Probabilistic Models in Cognitive Science and Artificial Intelligence

Transcription

1 Probabilistic Models in Cognitive Science and Artificial Intelligence

2 Very Brief History of Cog Sci and AI 1950 s-1980 s Symbolic models of cognition von Neumann computer architecture as metaphor 1980 s-1990 s Connectionist models of cognition Massively parallel neuron-like networks of simple processors as metaphor Late 1990 s -? Probabilistic / statistical models of cognition Formalizes the best of connectionist (subsymbolic) ideas

3 Relation of Probabilistic Models to Connectionist and Symbolic Models Connectionist models Probabilistic models Symbolic models weak (unknown) bias ad hoc, implicit incorporation of prior knowledge & assumptions strong bias principled, elegant incorporation of prior knowledge & assumptions via predicate calculus statistical learning (large # examples) rule learning from (small # examples) vector representations structured representations

4 Frequentist notion Two Notions of Probability Relative frequency obtained if event were observed many times (e.g., coin flip) Subjective notion Degree of belief in some hypothesis Analogous to connectionist activation Long philosophical battle between these two views Subjective notion makes sense for cog sci and AI given that probabilities represent mental states

5 Why Probability? Randomness in the brain and world introduces uncertainty, and uncertainty is well described in the language of random events. Currency of probability provides strong constraints (vs. neural net activation) It s the optimal thing to compute, in the sense that any other strategy will lead to lower expected returns e.g., I bet you $1 that roll of die will produce number < 3. How much are you willing to wager?

6 Why Probability? Leads to elegant theories to be based on premise that human performance is optimal Rational theories, ideal observer theories Probably true in some areas of cognition (e.g., vision) More interesting: bounded rationality Optimality is assumed to be subject to limitations on processing hardware and capacity, representation, experience with the world. Explicit qualitative assumptions

7 Basic Probability (most slides borrowed with permission from Andrew Moore of CMU and Google)

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10 Notation Digression P(A) is shorthand for P(A=true) P(~A) is shorthand for P(A=false) Similar notation applies to other binary RVs: P(Gender=M), P(Gender=F) Same notation applies to multivalued RVs: P(Major=history), P(Age=19), P(Q=c) Note: upper case letters/names for variables, lower case letters/names for values For RVs that have values other than true and false, P(Q) is shorthand for P(Q=q) for some unknown q

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16 Q R S P(H F) = R/(Q+R) P(F H) = R/(S+R)

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31 If you have joint distribution, you can perform any inference in the domain.

32 Simpler probabilistic facts and some algebra

33 Directed Graphical Model Concept learning What Is A Bayes Net? From = 31 parameters to =10 Earthquake Burglary A node is conditionally independent of its ancestors given its parents. E.g., C is conditionally independent of R, E, and B given A Notation: C? R,B,E A Radio Alarm Call

34 Tenenbaum (1999) Concept learning E.g., glorch vs. not glorch E.g., word meanings E.g., edible food glorch not glorch not glorch glorch Focus on Learning concepts from positive examples Learning from a small number of examples Contrast with machine learning approaches and psychological models at the time

35 Domain Two dimensional continuous feature space Categories defined by axis-parallel rectangles e.g., feature dimensions cholesterol level (x 1 ) insulin level (x 2 ) e.g., concept healthy (C)

36 Hypothesis (Model) Space H: all rectangles on the plane, parameterized by (l 1, l 2, s 1, s 2 ) h: one particular hypothesis Consider all hypotheses in parallel In contrast to non-bayesian approach of maintaining only the best hypothesis at any point in time.

37 Prediction via Model Averaging Generalization function for unknown input Y given a set of n examples X = {x 1, x 2, x 3,, x n } p(y X) = h p(y & h X) Marginalization p(y & h X) = p(y h, X) p(h X) Chain rule p(y h, X) = p(y h) = 1 if y is in h p(h X) ~ p(x h) p(h) likelihood prior

38 Priors, p(h) Priors and Likelihood Functions Location invariant Uninformative prior (prior depends only on area of rectangle) Expected size prior x Likelihood function, p(x h) X = set of n examples Size principle

39 Expected size prior

40 Generalization Gradients MIN: smallest hypothesis consistent with data weak Bayes: instead of using size principle, assumes examples are produced by process independent of the true class Dark line = 50% prob.

41 Experimental Design Subjects shown n dots on screen that are randomly chosen examples from some rectangle of healthy levels n {2, 3, 4, 6, 10, 50} Dots varied in horizontal and vertical range Task r {.25,.5, 1, 2, 4, 8} units in a 24 unit window draw the true rectangle around the dots

42 Experimental Results

43 Method Summary of Tenenbaum (1999) Pick prior distribution (includes hypothesis space) Pick likelihood function leads to predictions for generalization as a function of r (range) and n (number of examples) Claims people generalize optimally given assumptions about priors and likelihood Bayesian approach provides best description of how people generalize on rectangle task. Explains how people can learn from a small number of examples, and only positive examples.

44 Important Ideas in Bayesian Models Generative models Likelihood function, prior distribution Consideration of multiple models in parallel Potentially infinite model space Inference prediction via model averaging role of priors diminishes with amount of evidence Learning Just another form of inference Bayesian Occam's razor: trade off between model simplicity and fit to data

45 Important Technical Issues Representing structured data Grammars Relational schemas (e.g., paper authors and topics Hierarchical models Allows for weaker assumptions at the cost of more complex inference Nonparametric models Flexible models that grow in complexity as the data justifies Approximate inference Markov chain Monte Carlo, particle filters, variational approximations

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47 Griffiths and Tenenbaum (2006) Optimal Predictions in Everyday Cognition If you were assessing an insurance case for an 18- year-old man, what would you predict for his lifespan? If you phoned a box office to book tickets and had been on hold for 3 minutes, what would you predict for the total time you would be on hold? If your friend read you her favorite line of poetry, and told you it was line 5 of a poem, what would you predict for the total length of the poem? If you opened a book about the history of ancient Egypt to a page listing the reigns of the pharaohs, and noticed that in 4000 BC a particular pharaoh had been ruling for 11 years, what would you predict for the total duration of his reign?

48 Griffiths and Tenenbaum Conclusion Average responses reveal a close correspondence between peoples implicit probabilistic models and the statistics of the world. People show a statistical sophistication and optimality of reasoning generally assumed to be absent in the domain of higher-order cognition.

49 Griffiths and Tenenbaum Bayesian Model If an individual has lived for t cur =50 years, how many years t total do you expect them to live?

50 What Does Optimality Entail? Individuals have complete, accurate knowledge about the domain priors. Fairly sophisticated computation involving Bayesian integral

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52 From The Economist (1/5/2006) [Griffiths and Tenenbuam] put the idea of a Bayesian brain to a quotidian test. They found that it passed with flying colors. The key to successful Bayesian reasoning is in having an appropriate prior With the correct prior, even a single piece of data can be used to make meaningful Bayesian predictions.

53 My Caution Bayesian formalism is sufficiently broad that nearly any theory can be cast in Bayesian terms E.g., adding two numbers as Bayesian inference Emphasis on how cognition conforms to Bayesian principles often directs attention away from important memory and processing limitations.

54 Problem Latent Dirichlet Allocation (a.k.a. Topic Model) Given a set of text documents, can we infer the topics that are covered by the set, and can we assign topics to individual documents Unsupervised learning problem Technique Exploit statistical regularities in data E.g., documents that are on the topic of education will likely contain a set of words such as teacher, student, lesson, etc.

55 Generative Model of Text Each document is a collection of topics (e.g., education, finance, the arts) Each topic is characterized by a set of words that are likely to appear The string of words in a document is generated by: 1) Draw a topic from the probability distribution associated with a document 2) Draw a word from the probability distribution associated with a topic Bag of words approach

56 Inferring (Learning) Topics Input: set of unlabeled documents Learning task Infer distribution over topics for each document Infer distribution over words for each topic Distribution over topics can be helpful for classifying or clustering documents

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59 Dan Knights and Rob Lindsey s work at JDPA

60 Rob s Work: Phrase Discovery 0.17 new york 0.31 shuttle 0.27 non 0.19 minutes 0.16 new 0.23 lax 0.14 requested 0.13 waited 0.14 ny 0.16 flight 0.14 smoke vegas 0.12 early 0.12 room strip 0.11 sheraton 0.11 given york 0.09 sheraton gateway 0.09 smelled coaster 0.09 proximity 0.08 reserved 0.10 check 0.10 nyny 0.09 flights 0.08 change 0.10 min 0.08 roller 0.08 catch 0.07 told 0.10 waiting 0.08 las 0.08 morning 0.07 cigarette 0.09 arrived 0.07 it's 0.07 bus 0.07 assigned 0.09 wait 0.07 bars 0.07 pick 0.07 request 0.09 late 0.07 las vegas 0.07 shuttles 0.07 called fun 0.07 terminal 0.07 asked 0.08 arrival 0.06 drinks 0.06 layover 0.07 reservation 0.08 bell 0.06 mgm grand 0.06 international 0.06 advance 0.08 late night 0.06 you're 0.06 driver 0.06 resolve 0.08 pm 0.06 mgm 0.06 closeness 0.06 cigarette smoke 0.07 luggage 0.06 arcade 0.06 minutes 0.05 guaranteed 0.07 took forever 0.06 chin 0.06 pickup 0.05 smokers 0.07 told 0.06 italian 0.06 drop 0.05 prior 0.06 called 0.05 city 0.05 ride 0.05 upgrade 0.06 took care 0.05 island 0.05 marriott 0.05 ended skyline 0.05 terminals 0.05 checked 0.06 cleaned 0.05 big apple 0.05 convenience 0.05 smell 0.06 checkout 0.05 luxor 0.05 to/from 0.05 asking 0.05 took long

61 Bayesian Analysis Make inferences from data using probability models about quantities we want to predict E.g., expected age of death given 51 yr old E.g., latent topics in document 1. Set up full probability model that characterizes distribution over all quantities (observed and unobserved) 2. Condition model on observed data to compute posterior distribution 3. Evaluate fit of model to data