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**Bayesian models of inductive learning**

Josh Tenenbaum & Tom Griffiths MIT Computational Cognitive Science Group Department of Brain and Cognitive Sciences Computer Science and AI Lab (CSAIL)

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**What to expect What you’ll get out of this tutorial:**

Our view of what Bayesian models have to offer cognitive science. In-depth examples of basic and advanced models: how the math works & what it buys you. Some comparison to other approaches. Opportunities to ask questions. What you won’t get: Detailed, hands-on how-to. Where you can learn more:

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**Outline Morning Afternoon Introduction (Josh)**

Basic case study #1: Flipping coins (Tom) Basic case study #2: Rules and similarity (Josh) Afternoon Advanced case study #1: Causal induction (Tom) Advanced case study #2: Property induction (Josh) Quick tour of more advanced topics (Tom)

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**Outline Morning Afternoon Introduction (Josh)**

Basic case study #1: Flipping coins (Tom) Basic case study #2: Rules and similarity (Josh) Afternoon Advanced case study #1: Causal induction (Tom) Advanced case study #2: Property induction (Josh) Quick tour of more advanced topics (Tom)

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**Bayesian models in cognitive science**

Vision Motor control Memory Language Inductive learning and reasoning….

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**Everyday inductive leaps**

Learning concepts and words from examples “horse” “horse” I’m going to tell you about a broad research program… The problems that intrigue me are all things which people do effortlessly and for th most part quite well, but which we still don’t know how to get computers get do -- which is a sign that we don’t understand the computational basis of how people do these things. “horse”

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**Learning concepts and words**

“tufa” Can you pick out the tufas?

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**Inductive reasoning Input: Task: Judge how likely conclusion is to be**

Cows can get Hick’s disease. Gorillas can get Hick’s disease. All mammals can get Hick’s disease. (premises) (conclusion) Task: Judge how likely conclusion is to be true, given that premises are true.

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**Inferring causal relations**

Input: Took vitamin B23 Headache Day 1 yes no Day 2 yes yes Day 3 no yes Day 4 yes no Does vitamin B23 cause headaches? Task: Judge probability of a causal link given several joint observations.

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**Everyday inductive leaps**

How can we learn so much about . . . Properties of natural kinds Meanings of words Future outcomes of a dynamic process Hidden causal properties of an object Causes of a person’s action (beliefs, goals) Causal laws governing a domain . . . from such limited data?

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The Challenge How do we generalize successfully from very limited data? Just one or a few examples Often only positive examples Philosophy: Induction is a “problem”, a “riddle”, a “paradox”, a “scandal”, or a “myth”. Machine learning and statistics: Focus on generalization from many examples, both positive and negative.

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**Rational statistical inference (Bayes, Laplace)**

Likelihood Prior probability Posterior probability Sum over space of hypotheses

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**Bayesian models of inductive learning: some recent history**

Shepard (1987) Analysis of one-shot stimulus generalization, to explain the universal exponential law. Anderson (1990) Models of categorization and causal induction. Oaksford & Chater (1994) Model of conditional reasoning (Wason selection task). Heit (1998) Framework for category-based inductive reasoning.

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**Theory-Based Bayesian Models**

Rational statistical inference (Bayes): Learners’ domain theories generate their hypothesis space H and prior p(h). Well-matched to structure of the natural world. Learnable from limited data. Computationally tractable inference.

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**What is a theory? Working definition Analogy to grammar in language.**

An ontology and a system of abstract principles that generates a hypothesis space of candidate world structures along with their relative probabilities. Analogy to grammar in language. Example: Newton’s laws

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**Structure and statistics**

A framework for understanding how structured knowledge and statistical inference interact. How structured knowledge guides statistical inference, and is itself acquired through higher-order statistical learning. How simplicity trades off with fit to the data in evaluating structural hypotheses. How increasingly complex structures may grow as required by new data, rather than being pre-specified in advance.

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**Structure and statistics**

A framework for understanding how structured knowledge and statistical inference interact. How structured knowledge guides statistical inference, and is itself acquired through higher-order statistical learning. Hierarchical Bayes. How simplicity trades off with fit to the data in evaluating structural hypotheses. Bayesian Occam’s Razor. How increasingly complex structures may grow as required by new data, rather than being pre-specified in advance. Non-parametric Bayes.

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**Alternative approaches to inductive generalization**

Associative learning Connectionist networks Similarity to examples Toolkit of simple heuristics Constraint satisfaction Analogical mapping

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**Marr’s Three Levels of Analysis**

Computation: “What is the goal of the computation, why is it appropriate, and what is the logic of the strategy by which it can be carried out?” Representation and algorithm: Cognitive psychology Implementation: Neurobiology Cognitive psychology traditionally focuses on 2nd level. But this is unsatisying, for the same reason as in vision: - Ad hoc models, with arbitrary assumptions and free parameters. - Lots of different models with little sense of how they all fit together, or what the real differences are. Cognitive neuroscience has focused on the link between levels 2 and 3. But that doesn’t address what is to me the biggest mystery: how we are able to succeed in these inductive inference tasks, given that induction has been called a puzzle, a paradox, a scandal! Going back to plato, aristotle. Scandal though it might be, we do these things on a daily basis. That requires level 1. But outside of vision or language, not very much work on level 1, or the link from level 1 to level 2 and 3. No explanatory adequacy. Describe how the mind works, but don’t explain why it works that way. Why these inference strategies lead to success in the real world.

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**Why Bayes? A framework for explaining cognition.**

How people can learn so much from such limited data. Why process-level models work the way that they do. Strong quantitative models with minimal ad hoc assumptions. A framework for understanding how structured knowledge and statistical inference interact. How structured knowledge guides statistical inference, and is itself acquired through higher-order statistical learning. How simplicity trades off with fit to the data in evaluating structural hypotheses (Occam’s razor). How increasingly complex structures may grow as required by new data, rather than being pre-specified in advance.

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**Outline Morning Afternoon Introduction (Josh)**

Basic case study #1: Flipping coins (Tom) Basic case study #2: Rules and similarity (Josh) Afternoon Advanced case study #1: Causal induction (Tom) Advanced case study #2: Property induction (Josh) Quick tour of more advanced topics (Tom)

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Coin flipping

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Coin flipping HHTHT HHHHH What process produced these sequences?

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**Bayes’ rule For data D and a hypothesis H, we have:**

“Posterior probability”: “Prior probability”: “Likelihood”:

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**The origin of Bayes’ rule**

A simple consequence of using probability to represent degrees of belief For any two random variables:

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**Why represent degrees of belief with probabilities?**

Good statistics consistency, and worst-case error bounds. Cox Axioms necessary to cohere with common sense “Dutch Book” + Survival of the Fittest if your beliefs do not accord with the laws of probability, then you can always be out-gambled by someone whose beliefs do so accord. Provides a theory of learning a common currency for combining prior knowledge and the lessons of experience.

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**Bayes’ rule For data D and a hypothesis H, we have:**

“Posterior probability”: “Prior probability”: “Likelihood”:

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**Hypotheses in Bayesian inference**

Hypotheses H refer to processes that could have generated the data D Bayesian inference provides a distribution over these hypotheses, given D P(D|H) is the probability of D being generated by the process identified by H Hypotheses H are mutually exclusive: only one process could have generated D

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**Hypotheses in coin flipping**

Describe processes by which D could be generated D = HHTHT Fair coin, P(H) = 0.5 Coin with P(H) = p Markov model Hidden Markov model ... statistical models

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**Hypotheses in coin flipping**

Describe processes by which D could be generated D = HHTHT Fair coin, P(H) = 0.5 Coin with P(H) = p Markov model Hidden Markov model ... generative models

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**Representing generative models**

Graphical model notation Pearl (1988), Jordan (1998) Variables are nodes, edges indicate dependency Directed edges show causal process of data generation d d d d4 Fair coin, P(H) = 0.5 d d d d4 Markov model HHTHT d1 d2 d3 d4 d5

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**Models with latent structure**

d d d d4 P(H) = p p Not all nodes in a graphical model need to be observed Some variables reflect latent structure, used in generating D but unobserved d d d d4 Hidden Markov model s s s s4 HHTHT d1 d2 d3 d4 d5

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**Coin flipping Comparing two simple hypotheses**

P(H) = 0.5 vs. P(H) = 1.0 Comparing simple and complex hypotheses P(H) = 0.5 vs. P(H) = p Comparing infinitely many hypotheses P(H) = p Psychology: Representativeness

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**Coin flipping Comparing two simple hypotheses**

P(H) = 0.5 vs. P(H) = 1.0 Comparing simple and complex hypotheses P(H) = 0.5 vs. P(H) = p Comparing infinitely many hypotheses P(H) = p Psychology: Representativeness

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**Comparing two simple hypotheses**

Contrast simple hypotheses: H1: “fair coin”, P(H) = 0.5 H2:“always heads”, P(H) = 1.0 Bayes’ rule: With two hypotheses, use odds form

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**Bayes’ rule in odds form**

P(H1|D) P(D|H1) P(H1) P(H2|D) P(D|H2) P(H2) D: data H1, H2: models P(H1|D): posterior probability H1 generated the data P(D|H1): likelihood of data under model H1 P(H1): prior probability H1 generated the data = x

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Coin flipping HHTHT HHHHH What process produced these sequences?

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**Comparing two simple hypotheses**

P(H1|D) P(D|H1) P(H1) P(H2|D) P(D|H2) P(H2) D: HHTHT H1, H2: “fair coin”, “always heads” P(D|H1) = 1/25 P(H1) = 999/1000 P(D|H2) = 0 P(H2) = 1/1000 P(H1|D) / P(H2|D) = infinity = x

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**Comparing two simple hypotheses**

P(H1|D) P(D|H1) P(H1) P(H2|D) P(D|H2) P(H2) D: HHHHH H1, H2: “fair coin”, “always heads” P(D|H1) = 1/25 P(H1) = 999/1000 P(D|H2) = 1 P(H2) = 1/1000 P(H1|D) / P(H2|D) 30 = x

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**Comparing two simple hypotheses**

P(H1|D) P(D|H1) P(H1) P(H2|D) P(D|H2) P(H2) D: HHHHHHHHHH H1, H2: “fair coin”, “always heads” P(D|H1) = 1/210 P(H1) = 999/1000 P(D|H2) = 1 P(H2) = 1/1000 P(H1|D) / P(H2|D) 1 = x

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**Comparing two simple hypotheses**

Bayes’ rule tells us how to combine prior beliefs with new data top-down and bottom-up influences As a model of human inference predicts conclusions drawn from data identifies point at which prior beliefs are overwhelmed by new experiences But… more complex cases?

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**Coin flipping Comparing two simple hypotheses**

P(H) = 0.5 vs. P(H) = 1.0 Comparing simple and complex hypotheses P(H) = 0.5 vs. P(H) = p Comparing infinitely many hypotheses P(H) = p Psychology: Representativeness

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**Comparing simple and complex hypotheses**

d d d d4 P(H) = p p vs. d d d d4 Fair coin, P(H) = 0.5 Which provides a better account of the data: the simple hypothesis of a fair coin, or the complex hypothesis that P(H) = p?

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**Comparing simple and complex hypotheses**

P(H) = p is more complex than P(H) = 0.5 in two ways: P(H) = 0.5 is a special case of P(H) = p for any observed sequence X, we can choose p such that X is more probable than if P(H) = 0.5

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**Comparing simple and complex hypotheses**

Probability

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**Comparing simple and complex hypotheses**

Probability HHHHH p = 1.0

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**Comparing simple and complex hypotheses**

Probability HHTHT p = 0.6

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**Comparing simple and complex hypotheses**

P(H) = p is more complex than P(H) = 0.5 in two ways: P(H) = 0.5 is a special case of P(H) = p for any observed sequence X, we can choose p such that X is more probable than if P(H) = 0.5 How can we deal with this? frequentist: hypothesis testing information theorist: minimum description length Bayesian: just use probability theory!

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**Comparing simple and complex hypotheses**

P(H1|D) P(D|H1) P(H1) P(H2|D) P(D|H2) P(H2) Computing P(D|H1) is easy: P(D|H1) = 1/2N Compute P(D|H2) by averaging over p: = x

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**Comparing simple and complex hypotheses**

Probability Distribution is an average over all values of p

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**Comparing simple and complex hypotheses**

Probability Distribution is an average over all values of p

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**Comparing simple and complex hypotheses**

Simple and complex hypotheses can be compared directly using Bayes’ rule requires summing over latent variables Complex hypotheses are penalized for their greater flexibility: “Bayesian Occam’s razor” This principle is used in model selection methods in psychology (e.g. Myung & Pitt, 1997)

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**Coin flipping Comparing two simple hypotheses**

P(H) = 0.5 vs. P(H) = 1.0 Comparing simple and complex hypotheses P(H) = 0.5 vs. P(H) = p Comparing infinitely many hypotheses P(H) = p Psychology: Representativeness

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**Comparing infinitely many hypotheses**

Assume data are generated from a model: What is the value of p? each value of p is a hypothesis H requires inference over infinitely many hypotheses d d d d4 P(H) = p p

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**Comparing infinitely many hypotheses**

Flip a coin 10 times and see 5 heads, 5 tails. P(H) on next flip? 50% Why? 50% = 5 / (5+5) = 5/10. “Future will be like the past.” Suppose we had seen 4 heads and 6 tails. P(H) on next flip? Closer to 50% than to 40%. Why? Prior knowledge.

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**Integrating prior knowledge and data**

Posterior distribution P(p | D) is a probability density over p = P(H) Need to work out likelihood P(D | p) and specify prior distribution P(p) P(p | D) P(D | p) P(p)

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**? Likelihood and prior Likelihood: P(D | p) = pNH (1-p)NT Prior:**

NH: number of heads NT: number of tails Prior: P(p) pFH-1 (1-p)FT-1 ?

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**A simple method of specifying priors**

Imagine some fictitious trials, reflecting a set of previous experiences strategy often used with neural networks e.g., F ={1000 heads, 1000 tails} ~ strong expectation that any new coin will be fair In fact, this is a sensible statistical idea...

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**Likelihood and prior Likelihood: P(D | p) = pNH (1-p)NT Prior:**

NH: number of heads NT: number of tails Prior: P(p) pFH-1 (1-p)FT-1 FH: fictitious observations of heads FT: fictitious observations of tails Beta(FH,FT)

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**Conjugate priors Exist for many standard distributions**

formula for exponential family conjugacy Define prior in terms of fictitious observations Beta is conjugate to Bernoulli (coin-flipping) FH = FT = 1 FH = FT = 3 FH = FT = 1000

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**Likelihood and prior Likelihood: P(D | p) = pNH (1-p)NT Prior:**

NH: number of heads NT: number of tails Prior: P(p) pFH-1 (1-p)FT-1 FH: fictitious observations of heads FT: fictitious observations of tails

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**Comparing infinitely many hypotheses**

P(p | D) P(D | p) P(p) = pNH+FH-1 (1-p)NT+FT-1 Posterior is Beta(NH+FH,NT+FT) same form as conjugate prior Posterior mean: Posterior predictive distribution:

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Some examples e.g., F ={1000 heads, 1000 tails} ~ strong expectation that any new coin will be fair After seeing 4 heads, 6 tails, P(H) on next flip = 1004 / ( ) = 49.95% e.g., F ={3 heads, 3 tails} ~ weak expectation that any new coin will be fair After seeing 4 heads, 6 tails, P(H) on next flip = 7 / (7+9) = 43.75% Prior knowledge too weak

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**But… flipping thumbtacks**

e.g., F ={4 heads, 3 tails} ~ weak expectation that tacks are slightly biased towards heads After seeing 2 heads, 0 tails, P(H) on next flip = 6 / (6+3) = 67% Some prior knowledge is always necessary to avoid jumping to hasty conclusions... Suppose F = { }: After seeing 2 heads, 0 tails, P(H) on next flip = 2 / (2+0) = 100%

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**Origin of prior knowledge**

Tempting answer: prior experience Suppose you have previously seen 2000 coin flips: 1000 heads, 1000 tails By assuming all coins (and flips) are alike, these observations of other coins are as good as observations of the present coin

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**Problems with simple empiricism**

Haven’t really seen 2000 coin flips, or any flips of a thumbtack Prior knowledge is stronger than raw experience justifies Haven’t seen exactly equal number of heads and tails Prior knowledge is smoother than raw experience justifies Should be a difference between observing 2000 flips of a single coin versus observing 10 flips each for 200 coins, or 1 flip each for 2000 coins Prior knowledge is more structured than raw experience

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A simple theory “Coins are manufactured by a standardized procedure that is effective but not perfect.” Justifies generalizing from previous coins to the present coin. Justifies smoother and stronger prior than raw experience alone. Explains why seeing 10 flips each for 200 coins is more valuable than seeing 2000 flips of one coin. “Tacks are asymmetric, and manufactured to less exacting standards.”

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Limitations Can all domain knowledge be represented so simply, in terms of an equivalent number of fictional observations? Suppose you flip a coin 25 times and get all heads Something funny is going on… But with F ={1000 heads, 1000 tails}, P(H) on next flip = 1025 / ( ) = 50.6%. Looks like nothing unusual

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Hierarchical priors Higher-order hypothesis: is this coin fair or unfair? Example probabilities: P(fair) = 0.99 P(p|fair) is Beta(1000,1000) P(p|unfair) is Beta(1,1) 25 heads in a row propagates up, affecting p and then P(fair|D) fair p d d d d4 P(fair|25 heads) P(25 heads|fair) P(fair) P(unfair|25 heads) P(25 heads|unfair) P(unfair) = = 9 x 10-5

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**More hierarchical priors**

Latent structure can capture coin variability 10 flips from 200 coins is better than 2000 flips from a single coin: allows estimation of FH, FT p ~ Beta(FH,FT) FH,FT ... Coin 1 Coin 2 Coin 200 p p p d d d d4 d d d d4 d d d d4

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**Yet more hierarchical priors**

physical knowledge Discrete beliefs (e.g. symmetry) can influence estimation of continuous properties (e.g. FH, FT) FH,FT p p p d d d d4 d d d d4 d d d d4

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**Comparing infinitely many hypotheses**

Apply Bayes’ rule to obtain posterior probability density Requires prior over all hypotheses computation simplified by conjugate priors richer structure with hierarchical priors Hierarchical priors indicate how simple theories can inform statistical inferences one step towards structure and statistics

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**Coin flipping Comparing two simple hypotheses**

P(H) = 0.5 vs. P(H) = 1.0 Comparing simple and complex hypotheses P(H) = 0.5 vs. P(H) = p Comparing infinitely many hypotheses P(H) = p Psychology: Representativeness

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**Psychology: Representativeness**

Which sequence is more likely from a fair coin? HHTHT more representative of a fair coin (Kahneman & Tversky, 1972) HHHHH

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**What might representativeness mean?**

Evidence for a random generating process P(H1|D) P(D|H1) P(H1) P(H2|D) P(D|H2) P(H2) H1: random process (fair coin) H2: alternative processes = x likelihood ratio

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**A constrained hypothesis space**

Four hypotheses: h1 fair coin HHTHTTTH h2 “always alternates” HTHTHTHT h3 “mostly heads” HHTHTHHH h4 “always heads” HHHHHHHH

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**Representativeness judgments**

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Results Good account of representativeness data, with three pseudo-free parameters, = 0.91 “always alternates” means 99% of the time “mostly heads” means P(H) = 0.85 “always heads” means P(H) = 0.99 With scaling parameter, r = 0.95 (Tenenbaum & Griffiths, 2001)

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The role of theories The fact that HHTHT looks representative of a fair coin and HHHHH does not reflects our implicit theories of how the world works. Easy to imagine how a trick all-heads coin could work: high prior probability. Hard to imagine how a trick “HHTHT” coin could work: low prior probability.

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**Summary Three kinds of Bayesian inference Critical notions:**

comparing two simple hypotheses comparing simple and complex hypotheses comparing an infinite number of hypotheses Critical notions: generative models, graphical models Bayesian Occam’s razor priors: conjugate, hierarchical (theories)

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**Outline Morning Afternoon Introduction (Josh)**

Basic case study #1: Flipping coins (Tom) Basic case study #2: Rules and similarity (Josh) Afternoon Advanced case study #1: Causal induction (Tom) Advanced case study #2: Property induction (Josh) Quick tour of more advanced topics (Tom)

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Rules and similarity

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**Structure versus statistics**

Rules Logic Symbols Statistics Similarity Typicality

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A better metaphor

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A better metaphor

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**Structure and statistics**

Similarity Typicality Rules Logic Symbols

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**Structure and statistics**

Basic case study #1: Flipping coins Learning and reasoning with structured statistical models. Basic case study #2: Rules and similarity Statistical learning with structured representations.

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**The number game Program input: number between 1 and 100**

Program output: “yes” or “no”

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**The number game Learning task:**

Observe one or more positive (“yes”) examples. Judge whether other numbers are “yes” or “no”.

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**The number game Examples of “yes” numbers Generalization**

judgments (N = 20) 60 Diffuse similarity

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**The number game Examples of “yes” numbers Generalization**

judgments (n = 20) 60 Diffuse similarity Rule: “multiples of 10”

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**The number game Examples of “yes” numbers Generalization**

judgments (N = 20) 60 Diffuse similarity Rule: “multiples of 10” Focused similarity: numbers near 50-60

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**The number game Examples of “yes” numbers Generalization**

judgments (N = 20) 16 Diffuse similarity Rule: “powers of 2” Focused similarity: numbers near 20

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**The number game Main phenomena to explain:**

60 Diffuse similarity Rule: “multiples of 10” Focused similarity: numbers near 50-60 Main phenomena to explain: Generalization can appear either similarity-based (graded) or rule-based (all-or-none). Learning from just a few positive examples.

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**Rule/similarity hybrid models**

Category learning Nosofsky, Palmeri et al.: RULEX Erickson & Kruschke: ATRIUM

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**Divisions into “rule” and “similarity” subsystems**

Category learning Nosofsky, Palmeri et al.: RULEX Erickson & Kruschke: ATRIUM Language processing Pinker, Marcus et al.: Past tense morphology Reasoning Sloman Rips Nisbett, Smith et al.

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**Rule/similarity hybrid models**

Why two modules? Why do these modules work the way that they do, and interact as they do? How do people infer a rule or similarity metric from just a few positive examples?

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**Bayesian model H: Hypothesis space of possible concepts:**

h1 = {2, 4, 6, 8, 10, 12, …, 96, 98, 100} (“even numbers”) h2 = {10, 20, 30, 40, …, 90, 100} (“multiples of 10”) h3 = {2, 4, 8, 16, 32, 64} (“powers of 2”) h4 = {50, 51, 52, …, 59, 60} (“numbers between 50 and 60”) . . . Representational interpretations for H: Candidate rules Features for similarity “Consequential subsets” (Shepard, 1987)

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**Inferring hypotheses from similarity judgment**

Additive clustering (Shepard & Arabie, 1977): : similarity of stimuli i, j : weight of cluster k : membership of stimulus i in cluster k (1 if stimulus i in cluster k, 0 otherwise) Equivalent to similarity as a weighted sum of common features (Tversky, 1977).

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**Additive clustering for the integers 0-9:**

Rank Weight Stimuli in cluster Interpretation * * * powers of two * * * small numbers * * * multiples of three * * * * large numbers * * * * * middle numbers * * * * * odd numbers * * * * smallish numbers * * * * * largish numbers

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**Three hypothesis subspaces for number concepts**

Mathematical properties (24 hypotheses): Odd, even, square, cube, prime numbers Multiples of small integers Powers of small integers Raw magnitude (5050 hypotheses): All intervals of integers with endpoints between 1 and 100. Approximate magnitude (10 hypotheses): Decades (1-10, 10-20, 20-30, …)

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**Hypothesis spaces and theories**

Why a hypothesis space is like a domain theory: Represents one particular way of classifying entities in a domain. Not just an arbitrary collection of hypotheses, but a principled system. What’s missing? Explicit representation of the principles. Hypothesis spaces (and priors) are generated by theories. Some analogies: Grammars generate languages (and priors over structural descriptions) Hierarchical Bayesian modeling

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**Bayesian model H: Hypothesis space of possible concepts:**

Mathematical properties: even, odd, square, prime, Approximate magnitude: {1-10}, {10-20}, {20-30}, Raw magnitude: all intervals between 1 and 100. X = {x1, , xn}: n examples of a concept C. Evaluate hypotheses given data: p(h) [“prior”]: domain knowledge, pre-existing biases p(X|h) [“likelihood”]: statistical information in examples. p(h|X) [“posterior”]: degree of belief that h is the true extension of C.

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**Bayesian model H: Hypothesis space of possible concepts:**

Mathematical properties: even, odd, square, prime, Approximate magnitude: {1-10}, {10-20}, {20-30}, Raw magnitude: all intervals between 1 and 100. X = {x1, , xn}: n examples of a concept C. Evaluate hypotheses given data: p(h) [“prior”]: domain knowledge, pre-existing biases p(X|h) [“likelihood”]: statistical information in examples. p(h|X) [“posterior”]: degree of belief that h is the true extension of C.

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Likelihood: p(X|h) Size principle: Smaller hypotheses receive greater likelihood, and exponentially more so as n increases. Follows from assumption of randomly sampled examples. Captures the intuition of a representative sample.

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**Illustrating the size principle**

h2

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**Illustrating the size principle**

h2 Data slightly more of a coincidence under h1

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**Illustrating the size principle**

h2 Data much more of a coincidence under h1

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**Bayesian Occam’s Razor**

Law of “Conservation of Belief” M1 p(D = d | M ) M2 All possible data sets d For any model M,

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**Comparing simple and complex hypotheses**

Probability Distribution is an average over all values of p

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Prior: p(h) Choice of hypothesis space embodies a strong prior: effectively, p(h) ~ 0 for many logically possible but conceptually unnatural hypotheses. Prevents overfitting by highly specific but unnatural hypotheses, e.g. “multiples of 10 except 50 and 70”.

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Prior: p(h) Choice of hypothesis space embodies a strong prior: effectively, p(h) ~ 0 for many logically possible but conceptually unnatural hypotheses. Prevents overfitting by highly specific but unnatural hypotheses, e.g. “multiples of 10 except 50 and 70”. p(h) encodes relative weights of alternative theories: H: Total hypothesis space p(H1) = 1/5 p(H2) = 3/5 p(H3) = 1/5 H1: Math properties (24) even numbers powers of two multiples of three …. H2: Raw magnitude (5050) 10-15 20-32 37-54 …. H3: Approx. magnitude (10) 10-20 20-30 30-40 …. p(h) = p(H1) / 24 p(h) = p(H2) / 5050 p(h) = p(H3) / 10

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**A more complex approach to priors**

Start with a base set of regularities R and combination operators C. Hypothesis space = closure of R under C. C = {and, or}: H = unions and intersections of regularities in R (e.g., “multiples of 10 between 30 and 70”). C = {and-not}: H = regularities in R with exceptions (e.g., “multiples of 10 except 50 and 70”). Two qualitatively similar priors: Description length: number of combinations in C needed to generate hypothesis from R. Bayesian Occam’s Razor, with model classes defined by number of combinations: more combinations more hypotheses lower prior

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Posterior: X = {60, 80, 10, 30} Why prefer “multiples of 10” over “even numbers”? p(X|h). Why prefer “multiples of 10” over “multiples of 10 except 50 and 20”? p(h). Why does a good generalization need both high prior and high likelihood? p(h|X) ~ p(X|h) p(h)

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**Bayesian Occam’s Razor**

Probabilities provide a common currency for balancing model complexity with fit to the data.

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**Generalizing to new objects**

Given p(h|X), how do we compute , the probability that C applies to some new stimulus y? The judgments people make are not directly about which hypothesis is correct for the meaning of the word “blicket”, but about which things are blickets. How do we use our knowledge about which hypotheses are likely to correspond to the extension of the word blicket, encoded in the posterior, to generalize the word to new objects? Bayes tell us to compute….

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**Generalizing to new objects**

Hypothesis averaging: Compute the probability that C applies to some new object y by averaging the predictions of all hypotheses h, weighted by p(h|X): The judgments people make are not directly about which hypothesis is correct for the meaning of the word “blicket”, but about which things are blickets. How do we use our knowledge about which hypotheses are likely to correspond to the extension of the word blicket, encoded in the posterior, to generalize the word to new objects? Bayes tell us to compute….

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Examples: 16

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**Connection to feature-based similarity**

Additive clustering model of similarity: Bayesian hypothesis averaging: Equivalent if we identify features fk with hypotheses h, and weights wk with

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Examples: 16 8 2 64

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Examples: 16 23 19 20

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**Model fits Examples of “yes” numbers Generalization judgments (N = 20)**

Bayesian Model (r = 0.96) 60

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**Model fits Examples of “yes” numbers Generalization judgments (N = 20)**

Bayesian Model (r = 0.93) 16

124
**Summary of the Bayesian model**

How do the statistics of the examples interact with prior knowledge to guide generalization? Why does generalization appear rule-based or similarity-based? broad p(h|X): similarity gradient narrow p(h|X): all-or-none rule

125
**Summary of the Bayesian model**

How do the statistics of the examples interact with prior knowledge to guide generalization? Why does generalization appear rule-based or similarity-based? Many h of similar size: broad p(h|X) One h much smaller: narrow p(h|X)

126
**Alternative models Neural networks 60 even multiple of 10 power of 2**

80 10 30

127
**Alternative models Neural networks Hypothesis ranking and elimination**

even multiple of 10 multiple of 3 power of 2 …. 60 80 10 30

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**Alternative models Neural networks Hypothesis ranking and elimination**

Similarity to exemplars Average similarity: 60 Data Model (r = 0.80)

129
**Alternative models Neural networks Hypothesis ranking and elimination**

Similarity to exemplars Max similarity: 60 Data Model (r = 0.64)

130
**Alternative models Neural networks Hypothesis ranking and elimination**

Similarity to exemplars Average similarity Max similarity Flexible similarity? Bayes.

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**Alternative models Neural networks Hypothesis ranking and elimination**

Similarity to exemplars Toolbox of simple heuristics 60: “general” similarity : most specific rule (“subset principle”). : similarity in magnitude Why these heuristics? When to use which heuristic? Bayes.

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Summary Generalization from limited data possible via the interaction of structured knowledge and statistics. Structured knowledge: space of candidate rules, theories generate hypothesis space (c.f. hierarchical priors) Statistics: Bayesian Occam’s razor. Better understand the interactions between traditionally opposing concepts: Rules and statistics Rules and similarity Explains why central but notoriously slippery processing-level concepts work the way they do. Similarity Representativeness Rules and representativeness

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**Why Bayes? A framework for explaining cognition.**

How people can learn so much from such limited data. Why process-level models work the way that they do. Strong quantitative models with minimal ad hoc assumptions. A framework for understanding how structured knowledge and statistical inference interact. How structured knowledge guides statistical inference, and is itself acquired through higher-order statistical learning. How simplicity trades off with fit to the data in evaluating structural hypotheses (Occam’s razor). How increasingly complex structures may grow as required by new data, rather than being pre-specified in advance.

134
**Theory-Based Bayesian Models**

Rational statistical inference (Bayes): Learners’ domain theories generate their hypothesis space H and prior p(h). Well-matched to structure of the natural world. Learnable from limited data. Computationally tractable inference.

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**Looking towards the afternoon**

How do we apply these ideas to more natural and complex aspects of cognition? Where do the hypothesis spaces come from? Can we formalize the contributions of domain theories?

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**Outline Morning Afternoon Introduction (Josh)**

Basic case study #1: Flipping coins (Tom) Basic case study #2: Rules and similarity (Josh) Afternoon Advanced case study #1: Causal induction (Tom) Advanced case study #2: Property induction (Josh) Quick tour of more advanced topics (Tom)

138
**Outline Morning Afternoon Introduction (Josh)**

Basic case study #1: Flipping coins (Tom) Basic case study #2: Rules and similarity (Josh) Afternoon Advanced case study #1: Causal induction (Tom) Advanced case study #2: Property induction (Josh) Quick tour of more advanced topics (Tom)

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**Marr’s Three Levels of Analysis**

Computation: “What is the goal of the computation, why is it appropriate, and what is the logic of the strategy by which it can be carried out?” Representation and algorithm: Cognitive psychology Implementation: Neurobiology Cognitive psychology traditionally focuses on 2nd level. But this is unsatisying, for the same reason as in vision: - Ad hoc models, with arbitrary assumptions and free parameters. - Lots of different models with little sense of how they all fit together, or what the real differences are. Cognitive neuroscience has focused on the link between levels 2 and 3. But that doesn’t address what is to me the biggest mystery: how we are able to succeed in these inductive inference tasks, given that induction has been called a puzzle, a paradox, a scandal! Going back to plato, aristotle. Scandal though it might be, we do these things on a daily basis. That requires level 1. But outside of vision or language, not very much work on level 1, or the link from level 1 to level 2 and 3. No explanatory adequacy. Describe how the mind works, but don’t explain why it works that way. Why these inference strategies lead to success in the real world.

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**Working at the computational level**

statistical What is the computational problem? input: data output: solution

141
**Working at the computational level**

statistical What is the computational problem? input: data output: solution What knowledge is available to the learner? Where does that knowledge come from?

142
**Theory-Based Bayesian Models**

Rational statistical inference (Bayes): Learners’ domain theories generate their hypothesis space H and prior p(h). Well-matched to structure of the natural world. Learnable from limited data. Computationally tractable inference.

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Causality

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Bayes nets and beyond... Increasingly popular approach to studying human causal inferences (e.g. Glymour, 2001; Gopnik et al., 2004) Three reactions: Bayes nets are the solution! Bayes nets are missing the point, not sure why… what is a Bayes net?

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**Bayes nets and beyond... What are Bayes nets?**

graphical models causal graphical models An example: elemental causal induction Beyond Bayes nets… other knowledge in causal induction formalizing causal theories

146
**Bayes nets and beyond... What are Bayes nets?**

graphical models causal graphical models An example: elemental causal induction Beyond Bayes nets… other knowledge in causal induction formalizing causal theories

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Graphical models Express the probabilistic dependency structure among a set of variables (Pearl, 1988) Consist of a set of nodes, corresponding to variables a set of edges, indicating dependency a set of functions defined on the graph that defines a probability distribution

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**Undirected graphical models**

X3 X4 X1 Consist of a set of nodes a set of edges a potential for each clique, multiplied together to yield the distribution over variables Examples statistical physics: Ising model, spinglasses early neural networks (e.g. Boltzmann machines) X2 X5

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**Directed graphical models**

X3 X4 X1 Consist of a set of nodes a set of edges a conditional probability distribution for each node, conditioned on its parents, multiplied together to yield the distribution over variables Constrained to directed acyclic graphs (DAG) AKA: Bayesian networks, Bayes nets X2 X5

150
**Bayesian networks and Bayes**

Two different problems Bayesian statistics is a method of inference Bayesian networks are a form of representation There is no necessary connection many users of Bayesian networks rely upon frequentist statistical methods (e.g. Glymour) many Bayesian inferences cannot be easily represented using Bayesian networks

151
**Properties of Bayesian networks**

Efficient representation and inference exploiting dependency structure makes it easier to represent and compute with probabilities Explaining away pattern of probabilistic reasoning characteristic of Bayesian networks, especially early use in AI

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**Efficient representation and inference**

Three binary variables: Cavity, Toothache, Catch

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**Efficient representation and inference**

Three binary variables: Cavity, Toothache, Catch Specifying P(Cavity, Toothache, Catch) requires 7 parameters (1 for each set of values, minus 1 because it’s a probability distribution) With n variables, we need 2n -1 parameters Here n=3. Realistically, many more: X-ray, diet, oral hygiene, personality,

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**Conditional independence**

All three variables are dependent, but Toothache and Catch are independent given the presence or absence of Cavity In probabilistic terms: With n evidence variables, x1, …, xn, we need 2 n conditional probabilities:

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**A simple Bayesian network**

Graphical representation of relations between a set of random variables: Probabilistic interpretation: factorizing complex terms Cavity Toothache Catch

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**A more complex system Joint distribution sufficient for any inference:**

Battery Radio Ignition Gas Starts On time to work Joint distribution sufficient for any inference:

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**A more complex system Joint distribution sufficient for any inference:**

Battery Radio Ignition Gas Starts On time to work Joint distribution sufficient for any inference:

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**A more complex system Joint distribution sufficient for any inference:**

Battery Radio Ignition Gas Starts On time to work Joint distribution sufficient for any inference: General inference algorithm: local message passing (belief propagation; Pearl, 1988) efficiency depends on sparseness of graph structure

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Explaining away Rain Sprinkler Grass Wet Assume grass will be wet if and only if it rained last night, or if the sprinklers were left on:

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**Explaining away Rain Sprinkler Grass Wet**

Compute probability it rained last night, given that the grass is wet:

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**Explaining away Rain Sprinkler Grass Wet**

Compute probability it rained last night, given that the grass is wet:

162
**Explaining away Rain Sprinkler Grass Wet**

Compute probability it rained last night, given that the grass is wet:

163
**Explaining away Rain Sprinkler Grass Wet**

Compute probability it rained last night, given that the grass is wet:

164
**Explaining away Rain Sprinkler Grass Wet**

Compute probability it rained last night, given that the grass is wet: Between 1 and P(s)

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**Explaining away Rain Sprinkler Grass Wet**

Compute probability it rained last night, given that the grass is wet and sprinklers were left on: Both terms = 1

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**Explaining away Rain Sprinkler Grass Wet**

Compute probability it rained last night, given that the grass is wet and sprinklers were left on:

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**Explaining away Rain Sprinkler Grass Wet “Discounting” to**

prior probability.

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**Contrast w/ production system**

Rain Sprinkler Grass Wet Formulate IF-THEN rules: IF Rain THEN Wet IF Wet THEN Rain Rules do not distinguish directions of inference Requires combinatorial explosion of rules IF Wet AND NOT Sprinkler THEN Rain

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**Contrast w/ spreading activation**

Rain Sprinkler Grass Wet Excitatory links: Rain Wet, Sprinkler Wet Observing rain, Wet becomes more active. Observing grass wet, Rain and Sprinkler become more active. Observing grass wet and sprinkler, Rain cannot become less active. No explaining away!

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**Contrast w/ spreading activation**

Rain Sprinkler Grass Wet Excitatory links: Rain Wet, Sprinkler Wet Inhibitory link: Rain Sprinkler Observing grass wet, Rain and Sprinkler become more active. Observing grass wet and sprinkler, Rain becomes less active: explaining away.

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**Contrast w/ spreading activation**

Rain Burst pipe Sprinkler Grass Wet Each new variable requires more inhibitory connections. Interactions between variables are not causal. Not modular. Whether a connection exists depends on what other connections exist, in non-transparent ways. Big holism problem. Combinatorial explosion.

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**Graphical models Capture dependency structure in distributions**

Provide an efficient means of representing and reasoning with probabilities Allow kinds of inference that are problematic for other representations: explaining away hard to capture in a production system hard to capture with spreading activation

173
**Bayes nets and beyond... What are Bayes nets?**

graphical models causal graphical models An example: causal induction Beyond Bayes nets… other knowledge in causal induction formalizing causal theories

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**Causal graphical models**

Graphical models represent statistical dependencies among variables (ie. correlations) can answer questions about observations Causal graphical models represent causal dependencies among variables express underlying causal structure can answer questions about both observations and interventions (actions upon a variable)

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**Observation and intervention**

Battery Radio Ignition Gas Starts On time to work Graphical model: P(Radio|Ignition) Causal graphical model: P(Radio|do(Ignition))

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**Observation and intervention**

Battery Radio Ignition Gas Starts On time to work Graphical model: P(Radio|Ignition) Causal graphical model: P(Radio|do(Ignition)) “graph surgery” produces “mutilated graph”

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**Assessing interventions**

To compute P(Y|do(X=x)), delete all edges coming into X and reason with the resulting Bayesian network (“do calculus”; Pearl, 2000) Allows a single structure to make predictions about both observations and interventions

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**Causality simplifies inference**

Using a representation in which the direction of causality is correct produces sparser graphs Suppose we get the direction of causality wrong, thinking that “symptoms” causes “diseases”: Does not capture the correlation between symptoms: falsely believe P(Ache, Catch) = P(Ache) P(Catch). Ache Catch Cavity

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**Causality simplifies inference**

Using a representation in which the direction of causality is correct produces sparser graphs Suppose we get the direction of causality wrong, thinking that “symptoms” causes “diseases”: Inserting a new arrow allows us to capture this correlation. This model is too complex: do not believe that Ache Catch Cavity

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**Causality simplifies inference**

Using a representation in which the direction of causality is correct produces sparser graphs Suppose we get the direction of causality wrong, thinking that “symptoms” causes “diseases”: New symptoms require a combinatorial proliferation of new arrows. This reduces efficiency of inference. Ache X-ray Catch Cavity

181
**Learning causal graphical models**

Strength: how strong is a relationship? Structure: does a relationship exist? B E B C E B C B

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**Causal structure vs. causal strength**

Strength: how strong is a relationship? B E B C E B C B

183
**Causal structure vs. causal strength**

Strength: how strong is a relationship? requires defining nature of relationship B E B C w0 w1 E B C w0 B

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**Parameterization Generic Structures: h1 = h0 = Parameterization: C B B**

h1: P(E = 1 | C, B) h0: P(E = 1| C, B)

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**Parameterization Linear Structures: h1 = h0 = Parameterization: C B B**

w0 w1 w0, w1: strength parameters for B, C E E w1 w0 w1+ w0 Linear C B h1: P(E = 1 | C, B) h0: P(E = 1| C, B)

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**Parameterization “Noisy-OR” Structures: h1 = h0 = Parameterization: C**

B C B C w0 w1 w0, w1: strength parameters for B, C E E w1 w0 w1+ w0 – w1 w0 “Noisy-OR” C B h1: P(E = 1 | C, B) h0: P(E = 1| C, B)

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**maximize i P(bi,ci,ei; w0, w1)**

Parameter estimation Maximum likelihood estimation: maximize i P(bi,ci,ei; w0, w1) Bayesian methods: as in the “Comparing infinitely many hypotheses” example…

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**Causal structure vs. causal strength**

Structure: does a relationship exist? B E B C E B C B

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**Approaches to structure learning**

Constraint-based dependency from statistical tests (eg. 2) deduce structure from dependencies B B C E (Pearl, 2000; Spirtes et al., 1993)

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**Approaches to structure learning**

Constraint-based: dependency from statistical tests (eg. 2) deduce structure from dependencies B B C E (Pearl, 2000; Spirtes et al., 1993)

191
**Approaches to structure learning**

Constraint-based: dependency from statistical tests (eg. 2) deduce structure from dependencies B B C E (Pearl, 2000; Spirtes et al., 1993)

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**Approaches to structure learning**

Constraint-based: dependency from statistical tests (eg. 2) deduce structure from dependencies B B C E (Pearl, 2000; Spirtes et al., 1993) Attempts to reduce inductive problem to deductive problem

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**Approaches to structure learning**

Constraint-based: dependency from statistical tests (eg. 2) deduce structure from dependencies B B C E (Pearl, 2000; Spirtes et al., 1993) Bayesian: compute posterior probability of structures, given observed data B C B C E E P(S1|data) P(S0|data) P(S|data) P(data|S) P(S) (Heckerman, 1998; Friedman, 1999)

194
**Causal graphical models**

Extend graphical models to deal with interventions as well as observations Respecting the direction of causality results in efficient representation and inference Two steps in learning causal models parameter estimation structure learning

195
**Bayes nets and beyond... What are Bayes nets?**

graphical models causal graphical models An example: elemental causal induction Beyond Bayes nets… other knowledge in causal induction formalizing causal theories

196
**Elemental causal induction**

C present C absent E present a c E absent b d “To what extent does C cause E?”

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**Causal structure vs. causal strength**

Strength: how strong is a relationship? Structure: does a relationship exist? B E B C w0 w1 E B C w0 B

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**Causal strength Assume structure:**

Leading models (DP and causal power) are maximum likelihood estimates of the strength parameter w1, under different parameterizations for P(E|B,C): linear DP, Noisy-OR causal power B E B C w0 w1

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**Causal structure Hypotheses: h1 = h0 = Bayesian causal inference: B E**

support = B E B C E B C B

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**Buehner and Cheng (1997) People DP (r = 0.89) Power (r = 0.88)**

Support (r = 0.97)

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**The importance of parameterization**

Noisy-OR incorporates mechanism assumptions: generativity: causes increase probability of effects each cause is sufficient to produce the effect causes act via independent mechanisms (Cheng, 1997) Consider other models: statistical dependence: 2 test generic parameterization (Anderson, computer science)

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People Support (Noisy-OR) 2 Support (generic)

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**Generativity is essential**

P(e+|c+) 8/8 6/8 4/8 2/8 0/8 P(e+|c-) 100 50 Support Predictions result from “ceiling effect” ceiling effects only matter if you believe a cause increases the probability of an effect

204
**Bayes nets and beyond... What are Bayes nets?**

graphical models causal graphical models An example: elemental causal induction Beyond Bayes nets… other knowledge in causal induction formalizing causal theories

205
**Hamadeh et al. (2002) Toxicological sciences.**

chemicals genes Clofibrate Wyeth 14,643 Gemfibrozil Phenobarbital Carnitine Palmitoyl Transferase 1 p450 2B1 Hamadeh et al. (2002) Toxicological sciences.

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**Hamadeh et al. (2002) Toxicological sciences.**

chemicals genes X Clofibrate Wyeth 14,643 Gemfibrozil Phenobarbital Carnitine Palmitoyl Transferase 1 p450 2B1 Hamadeh et al. (2002) Toxicological sciences.

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**Hamadeh et al. (2002) Toxicological sciences.**

chemicals genes peroxisome proliferators Chemical X Clofibrate Wyeth 14,643 Gemfibrozil Phenobarbital + Carnitine Palmitoyl Transferase 1 p450 2B1 Hamadeh et al. (2002) Toxicological sciences.

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**Using causal graphical models**

Three questions (usually solved by researcher) what are the variables? what structures are plausible? how do variables interact? How are these questions answered if causal graphical models are used in cognition?

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**Bayes nets and beyond... What are Bayes nets?**

graphical models causal graphical models An example: elemental causal induction Beyond Bayes nets… other knowledge in causal induction formalizing causal theories

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**Theory-based causal induction**

Causal theory Ontology Plausible relations Functional form P(h|data) P(data|h) P(h) Evaluated by statistical inference Z B Y X h0: h1: P(h1) = r P(h0) =1 – r Hypothesis space of causal graphical models Generates

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**Blicket detector (Gopnik, Sobel, and colleagues)**

Oooh, it’s a blicket! Let’s put this one on the machine. See this? It’s a blicket machine. Blickets make it go.

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**“Blocking” Two objects: A and B**

Trial 1 Trial 2 Trials 3, 4 Two objects: A and B Trial 1: A on detector – detector active Trial 2: B on detector – detector inactive Trials 3,4: A B on detector – detector active 3, 4-year-olds judge whether each object is a blicket A: a blicket B: not a blicket

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A deductive inference? Causal law: detector activates if and only if one or more objects on top of it are blickets. Premises: Trial 1: A on detector – detector active Trial 2: B on detector – detector inactive Trials 3,4: A B on detector – detector active Conclusions deduced from premises and causal law: A: a blicket B: not a blicket

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**“Backwards blocking” (Sobel, Tenenbaum & Gopnik, 2004)**

Trial 1 Trial 2 Two objects: A and B Trial 1: A B on detector – detector active Trial 2: A on detector – detector active 4-year-olds judge whether each object is a blicket A: a blicket (100% of judgments) B: probably not a blicket (66% of judgments)

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**Theory Ontology Constraints on causal relations**

Types: Block, Detector, Trial Predicates: Contact(Block, Detector, Trial) Active(Detector, Trial) Constraints on causal relations For any Block b and Detector d, with prior probability q : Cause(Contact(b,d,t), Active(d,t)) Functional form of causal relations Causes of Active(d,t) are independent mechanisms, with causal strengths wi. A background cause has strength w0. Assume a near-deterministic mechanism: wi ~ 1, w0 ~ 0.

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**Theory Ontology Types: Block, Detector, Trial Predicates:**

Contact(Block, Detector, Trial) Active(Detector, Trial) E A B

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**Theory Ontology Types: Block, Detector, Trial Predicates:**

Contact(Block, Detector, Trial) Active(Detector, Trial) A B E A = 1 if Contact(block A, detector, trial), else 0 B = 1 if Contact(block B, detector, trial), else 0 E = 1 if Active(detector, trial), else 0

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**Theory h00 : h10 : h01 : h11 : Constraints on causal relations**

For any Block b and Detector d, with prior probability q : Cause(Contact(b,d,t), Active(d,t)) P(h00) = (1 – q)2 P(h10) = q(1 – q) h00 : h10 : h01 : h11 : B A B No hypotheses with E B, E A, A B, etc. A E E P(h01) = (1 – q) q P(h11) = q2 = “A is a blicket” E A A B E A B E

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**“Activation law”: E=1 if and only if A=1 or B=1.**

Theory Functional form of causal relations Causes of Active(d,t) are independent mechanisms, with causal strengths wb. A background cause has strength w0. Assume a near-deterministic mechanism: wb ~ 1, w0 ~ 0. P(h00) = (1 – q)2 P(h01) = (1 – q) q P(h10) = q(1 – q) P(h11) = q2 A B A B A B A B E E E E P(E=1 | A=0, B=0): P(E=1 | A=1, B=0): P(E=1 | A=0, B=1): P(E=1 | A=1, B=1): “Activation law”: E=1 if and only if A=1 or B=1.

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**Bayesian inference Evaluating causal models in light of data:**

Inferring a particular causal relation:

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**Modeling backwards blocking**

P(h00) = (1 – q)2 P(h01) = (1 – q) q P(h10) = q(1 – q) P(h11) = q2 A B A B A B A B E E E E P(E=1 | A=0, B=0): P(E=1 | A=1, B=0): P(E=1 | A=0, B=1): P(E=1 | A=1, B=1):

222
**Modeling backwards blocking**

P(h00) = (1 – q)2 P(h01) = (1 – q) q P(h10) = q(1 – q) P(h11) = q2 A B A B A B A B E E E E P(E=1 | A=1, B=1):

223
**Modeling backwards blocking**

P(h01) = (1 – q) q P(h10) = q(1 – q) P(h11) = q2 A B A B A B E E E P(E=1 | A=1, B=0): P(E=1 | A=1, B=1):

224
**Manipulating the prior**

I. Pre-training phase: Blickets are rare II. Backwards blocking phase: B A Trial 1 Trial 2 After each trial, adults judge the probability that each object is a blicket.

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**“Rare” condition: First observe 12 objects on detector, of which 2 set it off.**

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**“Common” condition: First observe 12 objects on detector, of which 10 set it off.**

227
**Inferences from ambiguous data**

I. Pre-training phase: Blickets are rare II. Two trials: A B detector, B C detector A B C Trial 1 Trial 2 After each trial, adults judge the probability that each object is a blicket.

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**Same domain theory generates hypothesis space for 3 objects:**

Hypotheses: h000 = h100 = h010 = h001 = h110 = h011 = h101 = h111 = Likelihoods: E E A B C A B C E E A B C A B C E E A B C A B C E E P(E=1| A, B, C; h) = 1 if A = 1 and A E exists, or B = 1 and B E exists, or C = 1 and C E exists, else 0.

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**“Rare” condition: First observe 12 objects on detector, of which 2 set it off.**

230
**The role of causal mechanism knowledge**

Is mechanism knowledge necessary? Constraint-based learning using c2 tests of conditional independence. How important is the deterministic functional form of causal relations? Bayes with “noisy sufficient causes” theory (c.f., Cheng’s causal power theory).

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**Bayes with correct theory:**

Bayes with “noisy sufficient causes” theory:

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**Theory-based causal induction**

Explains one-shot causal inferences about physical systems: blicket detectors Captures a spectrum of inferences: unambiguous data: adults and children make all-or-none inferences ambiguous data: adults and children make more graded inferences Extends to more complex cases with hidden variables, dynamic systems: come to my talk!

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Summary Causal graphical models provide a language for asking questions about causality Key issues in modeling causal induction: what do we mean by causal induction? how do knowledge and statistics interact? Bayesian approach allows exploration of different answers to these questions

234
**Outline Morning Afternoon Introduction (Josh)**

Basic case study #1: Flipping coins (Tom) Basic case study #2: Rules and similarity (Josh) Afternoon Advanced case study #1: Causal induction (Tom) Advanced case study #2: Property induction (Josh) Quick tour of more advanced topics (Tom)

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Property induction

236
**Collaborators Charles Kemp Neville Sanjana Lauren Schmidt Amy Perfors**

Fei Xu Liz Baraff Pat Shafto

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The Big Question How can we generalize new concepts reliably from just one or a few examples? Learning word meanings “horse” “horse” “horse”

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The Big Question How can we generalize new concepts reliably from just one or a few examples? Learning word meanings, causal relations, social rules, …. Property induction How probable is the the conclusion (target) given the premises (examples)? Gorillas have T4 cells. Squirrels have T4 cells. All mammals have T4 cells.

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The Big Question How can we generalize new concepts reliably from just one or a few examples? Learning word meanings, causal relations, social rules, …. Property induction More diverse examples stronger generalization Gorillas have T4 cells. Squirrels have T4 cells. All mammals have T4 cells. Gorillas have T4 cells. Chimps have T4 cells. All mammals have T4 cells.

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**Is rational inference the answer?**

Everyday induction often appears to follow principles of rational scientific inference. Could that explain its success? Goal of this work: a rational computational model of human inductive generalization. Explain people’s judgments as approximations to optimal inference in natural environments. Close quantitative fits to people’s judgments with a minimum of free parameters or assumptions.

241
**Theory-Based Bayesian Models**

Rational statistical inference (Bayes): Learners’ domain theories generate their hypothesis space H and prior p(h). Well-matched to structure of the natural world. Learnable from limited data. Computationally tractable inference.

242
**The plan Similarity-based models Theory-based model Bayesian models**

“Empiricist” Bayes Theory-based Bayes, with different theories Connectionist (PDP) models Advanced Theory-based Bayes Learning with multiple domain theories Learning domain theories Size principle: genericity, nonaccidental, Occam.

243
**The plan Similarity-based models Theory-based model Bayesian models**

“Empiricist” Bayes Theory-based Bayes, with different theories Connectionist (PDP) models Advanced Theory-based Bayes Learning with multiple domain theories Learning domain theories Size principle: genericity, nonaccidental, Occam.

244
**An experiment (Osherson et al., 1990)**

20 subjects rated the strength of 45 arguments: X1 have property P. X2 have property P. X3 have property P. All mammals have property P. 40 different subjects rated the similarity of all pairs of 10 mammals.

245
**Similarity-based models (Osherson et al.)**

strength(“all mammals” | X ) x x x Mammals: Examples: x

246
**Similarity-based models (Osherson et al.)**

strength(“all mammals” | X ) x x x Mammals: Examples: x

247
**Similarity-based models (Osherson et al.)**

strength(“all mammals” | X ) x x x Mammals: Examples: x

248
**Similarity-based models (Osherson et al.)**

strength(“all mammals” | X ) x x x Mammals: Examples: x

249
**Similarity-based models (Osherson et al.)**

Sum-Similarity: strength(“all mammals” | X ) x x S x Mammals: Examples: x

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**Similarity-based models (Osherson et al.)**

Max-Similarity: strength(“all mammals” | X ) x x max x Mammals: Examples: x

251
**Similarity-based models (Osherson et al.)**

Max-Similarity: strength(“all mammals” | X ) x x x Mammals: Examples: x

252
**Similarity-based models (Osherson et al.)**

Max-Similarity: strength(“all mammals” | X ) x x x Mammals: Examples: x

253
**Similarity-based models (Osherson et al.)**

Max-Similarity: strength(“all mammals” | X ) x x x Mammals: Examples: x

254
**Similarity-based models (Osherson et al.)**

Max-Similarity: strength(“all mammals” | X ) x x x Mammals: Examples: x

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**Sum-sim versus Max-sim**

Two models appear functionally similar: Both increase monotonically as new examples are observed. Reasons to prefer Sum-sim: Standard form of exemplar models of categorization, memory, and object recognition. Analogous to kernel density estimation techniques in statistical pattern recognition. Reasons to prefer Max-sim: Fit to generalization judgments

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**Data vs. models . Data Model Each “ ” represents one argument:**

X1 have property P. X2 have property P. X3 have property P. All mammals have property P. Each “ ” represents one argument:

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**Three data sets Max-sim Sum-sim Conclusion kind: “all mammals”**

“horses” “horses” Number of examples: , 2, or 3

258
**Feature rating data (Osherson and Wilkie)**

People were given 48 animals, 85 features, and asked to rate whether each animal had each feature. E.g., elephant: 'gray' 'hairless' 'toughskin' 'big' 'bulbous' 'longleg' 'tail' 'chewteeth' 'tusks' 'smelly' 'walks' 'slow' 'strong' 'muscle’ 'quadrapedal' 'inactive' 'vegetation' 'grazer' 'oldworld' 'bush' 'jungle' 'ground' 'timid' 'smart' 'group'

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**Compute similarity based on Hamming distance, or cosine. **

? Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features New property Compute similarity based on Hamming distance, or cosine. Generalize based on Max-sim or Sum-sim.

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**Three data sets r = 0.77 r = 0.75 r = 0.94 Max-Sim r = – 0.21 r = 0.63**

Sum-Sim Conclusion kind: “all mammals” “horses” “horses” Number of examples: , 2, or 3

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**Problems for sim-based approach**

No principled explanation for why Max-Sim works so well on this task, and Sum-Sim so poorly, when Sum-Sim is the standard in other similarity-based models. Free parameters mixing similarity and coverage terms, and possibly Max-Sim and Sum-Sim terms. Does not extend to induction with other kinds of properties, e.g., from Smith et al., 1993: Dobermanns can bite through wire. German shepherds can bite through wire. Poodles can bite through wire. German shepherds can bite through wire.

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**Marr’s Three Levels of Analysis**

Computation: “What is the goal of the computation, why is it appropriate, and what is the logic of the strategy by which it can be carried out?” Representation and algorithm: Max-sim, Sum-sim Implementation: Neurobiology Cognitive psychology traditionally focuses on 2nd level. But this is unsatisying, for the same reason as in vision: - Ad hoc models, with arbitrary assumptions and free parameters. - Lots of different models with little sense of how they all fit together, or what the real differences are. Cognitive neuroscience has focused on the link between levels 2 and 3. But that doesn’t address what is to me the biggest mystery: how we are able to succeed in these inductive inference tasks, given that induction has been called a puzzle, a paradox, a scandal! Going back to plato, aristotle. Scandal though it might be, we do these things on a daily basis. That requires level 1. But outside of vision or language, not very much work on level 1, or the link from level 1 to level 2 and 3. No explanatory adequacy. Describe how the mind works, but don’t explain why it works that way. Why these inference strategies lead to success in the real world.

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**The plan Similarity-based models Theory-based model Bayesian models**

“Empiricist” Bayes Theory-based Bayes, with different theories Connectionist (PDP) models Advanced Theory-based Bayes Learning with multiple domain theories Learning domain theories Size principle: genericity, nonaccidental, Occam.

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**Theory-based induction**

Scientific biology: species generated by an evolutionary branching process. A tree-structured taxonomy of species. Taxonomy also central in folkbiology (Atran).

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**Theory-based induction**

Begin by reconstructing intuitive taxonomy from similarity judgments: clustering chimp gorilla horse cow rhino seal elephant mouse squirrel dolphin

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**How taxonomy constrains induction**

Atran (1998): “Fundamental principle of systematic induction” (Warburton 1967, Bock 1973) Given a property found among members of any two species, the best initial hypothesis is that the property is also present among all species that are included in the smallest higher-order taxon containing the original pair of species.

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**Strong (0.76 [max = 0.82]) elephant squirrel chimp gorilla horse cow**

rhino mouse dolphin seal “all mammals” Cows have property P. Dolphins have property P. Squirrels have property P. All mammals have property P. Strong (0.76 [max = 0.82])

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**Strong: 0.76 [max = 0.82]) Weak: 0.17 [min = 0.14] elephant squirrel**

chimp gorilla horse cow rhino mouse dolphin seal “large herbivores” Cows have property P. Dolphins have property P. Squirrels have property P. All mammals have property P. Cows have property P. Horses have property P. Rhinos have property P. All mammals have property P. Strong: 0.76 [max = 0.82]) Weak: 0.17 [min = 0.14]

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**Strong: 0.76 [max = 0.82] Weak: 0.30 [min = 0.14] elephant squirrel**

chimp gorilla horse cow rhino mouse dolphin seal “all mammals” Cows have property P. Dolphins have property P. Squirrels have property P. All mammals have property P. Seals have property P. Dolphins have property P. Squirrels have property P. All mammals have property P. Strong: 0.76 [max = 0.82] Weak: 0.30 [min = 0.14]

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**Taxonomic distance Max-sim Sum-sim Conclusion kind: “all mammals”**

“horses” “horses” Number of examples: , 2, or 3

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The challenge Can we build models with the best of both traditional approaches? Quantitatively accurate predictions. Strong rational basis. Will require novel ways of integrating structured knowledge with statistical inference.

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**The plan Similarity-based models Theory-based model Bayesian models**

“Empiricist” Bayes Theory-based Bayes, with different theories Connectionist (PDP) models Advanced Theory-based Bayes Learning with multiple domain theories Learning domain theories Size principle: genericity, nonaccidental, Occam.

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**The Bayesian approach ? Features New property Species 1 Species 2 ?**

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**The Bayesian approach ? Features New property Generalization**

Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**The Bayesian approach ? Features New property Generalization**

Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**The Bayesian approach ? Features New property Generalization**

Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

277
**The Bayesian approach ? Features New property Generalization**

Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**The Bayesian approach ? Features New property Generalization**

Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

279
**The Bayesian approach ? Features New property Generalization**

Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**The Bayesian approach h d p(h) p(d |h) Features New property**

Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**Bayes’ rule: h d p(h) p(d |h) Features New property Generalization**

Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**Probability that property Q holds for species x:**

p(h) p(d |h) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**h d p(h) p(d |h) if d is consistent “Size principle”: with h**

|h | = # of positive instances of h otherwise p(h) p(d |h) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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The size principle h1 h2 “even numbers” “multiples of 10”

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**The size principle Data slightly more of a coincidence under h1 h1 h2**

“even numbers” “multiples of 10” Data slightly more of a coincidence under h1

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**The size principle Data much more of a coincidence under h1 h1 h2**

“even numbers” “multiples of 10” Data much more of a coincidence under h1

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**Illustrating the size principle**

Which argument is stronger? “Non-monotonicity” Grizzly bears have property P. All mammals have property P. Grizzly bears have property P. Brown bears have property P. Polar bears have property P. All mammals have property P.

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**... Probability that property Q holds for species x: p(Q(x)|d) h d**

p(h) p(d |h) p(Q(x)|d) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? ... Generalization Hypotheses New property

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**Probability that property Q holds for species x:**

p(h) p(d |h) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**Specifying the prior p(h)**

A good prior must focus on a small subset of all 2n possible hypotheses, in order to: Match the distribution of properties in the world. Be learnable from limited data. Be efficiently computationally. We consider two approaches: “Empiricist” Bayes: unstructured prior based directly on known features. “Theory-based” Bayes: structured prior based on rational domain theory, tuned to known features.

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**“Empiricist” Bayes: h d p(h) = (Heit, 1998) Features New property**

h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 p(h) = h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**Results r = 0.38 r = 0.16 r = 0.79 “Empiricist” Bayes r = 0.77**

Max-Sim

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**Why doesn’t “Empiricist” Bayes work?**

With no structural bias, requires too many features to estimate the prior reliably. An analogy: Estimating a smooth probability density function by local interpolation. N = 100 N = 500 N = 5

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**Why doesn’t “Empiricist” Bayes work?**

With no structural bias, requires too many features to estimate the prior reliably. An analogy: Estimating a smooth probability density function by local interpolation. Assuming an appropriately structured form for density (e.g., Gaussian) leads to better generalization from sparse data. N = 5 N = 5

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“Theory-based” Bayes Theory: Two principles based on the structure of species and properties in the natural world. 1. Species generated by an evolutionary branching process. A tree-structured taxonomy of species (Atran, 1998). 2. Features generated by stochastic mutation process and passed on to descendants. Novel features can appear anywhere in tree, but some distributions are more likely than others.

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**T h d Mutation process generates p(h|T): |b| = length of branch b**

Choose label for root. Probability that label mutates along branch b : l = mutation rate |b| = length of branch b s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 T p(h|T) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

297
**T h d Mutation process generates p(h|T): |b| = length of branch b**

Choose label for root. Probability that label mutates along branch b : l = mutation rate |b| = length of branch b x x x T p(h|T) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**Samples from the prior >**

Labelings that cut the data along fewer branches are more probable: > “monophyletic” “polyphyletic”

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**Samples from the prior >**

Labelings that cut the data along longer branches are more probable: > “more distinctive” “less distinctive”

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**T h d Mutation process over tree T generates p(h|T).**

Message passing over tree T efficiently sums over all h. How do we know which tree T to use? s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 T p(h|T) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**T h d The same mutation process generates p(Features|T): p(h|T)**

Assume each feature generated independently over the tree. Use MCMC to infer most likely tree T and mutation rate l given observed features. No free parameters! s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 T p(h|T) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**Results r = 0.91 r = 0.95 r = 0.91 “Theory-based” Bayes r = 0.38**

“Empiricist” Bayes r = 0.77 r = 0.75 r = 0.94 Max-Sim

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**Grounding in similarity**

Reconstruct intuitive taxonomy from similarity judgments: clustering chimp gorilla horse cow rhino seal elephant mouse squirrel dolphin

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**Theory-based Bayes Max-sim Sum-sim Conclusion kind: “all mammals”**

“horses” “horses” Number of examples: , 2, or 3

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**Explaining similarity**

Why does Max-sim fit so well? An efficient and accurate approximation to this Theory-Based Bayesian model. Theorem. Nearest neighbor classification approximates evolutionary Bayes in the limit of high mutation rate, if domain is tree-structured. Correlation with Bayes on three-premise general arguments, over 100 simulated trees: Mean r = 0.94 Correlation (r)

306
**Alternative feature-based models**

Taxonomic Bayes (strictly taxonomic hypotheses, with no mutation process) > “monophyletic” “polyphyletic”

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**Alternative feature-based models**

Taxonomic Bayes (strictly taxonomic hypotheses, with no mutation process) PDP network (Rogers and McClelland) Features Species

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**Results Note: PDP graph is mocked up, correlations OK. Tax Bayes**

Bias is too strong weak just right! Note: PDP graph is mocked up, correlations OK. Tax Bayes graphs OK, not sure about correlations. Theory-based Bayes r = 0.51 r = 0.53 r = 0.85 Taxonomic Bayes r = 0.41 r = 0.62 r = 0.71 PDP network

309
**Mutation principle versus pure Occam’s Razor**

Mutation principle provides a version of Occam’s Razor, by favoring hypotheses that span fewer disjoint clusters. Could we use a more generic Bayesian Occam’s Razor, without the biological motivation of mutation?

310
**T h d Mutation process generates p(h|T): |b| = length of branch b**

Choose label for root. Probability that label mutates along branch b : l = mutation rate |b| = length of branch b s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 T p(h|T) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

311
**T h d Mutation process generates p(h|T): |b| = length of branch b**

Choose label for root. Probability that label mutates along branch b : l = mutation rate |b| = length of branch b s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 T p(h|T) h d Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 ? Features Generalization Hypothesis New property

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**Premise typicality effect (Rips, 1975; Osherson et al., 1990): Strong:**

Bayes (taxonomy+ mutation) Premise typicality effect (Rips, 1975; Osherson et al., 1990): Strong: Weak: Bayes (taxonomy+ Occam) Horses have property P. All mammals have property P. Max-sim Seals have property P. All mammals have property P. Conclusion kind: “all mammals” Number of examples: 1

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**Typicality meets hierarchies**

Collins and Quillian: semantic memory structured hierarchically Traditional story: Simple hierarchical structure uncomfortable with typicality effects & exceptions. New story: Typicality & exceptions compatible with rational statistical inference over hierarchy.

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**Intuitive versus scientific theories of biology**

Same structure for how species are related. Tree-structured taxonomy. Same probabilistic model for traits Small probability of occurring along any branch at any time, plus inheritance. Different features Scientist: genes People: coarse anatomy and behavior

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**Induction in Biology: summary**

Theory-based Bayesian inference explains taxonomic inductive reasoning in folk biology. Insight into processing-level accounts. Why Max-sim over Sum-sim in this domain? How is hierarchical representation compatible with typicality effects & exceptions? Reveals essential principles of domain theory. Category structure: taxonomic tree. Feature distribution: stochastic mutation process inheritance.

316
**The plan Similarity-based models Theory-based model Bayesian models**

“Empiricist” Bayes Theory-based Bayes, with different theories Connectionist (PDP) models Advanced Theory-based Bayes Learning with multiple domain theories Learning domain theories Size principle: genericity, nonaccidental, Occam.

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**Property type Generic “essence” Theory Structure Taxonomic Tree . . .**

Lion Cheetah Hyena Giraffe Gazelle Gorilla Monkey Lion Cheetah Hyena Giraffe Gazelle Gorilla Monkey . . .

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**Generic “essence” Size-related Food-carried Theory Structure **

Property type Generic “essence” Size-related Food-carried Theory Structure Taxonomic Tree Dimensional Directed Acyclic Network Giraffe Lion Giraffe Cheetah Lion Lion Gorilla Hyena Gazelle Giraffe Hyena Hyena Cheetah Gazelle Gazelle Monkey Gorilla Cheetah Monkey Monkey Gorilla Lion Cheetah Hyena Giraffe Gazelle Gorilla Monkey . . . . . . . . .

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**One-dimensional predicates**

Q = “Have skins that are more resistant to penetration than most synthetic fibers”. Unknown relevant property: skin toughness Model influence of known properties via judged prior probability that each species has Q. threshold for Q Skin toughness House cat Camel Elephant Rhino

320
**One-dimensional predicates**

Bayes (taxonomy+ mutation) Max-sim Bayes (1D model)

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**Food web model fits (Shafto et al.)**

Disease r = 0.77 r = 0.82 Property r = -0.35 r = -0.05 Mammals Island

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**Taxonomic tree model fits (Shafto et al.)**

Disease r = -0.12 r = 0.16 Property r = 0.81 r = 0.62 Mammals Island

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**The plan Similarity-based models Theory-based model Bayesian models**

“Empiricist” Bayes Theory-based Bayes, with different theories Connectionist (PDP) models Advanced Theory-based Bayes Learning with multiple domain theories Learning domain theories Size principle: genericity, nonaccidental, Occam.

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**Theory Domain Structure Data**

Species organized in taxonomic tree structure Feature i generated by mutation process with rate li p(S|T) F9 Domain Structure F8 F7 F11 F14 F13 F6 F12 F14 F10 F3 F1 F2 F4 F5 F10 F10 S3 S4 S1 S2 S9 S10 S5 S6 S7 S8 p(D|S) l10 high ~ weight low Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Data

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**Theory Domain Structure Data**

Species organized in taxonomic tree structure Feature i generated by mutation process with rate li p(S|T) F9 Domain Structure F8 F7 F11 F14 F13 F6 F12 F14 F10 F3 F1 F2 F4 F5 F10 F10 S3 S4 S1 S2 S9 S10 S5 S6 S7 S8 p(D|S) Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Data ? ? ? ? ? ? ? ? ? ? ? ? ? Species X

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**Theory Domain Structure Data**

Species organized in taxonomic tree structure Feature i generated by mutation process with rate li p(S|T) F9 Domain Structure F8 F7 F11 F14 F13 F6 F12 F14 F10 F3 F1 F2 F4 F5 F10 SX F10 S3 S4 S1 S2 S9 S10 S5 S6 S7 S8 p(D|S) Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Data Species X

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**Where does the domain theory come from?**

Innate. Atran (1998): The tendency to group living kinds into hierarchies reflects an “innately determined cognitive structure”. Emerges (only approximately) through learning in unstructured connectionist networks. McClelland and Rogers (2003).

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**Bayesian inference to theories**

Challenge to the nativist-empiricist dichotomy. We really do have structured domain theories. We really do learn them. Bayesian framework applies over multiple levels: Given hypothesis space + data, infer concepts. Given theory + data, infer hypothesis space. Given X + data, infer theory.

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**Bayesian inference to theories**

Candidate theories for biological species and their features: T0: Features generated independently for each species. (c.f. naive Bayes, Anderson’s rational model.) T1: Features generated by mutation in tree-structured taxonomy of species. T2: Features generated by mutation in a one-dimensional chain of species. Score theories by likelihood on object-feature matrix:

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**Data T0: No organizational structure for species. Features distributed**

independently over species. F1 F2 F3 F5 F7 F8 F10 F12 F13 F1 F6 F7 F8 F9 F10 F13 F2 F4 F8 F9 F10 F11 F14 F2 F4 F6 F7 F9 F14 F2 F4 F7 F9 F12 F14 F2 F4 F5 F12 F13 F14 F1 F2 F5 F8 F9 F1 F5 F7 F13 F14 F2 F3 F6 F11 F13 F1 F6 F8 F9 F12 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Data Features

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**Data T0: No organizational structure for species. Features distributed**

independently over species. F1 F6 F7 F8 F9 F10 F11 F3 F7 F8 F9 F11 F12 F14 F3 F7 F8 F9 F11 F12 F14 F1 F6 F7 F8 F9 F11 F2 F6 F7 F8 F9 F11 F2 F6 F7 F8 F9 F11 F5 F9 F10 F13 F14 F5 F9 F10 F13 F14 F4 F8 F9 F4 F8 F9 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Features Data

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**Data T0: T1: No organizational structure Species organized in**

for species. Features distributed independently over species. T1: Species organized in taxonomic tree structure. Features distributed via stochastic mutation process. S3 S4 S1 S2 S9 S10 S5 S6 S7 S8 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F1 F6 F7 F8 F9 F10 F11 F3 F7 F8 F9 F11 F12 F14 F3 F7 F8 F9 F11 F12 F14 F1 F6 F7 F8 F9 F11 F2 F6 F7 F8 F9 F11 F2 F6 F7 F8 F9 F11 F5 F9 F10 F13 F14 F5 F9 F10 F13 F14 F4 F8 F9 F4 F8 F9 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Features Data

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**Data T0: p(Data|T1) ~ 1.83 x 10-41 T1: p(Data|T2) ~ 2.42 x 10-32**

No organizational structure for species. Features distributed independently over species. T1: p(Data|T2) ~ 2.42 x 10-32 Species organized in taxonomic tree structure. Features distributed via stochastic mutation process. S3 S4 S1 S2 S9 S10 S5 S6 S7 S8 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F1 F6 F7 F8 F9 F10 F11 F3 F7 F8 F9 F11 F12 F14 F3 F7 F8 F9 F11 F12 F14 F1 F6 F7 F8 F9 F11 F2 F6 F7 F8 F9 F11 F2 F6 F7 F8 F9 F11 F5 F9 F10 F13 F14 F5 F9 F10 F13 F14 F4 F8 F9 F4 F8 F9 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Features Data

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**Data T0: T1: No organizational structure Species organized in**

for species. Features distributed independently over species. T1: Species organized in taxonomic tree structure. Features distributed via stochastic mutation process. F1 F2 F3 F5 F7 F8 F10 F12 F13 F2 F4 F1 F5 F7 F13 F1 F6 F7 F8 F9 F10 F13 F2 F4 F8 F9 F10 F11 F14 F14 F8 F9 F12 F2 F4 F6 F7 F9 F14 F2 F4 F7 F9 F12 F14 F2 F4 F5 F12 F13 F14 F9 F13 F10 F1 F2 F5 F8 F9 F1 F5 F7 F13 F14 F2 F3 F6 F11 F13 F1 F6 F8 F9 F12 F7 F11 F13 F10 F8 F13 F10 F11 F12 F7 F3 F12 F12 F9 F6 F5 F6 F5 F8 F3 F2 F6 F6 F2 F14 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S2 S4 S7 S10 S8 S1 S9 S6 S3 S5 Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Data Features

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**Data T0: p(Data|T1) ~ 2.29 x 10-42 T1: p(Data|T2) ~ 4.38 x 10-53**

No organizational structure for species. Features distributed independently over species. T1: p(Data|T2) ~ 4.38 x 10-53 Species organized in taxonomic tree structure. Features distributed via stochastic mutation process. F1 F2 F3 F5 F7 F8 F10 F12 F13 F2 F4 F1 F5 F7 F13 F1 F6 F7 F8 F9 F10 F13 F2 F4 F8 F9 F10 F11 F14 F14 F8 F9 F12 F2 F4 F6 F7 F9 F14 F2 F4 F7 F9 F12 F14 F2 F4 F5 F12 F13 F14 F9 F13 F10 F1 F2 F5 F8 F9 F1 F5 F7 F13 F14 F2 F3 F6 F11 F13 F1 F6 F8 F9 F12 F7 F11 F13 F10 F8 F13 F10 F11 F12 F7 F3 F12 F12 F9 F6 F5 F6 F5 F8 F3 F2 F6 F6 F2 F14 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S2 S4 S7 S10 S8 S1 S9 S6 S3 S5 Species 1 Species 2 Species 3 Species 4 Species 5 Species 6 Species 7 Species 8 Species 9 Species 10 Data Features

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**Empirical tests Synthetic data: 32 objects, 120 features Real data**

tree-structured generative model linear chain generative model unconstrained (independent features). Real data Animal feature judgments: 48 species, 85 features. US Supreme Court decisions, : 9 people, 637 cases.

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Results Preferred Model Null Tree Linear

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**Theory acquisition: summary**

So far, just a computational proof of concept. Future work: Experimental studies of theory acquisition in the lab, with adult and child subjects. Modeling developmental or historical trajectories of theory change. Sources of hypotheses for candidate theories: What is innate? Role of analogy?

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**Outline Morning Afternoon Introduction (Josh)**

Basic case study #1: Flipping coins (Tom) Basic case study #2: Rules and similarity (Josh) Afternoon Advanced case study #1: Causal induction (Tom) Advanced case study #2: Property induction (Josh) Quick tour of more advanced topics (Tom)

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Advanced topics

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**Structure and statistics**

Statistical language modeling topic models Relational categorization attributes and relations

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**Structure and statistics**

Statistical language modeling topic models Relational categorization attributes and relations

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**Statistical language modeling**

A variety of approaches to statistical language modeling are used in cognitive science e.g. LSA (Landauer & Dumais, 1997) distributional clustering (Redington, Chater, & Finch, 1998) Generative models have unique advantages identify assumed causal structure of language make use of standard tools of Bayesian statistics easily extended to capture more complex structure

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**Generative models for language**

latent structure observed data

345
**Generative models for language**

meaning sentences

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**Topic models Each document a mixture of topics**

Each word chosen from a single topic Introduced by Blei, Ng, and Jordan (2001), reinterpretation of PLSI (Hofmann, 1999) Idea of probabilistic topics widely used (eg. Bigi et al., 1997; Iyer & Ostendorf, 1996; Ueda & Saito, 2003)

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**Generating a document q distribution over topics z z z**

topic assignments w w w observed words

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w P(w|z = 1) = f (1) w P(w|z = 2) = f (2) HEART 0.2 LOVE 0.2 SOUL 0.2 TEARS 0.2 JOY 0.2 SCIENTIFIC KNOWLEDGE 0.0 WORK 0.0 RESEARCH 0.0 MATHEMATICS 0.0 HEART 0.0 LOVE 0.0 SOUL 0.0 TEARS 0.0 JOY 0.0 SCIENTIFIC KNOWLEDGE 0.2 WORK 0.2 RESEARCH 0.2 MATHEMATICS 0.2 topic 1 topic 2

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**Choose mixture weights for each document, generate “bag of words”**

q = {P(z = 1), P(z = 2)} {0, 1} {0.25, 0.75} {0.5, 0.5} {0.75, 0.25} {1, 0} MATHEMATICS KNOWLEDGE RESEARCH WORK MATHEMATICS RESEARCH WORK SCIENTIFIC MATHEMATICS WORK SCIENTIFIC KNOWLEDGE MATHEMATICS SCIENTIFIC HEART LOVE TEARS KNOWLEDGE HEART MATHEMATICS HEART RESEARCH LOVE MATHEMATICS WORK TEARS SOUL KNOWLEDGE HEART WORK JOY SOUL TEARS MATHEMATICS TEARS LOVE LOVE LOVE SOUL TEARS LOVE JOY SOUL LOVE TEARS SOUL SOUL TEARS JOY

350
**A selection of topics (from 500)**

THEORY SCIENTISTS EXPERIMENT OBSERVATIONS SCIENTIFIC EXPERIMENTS HYPOTHESIS EXPLAIN SCIENTIST OBSERVED EXPLANATION BASED OBSERVATION IDEA EVIDENCE THEORIES BELIEVED DISCOVERED OBSERVE FACTS SPACE EARTH MOON PLANET ROCKET MARS ORBIT ASTRONAUTS FIRST SPACECRAFT JUPITER SATELLITE SATELLITES ATMOSPHERE SPACESHIP SURFACE SCIENTISTS ASTRONAUT SATURN MILES ART PAINT ARTIST PAINTING PAINTED ARTISTS MUSEUM WORK PAINTINGS STYLE PICTURES WORKS OWN SCULPTURE PAINTER ARTS BEAUTIFUL DESIGNS PORTRAIT PAINTERS STUDENTS TEACHER STUDENT TEACHERS TEACHING CLASS CLASSROOM SCHOOL LEARNING PUPILS CONTENT INSTRUCTION TAUGHT GROUP GRADE SHOULD GRADES CLASSES PUPIL GIVEN BRAIN NERVE SENSE SENSES ARE NERVOUS NERVES BODY SMELL TASTE TOUCH MESSAGES IMPULSES CORD ORGANS SPINAL FIBERS SENSORY PAIN IS CURRENT ELECTRICITY ELECTRIC CIRCUIT IS ELECTRICAL VOLTAGE FLOW BATTERY WIRE WIRES SWITCH CONNECTED ELECTRONS RESISTANCE POWER CONDUCTORS CIRCUITS TUBE NEGATIVE NATURE WORLD HUMAN PHILOSOPHY MORAL KNOWLEDGE THOUGHT REASON SENSE OUR TRUTH NATURAL EXISTENCE BEING LIFE MIND ARISTOTLE BELIEVED EXPERIENCE REALITY THIRD FIRST SECOND THREE FOURTH FOUR GRADE TWO FIFTH SEVENTH SIXTH EIGHTH HALF SEVEN SIX SINGLE NINTH END TENTH ANOTHER

351
**A selection of topics (from 500)**

JOB WORK JOBS CAREER EXPERIENCE EMPLOYMENT OPPORTUNITIES WORKING TRAINING SKILLS CAREERS POSITIONS FIND POSITION FIELD OCCUPATIONS REQUIRE OPPORTUNITY EARN ABLE DISEASE BACTERIA DISEASES GERMS FEVER CAUSE CAUSED SPREAD VIRUSES INFECTION VIRUS MICROORGANISMS PERSON INFECTIOUS COMMON CAUSING SMALLPOX BODY INFECTIONS CERTAIN WATER FISH SEA SWIM SWIMMING POOL LIKE SHELL SHARK TANK SHELLS SHARKS DIVING DOLPHINS SWAM LONG SEAL DIVE DOLPHIN UNDERWATER MIND WORLD DREAM DREAMS THOUGHT IMAGINATION MOMENT THOUGHTS OWN REAL LIFE IMAGINE SENSE CONSCIOUSNESS STRANGE FEELING WHOLE BEING MIGHT HOPE STORY STORIES TELL CHARACTER CHARACTERS AUTHOR READ TOLD SETTING TALES PLOT TELLING SHORT FICTION ACTION TRUE EVENTS TELLS TALE NOVEL FIELD MAGNETIC MAGNET WIRE NEEDLE CURRENT COIL POLES IRON COMPASS LINES CORE ELECTRIC DIRECTION FORCE MAGNETS BE MAGNETISM POLE INDUCED SCIENCE STUDY SCIENTISTS SCIENTIFIC KNOWLEDGE WORK RESEARCH CHEMISTRY TECHNOLOGY MANY MATHEMATICS BIOLOGY FIELD PHYSICS LABORATORY STUDIES WORLD SCIENTIST STUDYING SCIENCES BALL GAME TEAM FOOTBALL BASEBALL PLAYERS PLAY FIELD PLAYER BASKETBALL COACH PLAYED PLAYING HIT TENNIS TEAMS GAMES SPORTS BAT TERRY

352
**A selection of topics (from 500)**

JOB WORK JOBS CAREER EXPERIENCE EMPLOYMENT OPPORTUNITIES WORKING TRAINING SKILLS CAREERS POSITIONS FIND POSITION FIELD OCCUPATIONS REQUIRE OPPORTUNITY EARN ABLE DISEASE BACTERIA DISEASES GERMS FEVER CAUSE CAUSED SPREAD VIRUSES INFECTION VIRUS MICROORGANISMS PERSON INFECTIOUS COMMON CAUSING SMALLPOX BODY INFECTIONS CERTAIN WATER FISH SEA SWIM SWIMMING POOL LIKE SHELL SHARK TANK SHELLS SHARKS DIVING DOLPHINS SWAM LONG SEAL DIVE DOLPHIN UNDERWATER MIND WORLD DREAM DREAMS THOUGHT IMAGINATION MOMENT THOUGHTS OWN REAL LIFE IMAGINE SENSE CONSCIOUSNESS STRANGE FEELING WHOLE BEING MIGHT HOPE STORY STORIES TELL CHARACTER CHARACTERS AUTHOR READ TOLD SETTING TALES PLOT TELLING SHORT FICTION ACTION TRUE EVENTS TELLS TALE NOVEL FIELD MAGNETIC MAGNET WIRE NEEDLE CURRENT COIL POLES IRON COMPASS LINES CORE ELECTRIC DIRECTION FORCE MAGNETS BE MAGNETISM POLE INDUCED SCIENCE STUDY SCIENTISTS SCIENTIFIC KNOWLEDGE WORK RESEARCH CHEMISTRY TECHNOLOGY MANY MATHEMATICS BIOLOGY FIELD PHYSICS LABORATORY STUDIES WORLD SCIENTIST STUDYING SCIENCES BALL GAME TEAM FOOTBALL BASEBALL PLAYERS PLAY FIELD PLAYER BASKETBALL COACH PLAYED PLAYING HIT TENNIS TEAMS GAMES SPORTS BAT TERRY

353
**Learning topic hiearchies**

(Blei, Griffiths, Jordan, & Tenenbaum, 2004)

354
**Syntax and semantics from statistics**

Factorization of language based on statistical dependency patterns: long-range, document specific, dependencies short-range dependencies constant across all documents semantics: probabilistic topics q z z z w w w x x x syntax: probabilistic regular grammar (Griffiths, Steyvers, Blei, & Tenenbaum, submitted)

355
x = 2 x = 1 OF 0.6 FOR 0.3 BETWEEN 0.1 0.8 z = z = HEART 0.2 LOVE 0.2 SOUL 0.2 TEARS 0.2 JOY 0.2 SCIENTIFIC 0.2 KNOWLEDGE 0.2 WORK 0.2 RESEARCH 0.2 MATHEMATICS 0.2 0.7 0.3 0.1 0.2 x = 3 THE 0.6 A 0.3 MANY 0.1 0.9

356
**THE ……………………………… x = 2 x = 1 x = 3 0.8 z = 1 0.4 z = 2 0.6 0.7 0.3 0.1**

OF 0.6 FOR 0.3 BETWEEN 0.1 0.8 z = z = HEART 0.2 LOVE 0.2 SOUL 0.2 TEARS 0.2 JOY 0.2 SCIENTIFIC 0.2 KNOWLEDGE 0.2 WORK 0.2 RESEARCH 0.2 MATHEMATICS 0.2 0.7 0.3 0.1 0.2 x = 3 THE 0.6 A 0.3 MANY 0.1 0.9 THE ………………………………

357
**THE LOVE…………………… x = 2 x = 1 x = 3 0.8 z = 1 0.4 z = 2 0.6 0.7 0.3 0.1**

OF 0.6 FOR 0.3 BETWEEN 0.1 0.8 z = z = HEART 0.2 LOVE 0.2 SOUL 0.2 TEARS 0.2 JOY 0.2 SCIENTIFIC 0.2 KNOWLEDGE 0.2 WORK 0.2 RESEARCH 0.2 MATHEMATICS 0.2 0.7 0.3 0.1 0.2 x = 3 THE 0.6 A 0.3 MANY 0.1 0.9 THE LOVE……………………

358
**THE LOVE OF……………… x = 2 x = 1 x = 3 0.8 z = 1 0.4 z = 2 0.6 0.7 0.3**

FOR 0.3 BETWEEN 0.1 0.8 z = z = HEART 0.2 LOVE 0.2 SOUL 0.2 TEARS 0.2 JOY 0.2 SCIENTIFIC 0.2 KNOWLEDGE 0.2 WORK 0.2 RESEARCH 0.2 MATHEMATICS 0.2 0.7 0.3 0.1 0.2 x = 3 THE 0.6 A 0.3 MANY 0.1 0.9 THE LOVE OF………………

359
**THE LOVE OF RESEARCH …… x = 2 x = 1 x = 3 0.8 z = 1 0.4 z = 2 0.6 0.7**

FOR 0.3 BETWEEN 0.1 0.8 z = z = HEART 0.2 LOVE 0.2 SOUL 0.2 TEARS 0.2 JOY 0.2 SCIENTIFIC 0.2 KNOWLEDGE 0.2 WORK 0.2 RESEARCH 0.2 MATHEMATICS 0.2 0.7 0.3 0.1 0.2 x = 3 THE 0.6 A 0.3 MANY 0.1 0.9 THE LOVE OF RESEARCH ……

360
**Semantic categories FOOD FOODS BODY NUTRIENTS DIET FAT SUGAR ENERGY**

MILK EATING FRUITS VEGETABLES WEIGHT FATS NEEDS CARBOHYDRATES VITAMINS CALORIES PROTEIN MINERALS MAP NORTH EARTH SOUTH POLE MAPS EQUATOR WEST LINES EAST AUSTRALIA GLOBE POLES HEMISPHERE LATITUDE PLACES LAND WORLD COMPASS CONTINENTS DOCTOR PATIENT HEALTH HOSPITAL MEDICAL CARE PATIENTS NURSE DOCTORS MEDICINE NURSING TREATMENT NURSES PHYSICIAN HOSPITALS DR SICK ASSISTANT EMERGENCY PRACTICE BOOK BOOKS READING INFORMATION LIBRARY REPORT PAGE TITLE SUBJECT PAGES GUIDE WORDS MATERIAL ARTICLE ARTICLES WORD FACTS AUTHOR REFERENCE NOTE GOLD IRON SILVER COPPER METAL METALS STEEL CLAY LEAD ADAM ORE ALUMINUM MINERAL MINE STONE MINERALS POT MINING MINERS TIN BEHAVIOR SELF INDIVIDUAL PERSONALITY RESPONSE SOCIAL EMOTIONAL LEARNING FEELINGS PSYCHOLOGISTS INDIVIDUALS PSYCHOLOGICAL EXPERIENCES ENVIRONMENT HUMAN RESPONSES BEHAVIORS ATTITUDES PSYCHOLOGY PERSON CELLS CELL ORGANISMS ALGAE BACTERIA MICROSCOPE MEMBRANE ORGANISM FOOD LIVING FUNGI MOLD MATERIALS NUCLEUS CELLED STRUCTURES MATERIAL STRUCTURE GREEN MOLDS PLANTS PLANT LEAVES SEEDS SOIL ROOTS FLOWERS WATER FOOD GREEN SEED STEMS FLOWER STEM LEAF ANIMALS ROOT POLLEN GROWING GROW

361
**Syntactic categories BE MAKE GET HAVE GO TAKE DO FIND USE SEE HELP**

KEEP GIVE LOOK COME WORK MOVE LIVE EAT BECOME SAID ASKED THOUGHT TOLD SAYS MEANS CALLED CRIED SHOWS ANSWERED TELLS REPLIED SHOUTED EXPLAINED LAUGHED MEANT WROTE SHOWED BELIEVED WHISPERED THE HIS THEIR YOUR HER ITS MY OUR THIS THESE A AN THAT NEW THOSE EACH MR ANY MRS ALL MORE SUCH LESS MUCH KNOWN JUST BETTER RATHER GREATER HIGHER LARGER LONGER FASTER EXACTLY SMALLER SOMETHING BIGGER FEWER LOWER ALMOST ON AT INTO FROM WITH THROUGH OVER AROUND AGAINST ACROSS UPON TOWARD UNDER ALONG NEAR BEHIND OFF ABOVE DOWN BEFORE GOOD SMALL NEW IMPORTANT GREAT LITTLE LARGE * BIG LONG HIGH DIFFERENT SPECIAL OLD STRONG YOUNG COMMON WHITE SINGLE CERTAIN ONE SOME MANY TWO EACH ALL MOST ANY THREE THIS EVERY SEVERAL FOUR FIVE BOTH TEN SIX MUCH TWENTY EIGHT HE YOU THEY I SHE WE IT PEOPLE EVERYONE OTHERS SCIENTISTS SOMEONE WHO NOBODY ONE SOMETHING ANYONE EVERYBODY SOME THEN

362
**Statistical language modeling**

Generative models provide transparent assumptions about causal process opportunities to combine and extend models Richer generative models... probabilistic context-free grammars paragraph or sentence-level dependencies more complex semantics

363
**Structure and statistics**

Statistical language modeling topic models Relational categorization attributes and relations

364
**Relational categorization**

Most approaches to categorization in psychology and machine learning focus on attributes - properties of objects words in titles of CogSci posters But… a significant portion of knowledge is organized in terms of relations co-authors on posters who talks to whom (Kemp, Griffiths, & Tenenbaum, 2004)

365
**Attributes and relations**

Data Model objects P(X) = ik z P(xik|zi) i P(zi) attributes X mixture model (c.f. Anderson, 1990) objects Y P(Y) = ij z P(yij|zi) i P(zi) objects stochastic blockmodel

366
**Stochastic blockmodels**

For any pair of objects, (i,j), probability of relation is determined by classes, (zi, zj) Allows types of objects and class probabilities to be learned from data l21 l22 l23 l31 l32 l33 l11 l12 l13 = L From type i To type j Each entity has a type = Z P(Z,L|Y) P(Y|Z,L)P(Z)P(L)

367
**Stochastic blockmodels**

368
Categorizing words Relational data: word association norms (Nelson, McEvoy, & Schreiber, 1998) 5018 x 5018 matrix of associations symmetrized all words with < 50 and > 10 associates 2513 nodes, links

371
**Categorizing words BAND INSTRUMENT BLOW HORN FLUTE BRASS GUITAR PIANO**

TUBA TRUMPET TIE COAT SHOES ROPE LEATHER SHOE HAT PANTS WEDDING STRING SEW MATERIAL WOOL YARN WEAR TEAR FRAY JEANS COTTON CARPET WASH LIQUID BATHROOM SINK CLEANER STAIN DRAIN DISHES TUB SCRUB

372
Categorizing actors Internet Movie Database (IMDB) data, from the start of cinema to 1960 (Jeremy Kubica) Relational data: collaboration 5000 x 5000 matrix of most prolific actors all actors with < 400 and > 1 collaborators 2275 nodes, links

375
**Categorizing actors Albert Lieven Karel Stepanek Walter Rilla**

Anton Walbrook Moore Marriott Laurence Hanray Gus McNaughton Gordon Harker Helen Haye Alfred Goddard Morland Graham Margaret Lockwood Hal Gordon Bromley Davenport Gino Cervi Nadia Gray Enrico Glori Paolo Stoppa Bernardi Nerio Amedeo Nazzari Gina Lollobrigida Aldo Silvani Franco Interlenghi Guido Celano Archie Ricks Helen Gibson Oscar Gahan Buck Moulton Buck Connors Clyde McClary Barney Beasley Buck Morgan Tex Phelps George Sowards Germany UK British comedy Italian US Westerns

376
**Structure and statistics**

Bayesian approach allows us to specify structured probabilistic models Explore novel representations and domains topics for semantic representation relational categorization Use powerful methods for inference, developed in statistics and machine learning

377
**Other methods and tools...**

Inference algorithms belief propagation dynamic programming the EM algorithm and variational methods Markov chain Monte Carlo More complex models Dirichlet processes and Bayesian non-parametrics Gaussian processes and kernel methods Reading list at

378
Taking stock

379
**Bayesian models of inductive learning**

Inductive leaps can be explained with hierarchical Theory-based Bayesian models: Domain Theory Probabilistic Generative Model Bayesian inference Structural Hypotheses Data

380
**Bayesian models of inductive learning**

Inductive leaps can be explained with hierarchical Theory-based Bayesian models: T ... S S S ... D D D D D D D D D

381
**Bayesian models of inductive learning**

Inductive leaps can be explained with hierarchical Theory-based Bayesian models. What the approach offers: Strong quantitative models of generalization behavior. Flexibility to model different patterns of reasoning that in different tasks and domains, using differently structured theories, but the same general-purpose Bayesian engine. Framework for explaining why inductive generalization works, where knowledge comes from as well as how it is used.

382
**Bayesian models of inductive learning**

Inductive leaps can be explained with hierarchical Theory-based Bayesian models. Challenges: Theories are hard.

383
**Bayesian models of inductive learning**

Inductive leaps can be explained with hierarchical Theory-based Bayesian models: The interaction between structure and statistics is crucial. How structured knowledge supports statistical learning, by constraining hypothesis spaces. How statistics supports reasoning with and learning structured knowledge. How complex structures can grow from data, rather than being fully specified in advance.

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