August 2023, the company announced that it raised $235 million in a Series D funding, at a $4.5 billion valuation. The funding was led by Salesforce, and notable participation came from Goo... | Wikipedia - Hugging Face - History | 314 | 1,487 | null |
Section: Services and technologies > Hugging Face Hub. The Hugging Face Hub is a platform (centralized web service) for hosting: Git-based code repositories, including discussions and pull requests for projects. models, also with Git-based version control; datasets, mainly in text, images, and audio; web applications (... | Wikipedia - Hugging Face - Services and technologies > Hugging Face Hub | 163 | 825 | null |
Section: Services and technologies > Safetensors. The safetensors format was developed around 2021 to solve problems with the pickle format in python. It was designed for saving and loading tensors. Compared to pickle format, it allows lazy loading, and avoids security problems. After a security audit, it became the de... | Wikipedia - Hugging Face - Services and technologies > Safetensors | 175 | 629 | null |
Section: Simulation > Benefits. Human-in-the-loop allows the user to change the outcome of an event or process. The immersion effectively contributes to a positive transfer of acquired skills into the real world. This can be demonstrated by trainees utilizing flight simulators in preparation to become pilots. HITL also... | Wikipedia - Human-in-the-loop - Simulation > Benefits | 303 | 1,667 | null |
Article: Hyperparameter (machine learning). In machine learning, a hyperparameter is a parameter that can be set in order to define any configurable part of a model's learning process. Hyperparameters can be classified as either model hyperparameters (such as the topology and size of a neural network) or algorithm hype... | Wikipedia - Hyperparameter (machine learning) - Summary | 296 | 1,482 | null |
Section: Considerations > Robustness. An inherent stochasticity in learning directly implies that the empirical hyperparameter performance is not necessarily its true performance. Methods that are not robust to simple changes in hyperparameters, random seeds, or even different implementations of the same algorithm cann... | Wikipedia - Hyperparameter (machine learning) - Considerations > Robustness | 155 | 879 | null |
Section: Approaches > Grid search. The traditional method for hyperparameter optimization has been grid search, or a parameter sweep, which is simply an exhaustive searching through a manually specified subset of the hyperparameter space of a learning algorithm. A grid search algorithm must be guided by some performanc... | Wikipedia - Hyperparameter optimization - Approaches > Grid search | 347 | 1,505 | null |
Section: Approaches > Random search. Random Search replaces the exhaustive enumeration of all combinations by selecting them randomly. This can be simply applied to the discrete setting described above, but also generalizes to continuous and mixed spaces. A benefit over grid search is that random search can explore man... | Wikipedia - Hyperparameter optimization - Approaches > Random search | 171 | 946 | null |
Section: Approaches > Bayesian optimization. Bayesian optimization is a global optimization method for noisy black-box functions. Applied to hyperparameter optimization, Bayesian optimization builds a probabilistic model of the function mapping from hyperparameter values to the objective evaluated on a validation set. ... | Wikipedia - Hyperparameter optimization - Approaches > Bayesian optimization | 183 | 975 | null |
Section: Approaches > Gradient-based optimization. For specific learning algorithms, it is possible to compute the gradient with respect to hyperparameters and then optimize the hyperparameters using gradient descent. The first usage of these techniques was focused on neural networks. Since then, these methods have bee... | Wikipedia - Hyperparameter optimization - Approaches > Gradient-based optimization | 331 | 1,769 | null |
Section: Approaches > Evolutionary optimization. Evolutionary optimization is a methodology for the global optimization of noisy black-box functions. In hyperparameter optimization, evolutionary optimization uses evolutionary algorithms to search the space of hyperparameters for a given algorithm. Evolutionary hyperpar... | Wikipedia - Hyperparameter optimization - Approaches > Evolutionary optimization | 216 | 1,223 | null |
Section: Approaches > Population-based. Population Based Training (PBT) learns both hyperparameter values and network weights. Multiple learning processes operate independently, using different hyperparameters. As with evolutionary methods, poorly performing models are iteratively replaced with models that adopt modifi... | Wikipedia - Hyperparameter optimization - Approaches > Population-based | 175 | 937 | null |
Section: Approaches > Early stopping-based. A class of early stopping-based hyperparameter optimization algorithms is purpose built for large search spaces of continuous and discrete hyperparameters, particularly when the computational cost to evaluate the performance of a set of hyperparameters is high. Irace implemen... | Wikipedia - Hyperparameter optimization - Approaches > Early stopping-based | 209 | 1,112 | null |
Section: Issues with hyperparameter optimization. When hyperparameter optimization is done, the set of hyperparameters are often fitted on a training set and selected based on the generalization performance, or score, of a validation set. However, this procedure is at risk of overfitting the hyperparameters to the vali... | Wikipedia - Hyperparameter optimization - Issues with hyperparameter optimization | 223 | 1,109 | null |
Article: Prompt engineering. Prompt engineering is the process of structuring or crafting an instruction in order to produce the best possible output from a generative artificial intelligence (AI) model. A prompt is natural language text describing the task that an AI should perform. A prompt for a text-to-text languag... | Wikipedia - Prompt engineering - Summary | 210 | 999 | null |
Section: History. In 2018, researchers first proposed that all previously separate tasks in natural language processing (NLP) could be cast as a question-answering problem over a context. In addition, they trained a first single, joint, multi-task model that would answer any task-related question like "What is the sent... | Wikipedia - Prompt engineering - History | 260 | 1,197 | null |
Section: Text-to-text > Chain-of-thought. According to Google Research, chain-of-thought (CoT) prompting is a technique that allows large language models (LLMs) to solve a problem as a series of intermediate steps before giving a final answer. In 2022, Google Brain reported that chain-of-thought prompting improves reas... | Wikipedia - Prompt engineering - Text-to-text > Chain-of-thought | 349 | 1,562 | null |
Section: Text-to-text > In-context learning. In-context learning, refers to a model's ability to temporarily learn from prompts. For example, a prompt may include a few examples for a model to learn from, such as asking the model to complete "maison β house, chat β cat, chien β" (the expected response being dog), an ap... | Wikipedia - Prompt engineering - Text-to-text > In-context learning | 180 | 825 | null |
Section: Text-to-text > Prompting to estimate model sensitivity. Research consistently demonstrates that LLMs are highly sensitive to subtle variations in prompt formatting, structure, and linguistic properties. Some studies have shown up to 76 accuracy points across formatting changes in few-shot settings. Linguistic ... | Wikipedia - Prompt engineering - Text-to-text > Prompting to estimate model sensitivity | 194 | 1,139 | null |
Section: Text-to-text > Automatic prompt generation > Retrieval-augmented generation. Retrieval-augmented generation (RAG) is a technique that enables generative artificial intelligence (Gen AI) models to retrieve and incorporate new information. It modifies interactions with a large language model (LLM) so that the mo... | Wikipedia - Prompt engineering - Text-to-text > Automatic prompt generation > Retrieval-augmented generation | 248 | 1,265 | null |
Section: Text-to-text > Automatic prompt generation > Using language models to generate prompts. Large language models (LLM) themselves can be used to compose prompts for large language models. The automatic prompt engineer algorithm uses one LLM to beam search over prompts for another LLM: There are two LLMs. One is t... | Wikipedia - Prompt engineering - Text-to-text > Automatic prompt generation > Using language models to generate prompts | 295 | 1,446 | null |
Section: Text-to-image > Prompt formats. Early text-to-image models typically don't understand negation, grammar and sentence structure in the same way as large language models, and may thus require a different set of prompting techniques. The prompt "a party with no cake" may produce an image including a cake. As an a... | Wikipedia - Prompt engineering - Text-to-image > Prompt formats | 282 | 1,339 | null |
Section: Non-text prompts > Using gradient descent to search for prompts. In "prefix-tuning", "prompt tuning", or "soft prompting", floating-point-valued vectors are searched directly by gradient descent to maximize the log-likelihood on outputs. Formally, let E = { e 1 , β¦ , e k } {\displaystyle \mathbf {E} =\{\mathbf... | Wikipedia - Prompt engineering - Non-text prompts > Using gradient descent to search for prompts | 322 | 895 | null |
During training, the tunable embeddings, input, and output tokens are concatenated into a single sequence concat ( E ; X ; Y ) {\displaystyle {\text{concat}}(\mathbf {E} ;\mathbf {X} ;\mathbf {Y} )} , and fed to the LLMs. The losses are computed over the Y {\displaystyle \mathbf {Y} } tokens; the gradients are backprop... | Wikipedia - Prompt engineering - Non-text prompts > Using gradient descent to search for prompts | 241 | 808 | null |
Let an LLM be written as L L M ( X ) = F ( E ( X ) ) {\displaystyle LLM(X)=F(E(X))} , where X {\displaystyle X} is a sequence of linguistic tokens, E {\displaystyle E} is the token-to-vector function, and F {\displaystyle F} is the rest of the model. In prefix-tuning, one provides a set of input-output pairs { ( X i , ... | Wikipedia - Prompt engineering - Non-text prompts > Using gradient descent to search for prompts | 225 | 558 | null |
In prefix-tuning, one provides a set of input-output pairs { ( X i , Y i ) } i {\displaystyle \{(X^{i},Y^{i})\}_{i}} , and then use gradient descent to search for arg β‘ max Z ~ β i log β‘ P r [ Y i | Z ~ β E ( X i ) ] {\displaystyle \arg \max _{\tilde {Z}}\sum _{i}\log Pr[Y^{i}|{\tilde {Z}}\ast E(X^{i})]} . In words, lo... | Wikipedia - Prompt engineering - Non-text prompts > Using gradient descent to search for prompts | 337 | 864 | null |
For prefix tuning, it is similar, but the "prefix vector" Z ~ {\displaystyle {\tilde {Z}}} is pre-appended to the hidden states in every layer of the model. An earlier result uses the same idea of gradient descent search, but is designed for masked language models like BERT, and searches only over token sequences, rath... | Wikipedia - Prompt engineering - Non-text prompts > Using gradient descent to search for prompts | 182 | 599 | null |
Article: Inauthentic text. An inauthentic text is a computer-generated expository document meant to appear as genuine, but which is actually meaningless. Frequently they are created in order to be intermixed with genuine documents and thus manipulate the results of search engines, as with Spam blogs. They are also carr... | Wikipedia - Inauthentic text - Summary | 349 | 1,737 | null |
Article: Inception score. The Inception Score (IS) is an algorithm used to assess the quality of images created by a generative image model such as a generative adversarial network (GAN). The score is calculated based on the output of a separate, pretrained Inception v3 image classification model applied to a sample of... | Wikipedia - Inception score - Summary | 245 | 1,225 | null |
Section: Definition. Let there be two spaces, the space of images Ξ© X {\displaystyle \Omega _{X}} and the space of labels Ξ© Y {\displaystyle \Omega _{Y}} . The space of labels is finite. Let p g e n {\displaystyle p_{gen}} be a probability distribution over Ξ© X {\displaystyle \Omega _{X}} that we wish to judge. Let a d... | Wikipedia - Inception score - Definition | 279 | 873 | null |
The Inception Score of p g e n {\displaystyle p_{gen}} relative to p d i s {\displaystyle p_{dis}} is I S ( p g e n , p d i s ) := exp β‘ ( E x βΌ p g e n [ D K L ( p d i s ( β
| x ) β β« p d i s ( β
| x ) p g e n ( x ) d x ) ] ) {\displaystyle IS(p_{gen},p_{dis}):=\exp \left(\mathbb {E} _{x\sim p_{gen}}\left[D_{KL}\left(... | Wikipedia - Inception score - Definition | 347 | 666 | null |
p_{gen}}[p_{dis}(\cdot |x)]\right)\right]} ln β‘ I S ( p g e n , p d i s ) := H [ E x βΌ p g e n [ p d i s ( β
| x ) ] ] β E x βΌ p g e n [ H [ p d i s ( β
| x ) ] ] {\displaystyle \ln IS(p_{gen},p_{dis}):=H[\mathbb {E} _{x\sim p_{gen}}[p_{dis}(\cdot |x)]]-\mathbb {E} _{x\sim p_{gen}}[H[p_{dis}(\cdot |x)]]} ln β‘ I S {\dis... | Wikipedia - Inception score - Definition | 312 | 693 | null |
Compute p d i s ( β
| x i ) {\displaystyle p_{dis}(\cdot |x_{i})} , the probability distribution over labels conditional on image x i {\displaystyle x_{i}} . Sum up the results to obtain p ^ {\displaystyle {\hat {p}}} , an empirical estimate of β« p d i s ( β
| x ) p g e n ( x ) d x {\displaystyle \int p_{dis}(\cdot |x)... | Wikipedia - Inception score - Definition | 228 | 591 | null |
Section: Definition > Interpretation. A higher inception score is interpreted as "better", as it means that p g e n {\displaystyle p_{gen}} is a "sharp and distinct" collection of pictures. ln β‘ I S ( p g e n , p d i s ) β [ 0 , ln β‘ N ] {\displaystyle \ln IS(p_{gen},p_{dis})\in [0,\ln N]} , where N {\displaystyle N} i... | Wikipedia - Inception score - Definition > Interpretation | 350 | 895 | null |
That is, for any image x {\displaystyle x} sampled from p g e n {\displaystyle p_{gen}} , discriminator returns exactly the same label predictions p d i s ( β
| x ) {\displaystyle p_{dis}(\cdot |x)} . The highest inception score N {\displaystyle N} is achieved if and only if the two conditions are both true: For almost... | Wikipedia - Inception score - Definition > Interpretation | 335 | 973 | null |
Article: Inductive bias. The inductive bias (also known as learning bias) of a learning algorithm is the set of assumptions that the learner uses to predict outputs of given inputs that it has not encountered. Inductive bias is anything which makes the algorithm learn one pattern instead of another pattern (e.g., step-... | Wikipedia - Inductive bias - Summary | 333 | 1,756 | null |
Section: Types. The following is a list of common inductive biases in machine learning algorithms. Maximum conditional independence: if the hypothesis can be cast in a Bayesian framework, try to maximize conditional independence. This is the bias used in the Naive Bayes classifier. Minimum cross-validation error: when ... | Wikipedia - Inductive bias - Types | 300 | 1,545 | null |
Article: Inductive probability. Inductive probability attempts to give the probability of future events based on past events. It is the basis for inductive reasoning, and gives the mathematical basis for learning and the perception of patterns. It is a source of knowledge about the world. There are three sources of kno... | Wikipedia - Inductive probability - Summary | 256 | 1,322 | null |
Section: History. Probability and statistics was focused on probability distributions and tests of significance. Probability was formal, well defined, but limited in scope. In particular its application was limited to situations that could be defined as an experiment or trial, with a well defined population. Bayes's th... | Wikipedia - Inductive probability - History | 272 | 1,415 | null |
Section: History > Minimum description/message length. The program with the shortest length that matches the data is the most likely to predict future data. This is the thesis behind the minimum message length and minimum description length methods. At first sight Bayes' theorem appears different from the minimimum mes... | Wikipedia - Inductive probability - History > Minimum description/message length | 309 | 1,135 | null |
Section: History > Inference based on program complexity > Detecting patterns in the data. If all the bits are 1, then people infer that there is a bias in the coin and that it is more likely also that the next bit is 1 also. This is described as learning from, or detecting a pattern in the data. Such a pattern may be ... | Wikipedia - Inductive probability - History > Inference based on program complexity > Detecting patterns in the data | 208 | 866 | null |
Section: History > Inference based on program complexity > Considering all theories. The Greek philosopher Epicurus is quoted as saying "If more than one theory is consistent with the observations, keep all theories". As in a crime novel all theories must be considered in determining the likely murderer, so with induct... | Wikipedia - Inductive probability - History > Inference based on program complexity > Considering all theories | 223 | 1,045 | null |
Section: History > Inference based on program complexity > Universal priors. The programming language affects the predictions of the next bit in the string. The language acts as a prior probability. This is particularly a problem where the programming language codes for numbers and other data types. Intuitively we thin... | Wikipedia - Inductive probability - History > Inference based on program complexity > Universal priors | 307 | 1,658 | null |
Section: History > Universal artificial intelligence. The theory of universal artificial intelligence applies decision theory to inductive probabilities. The theory shows how the best actions to optimize a reward function may be chosen. The result is a theoretical model of intelligence. It is a fundamental theory of in... | Wikipedia - Inductive probability - History > Universal artificial intelligence | 235 | 1,229 | null |
Section: Probability. Probability is the representation of uncertain or partial knowledge about the truth of statements. Probabilities are subjective and personal estimates of likely outcomes based on past experience and inferences made from the data. This description of probability may seem strange at first. In natura... | Wikipedia - Inductive probability - Probability | 309 | 1,685 | null |
Section: Probability > Probability as estimation. The principle of indifference has played a key role in probability theory. It says that if N statements are symmetric so that one condition cannot be preferred over another then all statements are equally probable. Taken seriously, in evaluating probability this princip... | Wikipedia - Inductive probability - Probability > Probability as estimation | 301 | 1,621 | null |
Section: Probability and information. Whereas logic represents only two values; true and false as the values of statement, probability associates a number in [0,1] to each statement. If the probability of a statement is 0, the statement is false. If the probability of a statement is 1 the statement is true. In consider... | Wikipedia - Inductive probability - Probability and information | 217 | 873 | null |
Section: Probability and information > Combining information. Two statements A {\displaystyle A} and B {\displaystyle B} may be represented by two separate encodings. Then the length of the encoding is, L ( A β§ B ) = L ( A ) + L ( B ) {\displaystyle L(A\land B)=L(A)+L(B)} or in terms of probability, P ( A β§ B ) = P ( A... | Wikipedia - Inductive probability - Probability and information > Combining information | 185 | 612 | null |
Section: Probability and information > The internal language of information. The primary use of the information approach to probability is to provide estimates of the complexity of statements. Recall that Occam's razor states that "All things being equal, the simplest theory is the most likely to be correct". In order ... | Wikipedia - Inductive probability - Probability and information > The internal language of information | 231 | 1,262 | null |
Section: Probability and information > The internal language of information > Encoding expressions. An expression is constructed from sub expressions, Constants (including function identifier). Application of functions. quantifiers. A Huffman code must distinguish the 3 cases. The length of each code is based on the fr... | Wikipedia - Inductive probability - Probability and information > The internal language of information > Encoding expressions | 182 | 901 | null |
Section: Probability and frequency > Conditional probability. Each probability is always associated with the state of knowledge at a particular point in the argument. Probabilities before an inference are known as prior probabilities, and probabilities after are known as posterior probabilities. Probability depends on ... | Wikipedia - Inductive probability - Probability and frequency > Conditional probability | 216 | 906 | null |
Section: Probability and frequency > The frequentist approach applied to possible worlds. In the frequentist approach, probabilities are defined as the ratio of the number of outcomes within an event to the total number of outcomes. In the possible world model each possible world is an outcome, and statements about pos... | Wikipedia - Inductive probability - Probability and frequency > The frequentist approach applied to possible worlds | 173 | 717 | null |
The probability of a statement A {\displaystyle A} being true about possible worlds is then, P ( A ) = | { x : A ( x ) } | | x : β€ | {\displaystyle P(A)={\frac {|\{x:A(x)\}|}{|x:\top |}}} For a conditional probability. P ( B | A ) = | { x : A ( x ) β§ B ( x ) } | | x : A ( x ) | {\displaystyle P(B|A)={\frac {|\{x:A(x)\l... | Wikipedia - Inductive probability - Probability and frequency > The frequentist approach applied to possible worlds | 404 | 780 | null |
P ( B | A ) = | { x : A ( x ) β§ B ( x ) } | | x : A ( x ) | {\displaystyle P(B|A)={\frac {|\{x:A(x)\land B(x)\}|}{|x:A(x)|}}} then P ( A β§ B ) = | { x : A ( x ) β§ B ( x ) } | | x : β€ | = | { x : A ( x ) β§ B ( x ) } | | { x : A ( x ) } | | { x : A ( x ) } | | x : β€ | = P ( A ) P ( B | A ) {\displaystyle {\begin{aligned}... | Wikipedia - Inductive probability - Probability and frequency > The frequentist approach applied to possible worlds | 402 | 772 | null |
P ( A β§ B ) = P ( A ) P ( B | A ) = P ( B ) P ( A | B ) {\displaystyle P(A\land B)=P(A)P(B|A)=P(B)P(A|B)} This law describes the relationship between prior and posterior probabilities when new facts are learnt. Written as quantities of information Bayes' Theorem becomes, L ( A β§ B ) = L ( A ) + L ( B | A ) = L ( B ) + ... | Wikipedia - Inductive probability - Probability and frequency > The frequentist approach applied to possible worlds | 257 | 670 | null |
β i P ( A i | B ) = 1 {\displaystyle \sum _{i}{P(A_{i}|B)}=1} Substituting using Bayes' theorem gives the law of total probability β i P ( B | A i ) P ( A i ) = β i P ( A i | B ) P ( B ) {\displaystyle \sum _{i}{P(B|A_{i})P(A_{i})}=\sum _{i}{P(A_{i}|B)P(B)}} P ( B ) = β i P ( B | A i ) P ( A i ) {\displaystyle P(B)=\su... | Wikipedia - Inductive probability - Probability and frequency > The law of total of probability | 346 | 727 | null |
Section: Probability and frequency > Alternate possibilities. For mutually exclusive possibilities, the probabilities add. P ( A β¨ B ) = P ( A ) + P ( B ) , if P ( A β§ B ) = 0 {\displaystyle P(A\lor B)=P(A)+P(B),\qquad {\text{if }}P(A\land B)=0} Using A β¨ B = ( A β§ Β¬ ( A β§ B ) ) β¨ ( B β§ Β¬ ( A β§ B ) ) β¨ ( A β§ B ) {\disp... | Wikipedia - Inductive probability - Probability and frequency > Alternate possibilities | 241 | 581 | null |
Also, ( A β§ Β¬ ( A β§ B ) ) β¨ ( A β§ B ) = A {\displaystyle (A\land \neg (A\land B))\lor (A\land B)=A} P ( A β§ Β¬ ( A β§ B ) ) + P ( A β§ B ) = P ( A ) {\displaystyle P(A\land \neg (A\land B))+P(A\land B)=P(A)} P ( A β§ Β¬ ( A β§ B ) ) = P ( A ) β P ( A β§ B ) {\displaystyle P(A\land \neg (A\land B))=P(A)-P(A\land B)} so, puttin... | Wikipedia - Inductive probability - Probability and frequency > Alternate possibilities | 344 | 679 | null |
Section: Probability and frequency > Implication and condition probability. Implication is related to conditional probability by the following equation, A β B βΊ P ( B | A ) = 1 {\displaystyle A\to B\iff P(B|A)=1} Derivation, A β B βΊ P ( A β B ) = 1 βΊ P ( A β§ B β¨ Β¬ A ) = 1 βΊ P ( A β§ B ) + P ( Β¬ A ) = 1 βΊ P ( A β§ B ) = P... | Wikipedia - Inductive probability - Probability and frequency > Implication and condition probability | 268 | 580 | null |
Section: Bayesian hypothesis testing. Bayes' theorem may be used to estimate the probability of a hypothesis or theory H, given some facts F. The posterior probability of H is then P ( H | F ) = P ( H ) P ( F | H ) P ( F ) {\displaystyle P(H|F)={\frac {P(H)P(F|H)}{P(F)}}} or in terms of information, P ( H | F ) = 2 β (... | Wikipedia - Inductive probability - Bayesian hypothesis testing | 341 | 1,118 | null |
Given all mutually exclusive hypothesis H i {\displaystyle H_{i}} which give evidence, such that, L ( H i ) + L ( F | H i ) < L ( F ) {\displaystyle L(H_{i})+L(F|H_{i})<L(F)} and also the hypothesis R, that none of the hypothesis is true, then, P ( H i | F ) = P ( H i ) P ( F | H i ) P ( F | R ) + β j P ( H j ) P ( F |... | Wikipedia - Inductive probability - Bayesian hypothesis testing > Set of hypothesis | 350 | 677 | null |
) + β j 2 β ( L ( H j ) + L ( F | H j ) ) P ( R | F ) = 2 β L ( F | R ) 2 β L ( F | R ) + β j 2 β ( L ( H j ) + L ( F | H j ) ) {\displaystyle {\begin{aligned}P(H_{i}|F)&={\frac {2^{-(L(H_{i})+L(F|H_{i}))}}{2^{-L(F|R)}+\sum _{j}2^{-(L(H_{j})+L(F|H_{j}))}}}\\[8pt]P(R|F)&={\frac {2^{-L(F|R)}}{2^{-L(F|R)}+\sum _{j}{2^{-(L... | Wikipedia - Inductive probability - Bayesian hypothesis testing > Set of hypothesis | 350 | 625 | null |
( H j ) + L ( F | H j ) ) P ( R | F ) β 2 β L ( F ) 2 β L ( F ) + β j 2 β ( L ( H j ) + L ( F | H j ) ) {\displaystyle {\begin{aligned}P(H_{i}|F)&\approx {\frac {2^{-(L(H_{i})+L(F|H_{i}))}}{2^{-L(F)}+\sum _{j}{2^{-(L(H_{j})+L(F|H_{j}))}}}}\\[8pt]P(R|F)&\approx {\frac {2^{-L(F)}}{2^{-L(F)}+\sum _{j}{2^{-(L(H_{j})+L(F|H_... | Wikipedia - Inductive probability - Bayesian hypothesis testing > Set of hypothesis | 238 | 344 | null |
Section: Boolean inductive inference. Abductive inference starts with a set of facts F which is a statement (Boolean expression). Abductive reasoning is of the form, A theory T implies the statement F. As the theory T is simpler than F, abduction says that there is a probability that the theory T is implied by F. The t... | Wikipedia - Inductive probability - Boolean inductive inference | 285 | 1,142 | null |
Section: Boolean inductive inference > Generalization and specialization. Inductive inference is related to generalization. Generalizations may be formed from statements by replacing a specific value with membership of a category, or by replacing membership of a category with membership of a broader category. In deduct... | Wikipedia - Inductive probability - Boolean inductive inference > Generalization and specialization | 307 | 1,662 | null |
Section: Boolean inductive inference > Newton's use of induction. Isaac Newton used inductive arguments in constructing his law of universal gravitation. Starting with the statement, The center of an apple falls towards the center of the Earth. Generalizing by replacing apple for object, and Earth for object gives, in ... | Wikipedia - Inductive probability - Boolean inductive inference > Newton's use of induction | 333 | 1,143 | null |
Using Galileo's observation that all objects drop with the same speed, F 1 = m 1 a 1 = m 1 k 1 r 2 i 1 {\displaystyle F_{1}=m_{1}a_{1}={\frac {m_{1}k_{1}}{r^{2}}}i_{1}} F 2 = m 2 a 2 = m 2 k 2 r 2 i 2 {\displaystyle F_{2}=m_{2}a_{2}={\frac {m_{2}k_{2}}{r^{2}}}i_{2}} where i 1 {\displaystyle i_{1}} and i 2 {\displaystyl... | Wikipedia - Inductive probability - Boolean inductive inference > Newton's use of induction | 254 | 512 | null |
Section: Boolean inductive inference > Probabilities for inductive inference. Implication determines condition probability as, T β F βΊ P ( F | T ) = 1 {\displaystyle T\to F\iff P(F|T)=1} So, P ( F | T ) = 1 {\displaystyle P(F|T)=1} L ( F | T ) = 0 {\displaystyle L(F|T)=0} This result may be used in the probabilities gi... | Wikipedia - Inductive probability - Boolean inductive inference > Probabilities for inductive inference | 285 | 758 | null |
Then for a set of theories T i = H i {\displaystyle T_{i}=H_{i}} , such that L ( T i ) < L ( F ) {\displaystyle L(T_{i})<L(F)} , P ( T i | F ) = P ( T i ) P ( F | R ) + β j P ( T j ) {\displaystyle P(T_{i}|F)={\frac {P(T_{i})}{P(F|R)+\sum _{j}{P(T_{j})}}}} P ( R | F ) = P ( F | R ) P ( F | R ) + β j P ( T j ) {\display... | Wikipedia - Inductive probability - Boolean inductive inference > Probabilities for inductive inference | 350 | 598 | null |
Section: Derivations > Derivation of inductive probability. Make a list of all the shortest programs K i {\displaystyle K_{i}} that each produce a distinct infinite string of bits, and satisfy the relation, T n ( R ( K i ) ) = x {\displaystyle T_{n}(R(K_{i}))=x} where R ( K i ) {\displaystyle R(K_{i})} is the result of... | Wikipedia - Inductive probability - Derivations > Derivation of inductive probability | 322 | 921 | null |
This is represented by the conditional probability, P ( s = R ( K i ) | T n ( s ) = x ) {\displaystyle P(s=R(K_{i})|T_{n}(s)=x)} Using the extended form of Bayes' theorem P ( s = R ( K i ) | T n ( s ) = x ) = P ( T n ( s ) = x | s = R ( K i ) ) P ( s = R ( K i ) ) β j P ( T n ( s ) = x | s = R ( K j ) ) P ( s = R ( K j... | Wikipedia - Inductive probability - Derivations > Derivation of inductive probability | 330 | 720 | null |
This must be true, as in the limit as n β β , {\displaystyle n\to \infty ,} there is always at least one program that produces T n ( s ) {\displaystyle T_{n}(s)} . As K i {\displaystyle K_{i}} are chosen so that T n ( R ( K i ) ) = x , {\displaystyle T_{n}(R(K_{i}))=x,} then, P ( T n ( s ) = x | s = R ( K i ) ) = 1 {\d... | Wikipedia - Inductive probability - Derivations > Derivation of inductive probability | 278 | 666 | null |
As K i {\displaystyle K_{i}} are chosen so that T n ( R ( K i ) ) = x , {\displaystyle T_{n}(R(K_{i}))=x,} then, P ( T n ( s ) = x | s = R ( K i ) ) = 1 {\displaystyle P(T_{n}(s)=x|s=R(K_{i}))=1} The apriori probability of the string being produced from the program, given no information about the string, is based on th... | Wikipedia - Inductive probability - Derivations > Derivation of inductive probability | 313 | 679 | null |
{\displaystyle P(s=R(K_{i})|T_{n}(s)=x)={\frac {2^{-I(K_{i})}}{\sum _{j}2^{-I(K_{j})}}}.} Programs that are the same or longer than the length of x provide no predictive power. Separate them out giving, P ( s = R ( K i ) | T n ( s ) = x ) = 2 β I ( K i ) β j : I ( K j ) < n 2 β I ( K j ) + β j : I ( K j ) β©Ύ n 2 β I ( K... | Wikipedia - Inductive probability - Derivations > Derivation of inductive probability | 157 | 326 | null |
Separate them out giving, P ( s = R ( K i ) | T n ( s ) = x ) = 2 β I ( K i ) β j : I ( K j ) < n 2 β I ( K j ) + β j : I ( K j ) β©Ύ n 2 β I ( K j ) . {\displaystyle P(s=R(K_{i})|T_{n}(s)=x)={\frac {2^{-I(K_{i})}}{\sum _{j:I(K_{j})<n}2^{-I(K_{j})}+\sum _{j:I(K_{j})\geqslant n}2^{-I(K_{j})}}}.} Then identify the two prob... | Wikipedia - Inductive probability - Derivations > Derivation of inductive probability | 370 | 693 | null |
{\displaystyle P(s=R(K_{i})|T_{n}(s)=x)={\frac {2^{-I(K_{i})}}{\sum _{j:I(K_{j})<n}2^{-I(K_{j})}+\sum _{j:I(K_{j})\geqslant n}2^{-I(K_{j})}}}.} Then identify the two probabilities as, P ( x has pattern ) = β j : I ( K j ) < n 2 β I ( K j ) {\displaystyle P(x{\text{ has pattern}})=\sum _{j:I(K_{j})<n}2^{-I(K_{j})}} P ( ... | Wikipedia - Inductive probability - Derivations > Derivation of inductive probability | 353 | 643 | null |
So, P ( s = R ( K i ) | T n ( s ) = x ) = 2 β I ( K i ) 2 β n + β j : I ( K j ) < n 2 β I ( K j ) . {\displaystyle P(s=R(K_{i})|T_{n}(s)=x)={\frac {2^{-I(K_{i})}}{2^{-n}+\sum _{j:I(K_{j})<n}2^{-I(K_{j})}}}.} The probability that the source is random, or unpredictable is, P ( random β‘ ( s ) | T n ( s ) = x ) = 2 β n 2 β... | Wikipedia - Inductive probability - Derivations > Derivation of inductive probability | 275 | 477 | null |
Section: Derivations > A model for inductive inference. A model of how worlds are constructed is used in determining the probabilities of theories, A random bit string is selected. A condition is constructed from the bit string. A world is constructed that is consistent with the condition. If w is the bit string then t... | Wikipedia - Inductive probability - Derivations > A model for inductive inference | 255 | 870 | null |
Section: Derivations > A model for inductive inference > Applying Bayes' theorem. extended form of Bayes' theorem may be applied P ( A i | B ) = P ( B | A i ) P ( A i ) β j P ( B | A j ) P ( A j ) , {\displaystyle P(A_{i}|B)={\frac {P(B|A_{i})\,P(A_{i})}{\sum _{j}P(B|A_{j})\,P(A_{j})}},} where, B = E ( C ) {\displaysty... | Wikipedia - Inductive probability - Derivations > A model for inductive inference > Applying Bayes' theorem | 237 | 577 | null |
For T ( C ) {\displaystyle T(C)} to be a partition, no bit string n may belong to two theories. To prove this assume they can and derive a contradiction, ( N β T ) β§ ( N β M ) β§ ( N β M ) β§ ( n β E ( N ) β§ n β E ( M ) ) {\displaystyle (N\in T)\land (N\in M)\land (N\neq M)\land (n\in E(N)\land n\in E(M))} βΉ ( N β M ) β§ ... | Wikipedia - Inductive probability - Derivations > A model for inductive inference > Applying Bayes' theorem | 228 | 628 | null |
So Bayes theorem may be applied as specified giving, β t β T ( C ) , P ( E ( t ) | E ( C ) ) = P ( E ( t ) ) β
P ( E ( C ) | E ( t ) ) β j β T ( C ) P ( E ( j ) ) β
P ( E ( C ) | E ( j ) ) {\displaystyle \forall t\in T(C),P(E(t)|E(C))={\frac {P(E(t))\cdot P(E(C)|E(t))}{\sum _{j\in T(C)}P(E(j))\cdot P(E(C)|E(j))}}} Usin... | Wikipedia - Inductive probability - Derivations > A model for inductive inference > Applying Bayes' theorem | 350 | 718 | null |
) , P ( E ( t ) | E ( C ) ) = β n : R ( n ) β‘ t 2 β L ( n ) β j β T ( C ) β m : R ( m ) β‘ j 2 β L ( m ) {\displaystyle \forall t\in T(C),P(E(t)|E(C))={\frac {\sum _{n:R(n)\equiv t}2^{-L(n)}}{\sum _{j\in T(C)}\sum _{m:R(m)\equiv j}2^{-L(m)}}}} Finally the probabilities of the events may be identified with the probabilit... | Wikipedia - Inductive probability - Derivations > A model for inductive inference > Applying Bayes' theorem | 350 | 686 | null |
Section: Derivations > A model for inductive inference > Removing theories without predictive power. Theories that are less probable than the condition C have no predictive power. Separate them out giving, β t β T ( C ) , P ( t | C ) = P ( E ( t ) ) ( β j : j β T ( C ) β§ P ( E ( j ) ) > P ( E ( C ) ) P ( E ( j ) ) ) + ... | Wikipedia - Inductive probability - Derivations > A model for inductive inference > Removing theories without predictive power | 267 | 641 | null |
So, P ( E ( C ) ) = β j : j β T ( C ) β§ P ( E ( j ) ) β€ P ( E ( C ) ) P ( j ) {\displaystyle P(E(C))=\sum _{j:j\in T(C)\land P(E(j))\leq P(E(C))}P(j)} So the probability β t β T ( C ) , P ( t | C ) = P ( E ( t ) ) P ( E ( C ) ) + β j : j β T ( C ) β§ P ( E ( j ) ) > P ( E ( C ) ) P ( E ( j ) ) {\displaystyle \forall t\i... | Wikipedia - Inductive probability - Derivations > A model for inductive inference > Removing theories without predictive power | 350 | 689 | null |
{P(E(C))}{P(E(C))+\sum _{j:j\in T(C)\land P(E(j))>P(E(C))}P(E(j))}}} The probability of a condition was given as, β t , P ( E ( t ) ) = β n : R ( n ) β‘ t 2 β L ( n ) {\displaystyle \forall t,P(E(t))=\sum _{n:R(n)\equiv t}2^{-L(n)}} Bit strings for theories that are more complex than the bit string given to the agent as... | Wikipedia - Inductive probability - Derivations > A model for inductive inference > Removing theories without predictive power | 173 | 412 | null |
To implement this a new definition is given as F in, β t , P ( F ( t , c ) ) = β n : R ( n ) β‘ t β§ L ( n ) < L ( c ) 2 β L ( n ) {\displaystyle \forall t,P(F(t,c))=\sum _{n:R(n)\equiv t\land L(n)<L(c)}2^{-L(n)}} Using F, an improved version of the abductive probabilities is, β t β T ( C ) , P ( t | C ) = P ( F ( t , c ... | Wikipedia - Inductive probability - Derivations > A model for inductive inference > Removing theories without predictive power | 350 | 674 | null |
Article: Inductive programming. Inductive programming (IP) is a special area of automatic programming, covering research from artificial intelligence and programming, which addresses learning of typically declarative (logic or functional) and often recursive programs from incomplete specifications, such as input/output... | Wikipedia - Inductive programming - Summary | 155 | 858 | null |
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