# Bayesian models of inductive lea

Document Sample

```					        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)
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.
• What you won’t get:
– Detailed, hands-on how-to.
http://bayesiancognition.com
Outline
• Morning
– 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)
Outline
• Morning
– 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)
Bayesian models in
cognitive science
•   Vision
•   Motor control
•   Memory
•   Language
•   Inductive learning and reasoning….
Everyday inductive leaps
• Learning concepts and words from examples

“horse”

“horse”

“horse”
Learning concepts and words
“tufa”

“tufa”

“tufa”

Can you pick out the tufas?
Inductive reasoning
Input:
Cows can get Hick’s disease.
(premises)
Gorillas can get Hick’s disease.

All mammals can get Hick’s disease.   (conclusion)

Task: Judge how likely conclusion is to be
true, given that premises are true.
Inferring causal relations
Input:
Day 1        yes                    no
Day 2        yes                    yes
Day 3        no                     yes
Day 4        yes                    no
...         ...                    ...

given several joint observations.
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?
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.
Rational statistical inference
(Bayes, Laplace)

Posterior         Likelihood      Prior
probability                       probability

p ( d | h) p ( h)
p(h | d ) 
 p(d | h) p(h)
hH
Sum over space
of hypotheses
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.
Theory-Based Bayesian Models
• Rational statistical inference (Bayes):
p ( d | h) p ( h)
p(h | d ) 
 p(d | h) p(h)
hH

• 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.
What is a theory?
• Working definition
– 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
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.
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.
Alternative approaches to
inductive generalization
•   Associative learning
•   Connectionist networks
•   Similarity to examples
•   Toolkit of simple heuristics
•   Constraint satisfaction
•   Analogical mapping
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
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.
Outline
• Morning
– 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)
Coin flipping
Coin flipping

HHTHT
HHHHH
What process produced these sequences?
Bayes’ rule
For data D and a hypothesis H, we have:

P( H ) P( D | H )
P( H | D) 
P( D)

• “Posterior probability”: P( H | D)
• “Prior probability”: P(H )
• “Likelihood”: P( D | H )
The origin of Bayes’ rule
• A simple consequence of using probability
to represent degrees of belief
• For any two random variables:
p ( A & B)  p ( A) p ( B | A)
p( A & B)  p( B) p( A | B)
p( B) p( A | B)  p( A) p( B | A)
p( A) p( B | A)
p( A | B) 
p( B)
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.
Bayes’ rule
For data D and a hypothesis H, we have:

P( H ) P( D | H )
P( H | D) 
P( D)

• “Posterior probability”: P( H | D)
• “Prior probability”: P(H )
• “Likelihood”: P( D | H )
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
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       statistical
models
• Markov model
• Hidden Markov model
• ...
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       generative
models
• Markov model
• Hidden Markov model
• ...
Representing generative models
• Graphical model notation
– Pearl (1988), Jordan (1998)   d1     d2   d3    d4
• Variables are nodes, edges      Fair coin, P(H) = 0.5
indicate dependency
• Directed edges show causal
d1    d2    d3    d4
process of data generation
Markov model

HHTHT
d1 d2 d3 d4 d5
Models with latent structure
p
• Not all nodes in a graphical
model need to be observed
• Some variables reflect latent   d1   d2        d3   d4
structure, used in generating        P(H) = p
D but unobserved
s1    s2       s3   s4

HHTHT                     d1   d2        d3   d4
d1 d2 d3 d4 d5           Hidden Markov model
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
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
Comparing two simple hypotheses
• Contrast simple hypotheses:
– H1: “fair coin”, P(H) = 0.5
– H2:“always heads”, P(H) = 1.0
• Bayes’ rule:
P( H ) P( D | H )
P( H | D) 
P( D)
• With two hypotheses, use odds form
Bayes’ rule in odds form
P(H1|D)        P(D|H1)          P(H1)
=            x
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
Coin flipping

HHTHT
HHHHH
What process produced these sequences?
Comparing two simple hypotheses
P(H1|D)         P(D|H1)         P(H1)
=            x
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
Comparing two simple hypotheses
P(H1|D)         P(D|H1)          P(H1)
=            x
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
Comparing two simple hypotheses
P(H1|D)         P(D|H1)         P(H1)
=            x
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
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?
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
Comparing simple and complex hypotheses
p

d1   d2    d3     d4    vs.   d1   d2       d3   d4
Fair coin, P(H) = 0.5              P(H) = p

• Which provides a better account of the data:
the simple hypothesis of a fair coin, or the
complex hypothesis that P(H) = p?
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
Comparing simple and complex hypotheses
Probability
Comparing simple and complex hypotheses
Probability

HHHHH     p = 1.0
Comparing simple and complex hypotheses
Probability

HHTHT     p = 0.6
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!
Comparing simple and complex hypotheses

P(H1|D)      P(D|H1)      P(H1)
=            x
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:
Comparing simple and complex hypotheses
Probability

Distribution is an average over all values of p
Comparing simple and complex hypotheses
Probability

Distribution is an average over all values of p
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)
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
Comparing infinitely many hypotheses
• Assume data are generated from a model:
p

d1    d2       d3   d4
P(H) = p

• What is the value of p?
– each value of p is a hypothesis H
– requires inference over infinitely many hypotheses
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.”

• P(H) on next flip? Closer to 50% than to 40%.
• Why? Prior knowledge.
Integrating prior knowledge and data
P( H ) P( D | H )
P( H | D) 
P( D)

P(p | D)  P(D | p) P(p)

• 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)
Likelihood and prior
• Likelihood:
P(D | p) = pNH (1-p)NT
– NT: number of tails
• Prior:
P(p)  pFH-1 (1-p)FT-1
?
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...
Likelihood and prior
• Likelihood:
P(D | p) = pNH (1-p)NT
– NT: number of tails
• Prior:
P(p)  pFH-1 (1-p)FT-1 Beta(FH,FT)
– FH: fictitious observations of heads
– FT: fictitious observations of tails
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
Likelihood and prior
• Likelihood:
P(D | p) = pNH (1-p)NT
– NT: number of tails
• Prior:
P(p)  pFH-1 (1-p)FT-1
– FH: fictitious observations of heads
– FT: fictitious observations of tails
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:
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 / (1004+1006) = 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
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%
Origin of prior knowledge
• 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
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
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.”
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 / (1025+1000) = 50.6%.
Looks like nothing unusual
Hierarchical priors

• Higher-order hypothesis: is this
coin fair or unfair?
fair
• Example probabilities:
– P(fair) = 0.99
p
– P(p|fair) is Beta(1000,1000)
– P(p|unfair) is Beta(1,1)
d1   d2          d3   d4
• 25 heads in a row propagates up,
affecting p and then P(fair|D)
=
More hierarchical priors

• Latent structure can capture coin variability
p ~ Beta(FH,FT)
FH,FT

Coin 1        p             Coin 2     p        ...             p Coin 200

d1       d2       d3   d4   d1   d2        d3   d4    d1   d2     d3    d4

• 10 flips from 200 coins is better than 2000 flips
from a single coin: allows estimation of FH, FT
Yet more hierarchical priors
physical knowledge

FH,FT

p                       p                       p

d1   d2       d3   d4   d1   d2       d3   d4   d1   d2       d3   d4

• Discrete beliefs (e.g. symmetry) can influence
estimation of continuous properties (e.g. FH, FT)
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
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
Psychology: Representativeness
Which sequence is more likely from a fair coin?

HHTHT                  more representative
of a fair coin
(Kahneman & Tversky, 1972)

HHHHH
What might representativeness mean?

Evidence for a random generating process
P(H1|D)       P(D|H1)              P(H1)
=                    x
P(H2|D)       P(D|H2)              P(H2)
likelihood ratio

H1: random process (fair coin)
H2: alternative processes
A constrained hypothesis space
Four hypotheses:

h1       fair coin             HHTHTTTH
h2       “always alternates”   HTHTHTHT
Representativeness judgments
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)
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.
Summary
• Three kinds of Bayesian inference
– 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)
Outline
• Morning
– 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)
Rules and similarity
Structure versus statistics
Rules                               Statistics
Logic                               Similarity
Symbols                             Typicality
A better metaphor
A better metaphor
Structure and statistics

Statistics
Similarity
Typicality

Rules
Logic
Symbols
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.
The number game

• Program input: number between 1 and 100
• Program output: “yes” or “no”
The number game

– Observe one or more positive (“yes”) examples.
– Judge whether other numbers are “yes” or “no”.
The number game
Examples of      Generalization
“yes” numbers    judgments (N = 20)

60
Diffuse similarity
The number game
Examples of      Generalization
“yes” numbers    judgments (n = 20)

60
Diffuse similarity

60 80 10 30                           Rule:
“multiples of 10”
The number game
Examples of      Generalization
“yes” numbers    judgments (N = 20)

60
Diffuse similarity

60 80 10 30                           Rule:
“multiples of 10”

60 52 57 55                           Focused similarity:
numbers near 50-60
The number game
Examples of      Generalization
“yes” numbers    judgments (N = 20)

16
Diffuse similarity

16 8 2 64                             Rule:
“powers of 2”

16 23 19 20                           Focused similarity:
numbers near 20
The number game
60
Diffuse similarity

60 80 10 30                             Rule:
“multiples of 10”

60 52 57 55                             Focused similarity:
numbers near 50-60

Main phenomena to explain:
– Generalization can appear either similarity-
– Learning from just a few positive examples.
Rule/similarity hybrid models
• Category learning
– Nosofsky, Palmeri et al.: RULEX
– Erickson & Kruschke: ATRIUM
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.
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?
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)
Inferring hypotheses from
similarity judgment
Additive clustering (Shepard & Arabie, 1977):
sij   wk fik f jk
k
sij : similarity of stimuli i, j
wk : weight of cluster k
f ik : 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).
Additive clustering for the integers 0-9:

sij   wk fik f jk
k

Rank   Weight   Stimuli in cluster                     Interpretation
0 1 2 3 4 5 6 7 8 9
1      .444            *       *               *       powers of two
2      .345     * * *                                  small numbers
3      .331                *           *           *   multiples of three
4      .291                            * * * *         large numbers
5      .255            * * * * *                       middle numbers
6      .216        *       *       *       *       *   odd numbers
7      .214        * * * *                             smallish numbers
8      .172                    * * * * *               largish numbers
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, …)
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
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 ( X | h) p ( h)
p(h | X ) 
p( X )

– 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.
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 ( X | h) p ( h)
p(h | X ) 
 p( X | h) p(h)
hH
– 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.
Likelihood: p(X|h)
• Size principle: Smaller hypotheses receive greater
likelihood, and exponentially more so as n increases.
n
 1 
p ( X | h)            if x1 ,  , xn  h
 size(h) 
 0 if any xi  h

• Follows from assumption of randomly sampled examples.
• Captures the intuition of a representative sample.
Illustrating the size principle
h1   2    4    6    8 10     h2
12   14   16   18 20
22   24   26   28 30
32   34   36   38 40
42   44   46   48 50
52   54   56   58 60
62   64   66   68 70
72   74   76   78 80
82   84   86   88 90
92   94   96   98 100
Illustrating the size principle
h1       2    4    6    8 10         h2
12   14   16   18 20
22   24   26   28 30
32   34   36   38 40
42   44   46   48 50
52   54   56   58 60
62   64   66   68 70
72   74   76   78 80
82   84   86   88 90
92   94   96   98 100

Data slightly more of a coincidence under h1
Illustrating the size principle
h1       2    4    6    8 10       h2
12   14   16   18 20
22   24   26   28 30
32   34   36   38 40
42   44   46   48 50
52   54   56   58 60
62   64   66   68 70
72   74   76   78 80
82   84   86   88 90
92   94   96   98 100

Data much more of a coincidence under h1
Bayesian Occam’s Razor
Law of
M1         “Conservation
of Belief”
p(D = d | M )

M2

All possible data sets d

For any model M,                     p(D  d | M )  1
all d D
Comparing simple and complex hypotheses
Probability

Distribution is an average over all values of p
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”.
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(H3) = 1/5
p(H2) = 3/5

H1: Math properties (24)   H2: Raw magnitude (5050)    H3: Approx. magnitude (10)
• even numbers             • 10-15                     • 10-20
• powers of two            • 20-32                     • 20-30
• multiples of three       • 37-54                     • 30-40
…. p(h) = p(H1) / 24       …. p(h) = p(H2) / 5050      …. p(h) = p(H3) / 10
A more complex approach to priors
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
p ( X | h) p ( h)
Posterior:   p(h | X ) 
 p( X | h) p(h)
hH

• 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)
Bayesian Occam’s Razor
Probabilities provide a common currency for
balancing model complexity with fit to the data.
Generalizing to new objects
Given p(h|X), how do we compute p( y  C | X ) ,  p( y 

the probability that C applies to some new       hH
stimulus y?
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):

p( y  C | X )      p(h) p(h | X )

y C |

hH       1 if yh

 0 if yh

                  p(h | X )
h { y , X }
Examples:
16
Connection to feature-based
similarity
• Additive clustering model of similarity:
sij   wk fik f jk
k

• Bayesian hypothesis averaging:
p( y  C | X )   y  C ( h ) |X (hp|(X )
 p(     p | X| p ) h )
h
h{ y , X }
hH

• Equivalent if we identify features fk with
hypotheses h, and weights wk with p(h | X ) .
Examples:
16
8
2
64
Examples:
16
23
19
20
Model fits
Examples of     Generalization       Bayesian Model
“yes” numbers   judgments (N = 20)   (r = 0.96)

60

60 80 10 30

60 52 57 55
Model fits
Examples of     Generalization       Bayesian Model
“yes” numbers   judgments (N = 20)   (r = 0.93)

16

16 8 2 64

16 23 19 20
Summary of the Bayesian model
• How do the statistics of the examples interact with
prior knowledge to guide generalization?

posterior likelihood prior

• Why does generalization appear rule-based or
similarity-based?
hypothesisaveraging size principle

narrow p(h|X): all-or-none rule
Summary of the Bayesian model
• How do the statistics of the examples interact with
prior knowledge to guide generalization?

posterior likelihood prior

• Why does generalization appear rule-based or
similarity-based?
hypothesisaveraging size principle

Many h of similar size: broad p(h|X)
One h much smaller: narrow p(h|X)
Alternative models
• Neural networks

even   multiple   multiple   power
of 10      of 3       of 2
60
80
10
30
Alternative models
• Neural networks
• Hypothesis ranking and elimination
Hypothesis
ranking:      1        2          3        4      ….
even   multiple   multiple   power   ….
of 10      of 3       of 2
60
80
10
30
Alternative models
• Neural networks
• Hypothesis ranking and elimination
• Similarity to exemplars
1
– Average similarity: p( y  C | X )          sim( y, x j )
| X | x X
j

60

60 80 10 30

60 52 57 55

Data                Model (r = 0.80)
Alternative models
• Neural networks
• Hypothesis ranking and elimination
• Similarity to exemplars
– Max similarity: p( y  C | X )  max sim( y, x j )
x j X
60

60 80 10 30

60 52 57 55

Data              Model (r = 0.64)
Alternative models
• Neural networks
• Hypothesis ranking and elimination
• Similarity to exemplars
– Average similarity
– Max similarity
– Flexible similarity? Bayes.
Alternative models
•    Neural networks
•    Hypothesis ranking and elimination
•    Similarity to exemplars
•    Toolbox of simple heuristics
– 60: “general” similarity
– 60 80 10 30: most specific rule (“subset principle”).
– 60 52 57 55: similarity in magnitude
Why these heuristics? When to use which heuristic?
Bayes.
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
– Rules and statistics   – Rules and representativeness
– Rules and similarity
• Explains why central but notoriously slippery
processing-level concepts work the way they do.
– Similarity
– Representativeness
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.
Theory-Based Bayesian Models
• Rational statistical inference (Bayes):
p ( d | h) p ( h)
p(h | d ) 
 p(d | h) p(h)
hH

• 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.
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?
Outline
• Morning
– 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)
Outline
• Morning
– 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)
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
Working at the computational level
statistical
• What is the computational problem?
– input: data
– output: solution
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?
Theory-Based Bayesian Models
• Rational statistical inference (Bayes):
p ( d | h) p ( h)
p(h | d ) 
 p(d | h) p(h)
hH

• 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.
Causality
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?
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
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
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
Undirected graphical models
X3               X4
X1
• Consist of
– a set of nodes             X2            X5
– 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)
Directed graphical models
X3           X4
X1
• Consist of
– a set of nodes              X2           X5
– 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
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
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
Efficient representation and inference
• Three binary variables: Cavity, Toothache, Catch
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, . . . .
Conditional independence
• All three variables are dependent, but Toothache
and Catch are independent given the presence or
absence of Cavity
• In probabilistic terms:
P(ache  catch | cav)  P(ache | cav) P(catch | cav)
P(ache  catch | cav)  P(ache | cav) P(catch | cav)
 1  P(ache | cav)P(catch | cav)
• With n evidence variables, x1, …, xn, we need 2 n
conditional probabilities: P( xi | cav), P( xi | cav)
A simple Bayesian network
• Graphical representation of relations between a set of
random variables:
Cavity

Toothache          Catch

• Probabilistic interpretation: factorizing complex terms
P( A, B, C )              P(V | parents[V ])
V { A, B,C}

P( Ache, Catch, Cav)  P( Ache, Catch | Cav) P(Cav)
 P( Ache | Cav) P(Catch | Cav) P(Cav)
A more complex system
Battery

Starts

On time to work

• Joint distribution sufficient for any inference:
P( B, R, I , G, S , O)  P( B) P( R | B) P( I | B) P(G) P(S | I , G) P(O | S )

 P( B) P( R | B) P( I | B) P(G) P(S | I , G) P(O | S )
P(O, G ) B, R, I , S
P(O | G )          
P(G )                                   P(G )
A more complex system
Battery

Starts

On time to work

• Joint distribution sufficient for any inference:
P( B, R, I , G, S , O)  P( B) P( R | B) P( I | B) P(G) P(S | I , G) P(O | S )

                                  
    P( B) P( I | B) P( S | I , G )  P(O | S )
P(O, G )
P(O | G ) 
P(G )                                         
S  B, I                             
A more complex system
Battery

Starts

On time to work

• Joint distribution sufficient for any inference:
P( B, R, I , G, S , O)  P( B) P( R | B) P( I | B) P(G) P(S | I , G) P(O | S )
• General inference algorithm: local message passing
(belief propagation; Pearl, 1988)
– efficiency depends on sparseness of graph structure
Explaining away
Rain               Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

• Assume grass will be wet if and only if it rained last
night, or if the sprinklers were left on:
P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.
Explaining away
Rain               Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.

Compute probability it                         P( w | r ) P(r )
P (r | w) 
rained last night, given                           P ( w)
that the grass is wet:
Explaining away
Rain                Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.

Compute probability it                        P( w | r ) P(r )
P(r | w) 
rained last night, given
that the grass is wet:
 P(w | r, s) P(r, s)
r , s 
Explaining away
Rain                Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.

Compute probability it                                   P(r )
P(r | w) 
rained last night, given                   P(r , s )  P(r , s )  P(r , s)
that the grass is wet:
Explaining away
Rain                Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.

Compute probability it                           P(r )
P(r | w) 
rained last night, given                   P(r )  P(r , s)
that the grass is wet:
Explaining away
Rain                Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.

Compute probability it                               P(r )
P(r | w)                                  P(r )
rained last night, given                   P ( r )  P ( r ) P ( s )
that the grass is wet:
Between 1 and P(s)
Explaining away
Rain                  Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.

Compute probability it                       P( w | r , s) P(r | s)
P(r | w, s ) 
rained last night, given                          P( w | s)
that the grass is wet and
sprinklers were left on:                       Both terms = 1
Explaining away
Rain               Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.

Compute probability it
rained last night, given    P(r | w, s)  P(r | s)  P(r )
that the grass is wet and
sprinklers were left on:
Explaining away
Rain                  Sprinkler

Grass Wet

P( R, S ,W )  P( R) P(S ) P(W | S , R)

P(W  w | S , R)  1 if S  s or R  r
 0 if R  r and S  s.

P(r )
P(r | w)                                 P(r )
P ( r )  P ( r ) P ( s )
“Discounting” to
P(r | w, s)  P(r | s)  P(r )                 prior probability.
Contrast w/ production system
Rain               Sprinkler

Grass Wet

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

Grass Wet

• Excitatory links: Rain       Wet, Sprinkler   Wet
• Observing grass wet, Rain and Sprinkler become
more active.
• Observing grass wet and sprinkler, Rain becomes
less active: explaining away.
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.
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
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
Causal graphical models
• Graphical models represent statistical
dependencies among variables (ie. correlations)
• Causal graphical models represent causal
dependencies among variables
– express underlying causal structure
interventions (actions upon a variable)
Observation and intervention
Battery

Starts

On time to work

Observation and intervention
Battery

Starts

On time to work

“graph surgery” produces “mutilated graph”
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
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”:

Ache            Catch

Cavity
• Does not capture the correlation between symptoms:
falsely believe P(Ache, Catch) = P(Ache) P(Catch).
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”:

Ache              Catch

Cavity
• Inserting a new arrow allows us to capture this
correlation.
• This model is too complex: do not believe that
P( Ache, Catch | Cav)  P( Ache | Cav) P(Catch | Cav)
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”:

Ache                     X-ray
Catch

Cavity

• New symptoms require a combinatorial proliferation
of new arrows. This reduces efficiency of inference.
Learning causal graphical models

B       C           B       C

E                   E

• Strength: how strong is a relationship?
• Structure: does a relationship exist?
Causal structure vs. causal strength

B       C           B       C

E                   E

• Strength: how strong is a relationship?
Causal structure vs. causal strength

B        C             B         C
w0       w1             w0
E                       E

• Strength: how strong is a relationship?
– requires defining nature of relationship
Parameterization
• Structures: h1 =      B           C     h0 =        B       C

E                         E

• Parameterization:      Generic
C B      h1: P(E = 1 | C, B)           h0: P(E = 1| C, B)
0   0           p00                             p0
1   0           p10                             p0
0   1           p01                             p1
1   1           p11                             p1
Parameterization
• Structures: h1 =      B           C      h0 =        B        C
w0          w1                 w0
E                           E
w0, w1: strength parameters for B, C

• Parameterization:      Linear
C B      h1: P(E = 1 | C, B)            h0: P(E = 1| C, B)
0   0           0                                0
1   0           w1                               0
0   1           w0                               w0
1   1         w1+ w0                             w0
Parameterization
• Structures: h1 =      B           C      h0 =        B        C
w0          w1                 w0
E                           E
w0, w1: strength parameters for B, C

• Parameterization:      “Noisy-OR”
C B      h1: P(E = 1 | C, B)            h0: P(E = 1| C, B)
0   0            0                               0
1   0           w1                               0
0   1           w0                               w0
1   1      w1+ w0 – w1 w0                        w0
Parameter estimation

• Maximum likelihood estimation:

maximize i P(bi,ci,ei; w0, w1)

• Bayesian methods: as in the “Comparing
infinitely many hypotheses” example…
Causal structure vs. causal strength

B       C           B       C

E                   E

• Structure: does a relationship exist?
Approaches to structure learning
• Constraint-based                               B         C
– dependency from statistical tests (eg. 2)
– deduce structure from dependencies                E
(Pearl, 2000; Spirtes et al., 1993)
Approaches to structure learning
• Constraint-based:                              B         C
– dependency from statistical tests (eg. 2)
– deduce structure from dependencies                E
(Pearl, 2000; Spirtes et al., 1993)
Approaches to structure learning
• Constraint-based:                              B         C
– dependency from statistical tests (eg. 2)
– deduce structure from dependencies                E
(Pearl, 2000; Spirtes et al., 1993)
Approaches to structure learning
• Constraint-based:                              B         C
– dependency from statistical tests (eg. 2)
– deduce structure from dependencies                E
(Pearl, 2000; Spirtes et al., 1993)

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

• Bayesian:                          B         C       B         C
– compute posterior
probability of structures,
E                 E
given observed data
P(S1|data)         P(S0|data)
P(S|data)  P(data|S) P(S)        (Heckerman, 1998; Friedman, 1999)
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
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
Elemental causal induction

C present   C absent

E present       a          c

E absent        b          d

“To what extent does C cause E?”
Causal structure vs. causal strength

B        C          B        C
w0       w1         w0
E                   E

• Strength: how strong is a relationship?
• Structure: does a relationship exist?
Causal strength
• Assume structure:        B        C
w0       w1
E

• 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
Causal structure
• Hypotheses: h1 =       B         C        h0 =    B         C

E                          E

• Bayesian causal inference:
1 1
P(data | h1)     0 P(data | w0 , w1) p(w0 , w1 | h1
support =                    0
1
P(data | h0 )   P(data | w0 ) p(w0 | h0 ) dw0
0
Buehner and Cheng (1997)

People

DP (r = 0.89)

Power (r = 0.88)

Support (r = 0.97)
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)
People

Support (Noisy-OR)

2

Support (generic)
Generativity is essential
P(e+|c+)      8/8      6/8      4/8      2/8      0/8
P(e+|c-)      8/8      6/8      4/8      2/8      0/8
100                         Support
50
0

• Predictions result from “ceiling effect”
– ceiling effects only matter if you believe a cause
increases the probability of an effect
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
chemicals
genes

Clofibrate   Wyeth 14,643    Gemfibrozil      Phenobarbital

p450 2B1          Carnitine Palmitoyl Transferase 1

Hamadeh et al. (2002) Toxicological sciences.
chemicals
X                     genes

Clofibrate   Wyeth 14,643       Gemfibrozil   Phenobarbital

p450 2B1          Carnitine Palmitoyl Transferase 1

Hamadeh et al. (2002) Toxicological sciences.
chemicals
Chemical X                   genes

peroxisome proliferators

Clofibrate     Wyeth 14,643      Gemfibrozil      Phenobarbital

+          +        +

p450 2B1              Carnitine Palmitoyl Transferase 1

Hamadeh et al. (2002) Toxicological sciences.
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?
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
Theory-based causal induction
Causal theory
– Ontology                  P(h|data)  P(data|h) P(h)
– Plausible relations       Evaluated by statistical inference
– Functional form
P(h1) =                 P(h0) =1 – 
h1:      B
X         Y        h0:   X
B         Y

Generates                        Z                        Z

Hypothesis space of causal graphical models
Blicket detector
Both objects activate     Object A does not       Chi
the detector        activate the detector   each
by itself        Then
they
maket

(Gopnik, Sobel, and colleagues)
Procedure used in Sobel et al. (2002), Experiment 2
Backward Blocking Condition
e Condition

Both objects activate   Object Aactivates the     Chi

See this? It’s a                       Let’s put this one                          Oooh, it’s a
s activate          Object A does not          Children are asked if   the detector         detector by itself     each
tector             activate the detector      each is a blicket                                                    Then
they
by itself           Then
blicket machine.                       on the machine.
makehe machine go
t                                       blicket!
Blickets
Blocking Condition make it go.

s activate        Object Aactivates the        Children are asked if
tector              detector by itself        each is a blicket
Then
Both objects activate                Object A does not                 Children are asked if     Both objects activate
the detector                   activate the detector             each is a blicket               the detector
by itself                  Then
makehe machine go

“Blocking”
t
Experiment Sobel et al. (2002), Experiment 2
edure used in2
Backward Blocking Condition                                                                       Backward Blocking C
dition

e         A
B
activate
Children areObject if does not
detector A
theblicket
Trial 1
Object Aactivates the
by itself                          Trial 2
each is a blicket
Trials 3, 4
Both objects activate
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each is a activate the detector              each is a blicket
Then                                                                   Then
they are asked to by itself                  Then
makehe machine go                                                      makehe machine go
t
t                                        makehe machine go
t
– Two objects: A and B
king Condition
– Trial 1: A on detector – detector active
– Trial 2: B on detector – detector inactive
– Trials 3,4: A B on detector – detector active
e
Then
– 3, 4-year-olds judge whether each object is a blicket
Object Aactivates the
each is a blicket
detector by itself
each is a blicket
makehe machine go
t
• A: a blicket             makehe machine go
t

• B: not a blicket
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
Both objects activate                Object A does not                Children are a

“Backwards blocking”
the detector                   activate the detector            each is a blicke
by itself                 Then
makehe machin
t
el et al. (2002), Experiment 2
Figure 13: Procedure used in Sobel et al. (2002), Experiment 2
(Sobel, Tenenbaum & Gopnik, 2004)
Backward Blocking Condition
One-Cause Condition

t A does not                A                B
Children are asked if objects activate
Both                    Trial 1
Both objects activate
Object A does not
the detector
Trial 2
Object Aactivates the
detector by itself
Children are a
each is a blicke
e the detector          each is a blicket      the detector                  activate the detector            each is a blicket
by itself               Then
by itself                 Then
makehe machine go
t                                                                                                          makehe machin
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makehe machine go
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–                   A and B
Two objects:Blocking Condition
Backward
–   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
• each isablicket Both the detector
Aactivates the           Children are asked if
tor by itself           A: a blicket (100% of judgments)
Then
objects activate Object Aactivates the
detector by itself
each is a blicket
• B: probably not a blicket (66% of judgments)
makehe machine go
t                                                       makehe machine go
t
Theory
• Ontology
– 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.
Theory
• Ontology
– Types: Block, Detector, Trial
– Predicates:
A       B
Contact(Block, Detector, Trial)
Active(Detector, Trial)
E
Theory
• Ontology
– Types: Block, Detector, Trial
– Predicates:
A       B
Contact(Block, Detector, Trial)
Active(Detector, Trial)
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
Theory
• 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)

No hypotheses with          h00 :      A       B         h10 :   A        B
E    B, E A,
A    B, etc.                               E                          E

P(h01) = (1 – q) q           P(h11) = q2
A
= “A is a blicket”                 A       B                  A       B
h01 :                        h11 :
E
E                          E
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):       0             0                    0                  0
P(E=1 | A=1, B=0):       0             0                    1                  1
P(E=1 | A=0, B=1):       0             1                    0                  1
P(E=1 | A=1, B=1):       0             1                    1                  1

“Activation law”: E=1 if and only if A=1 or B=1.
Bayesian inference
• Evaluating causal models in light of data:
P(d | hi ) P(hi )
P(hi | d ) 
 P(d | h j ) P(h j )
h H
j
• Inferring a particular causal relation:

P( A  E | d )     P( A  E | h j ) P ( h j | d )
h H
j
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):       0             0                    0                  0
P(E=1 | A=1, B=0):       0             0                    1                  1
P(E=1 | A=0, B=1):       0             1                    0                  1
P(E=1 | A=1, B=1):       0             1                    1                  1

P ( B  E | d ) P(h01 )  P(h11 )    q
                  
P( B    E | d ) P (h00 )  P(h10 ) 1  q
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):       0             1                    1                  1

P ( B  E | d ) P (h01 )  P(h11 )    1
                   
P( B    E | d)       P (h10 )        1 q
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):         0                    1                  1

P(E=1 | A=1, B=1):         1                    1                  1

P( B  E | d ) P(h11 )   q
        
P( B   E | d ) P(h10 ) 1  q
Both objects activate                         Object A does not                          Children are asked if
the detector

et al. 13:
Manipulating the prior
One-Cause
Figure 13: Condition (2002), Experiment 2
esed in Sobel et al. (2002), Experiment 2Procedure used 2Procedure al.13: in Sobel et used in Sobel et al. (2002),
13: Procedure used in SobelFigure(2002), Experiment in Sobel et used Procedure al. (2002), Experiment 2
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Figure 13: Procedure used in Sobel et al. (2002), Experiment 2
activate the detector
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each is a blicket
Then
makehe machine go
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Backward Blocking Condition One-Cause Condition
Cause Condition           One-Cause Condition       One-Cause Condition
I. Pre-training phase: Blickets are rare . . . .

Both objects activate                       Object A does not                          Children are a
the detector                          activate the detector                      each is a blicke
by itself                           Then
Figure 13: Procedure used in Sobel et al. (2002), Experiment 2
Object      Both objects activate A does notobjects activateAactivates the areactivate A does not are asked if A does not
h objects activate A does not            Object       Both
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II. Backwards blocking phase: Backward Blocking Condition
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t                     makehe machine go
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ward Blocking Condition
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Backward Blocking Condition Backward Blocking Condition

Both objects activate                        Object A does not                           Children are asked if
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• “Rare” condition: First observe 12 objects
on detector, of which 2 set it off.
• “Common” condition: First observe 12
objects on detector, of which 10 set it off.
Both objects activate                         Object A does not                          Children are asked if
Figure 13: Procedure usedeach is a blicketFigure 13: Procedure used in Sobel
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Backward Blocking Condition One-Cause Condition
Cause Condition           One-Cause Condition       One-Cause Condition
I. Pre-training phase: Blickets are rare . . . .

Both objects activate                       Object A doesBoth objects activate
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object isBlocking Condition
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Same domain theory generates hypothesis
space for 3 objects:
A   B   C            A    B      C

• Hypotheses:       h000 =       E       h100 =        E
A   B   C            A    B      C
h010 =       E
h001 =        E
A   B   C            A    B      C
h110 =               h011 =
E                     E
A   B   C            A    B      C
h101 =               h111 =
E                     E
• Likelihoods: 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.
• “Rare” condition: First observe 12 objects
on detector, of which 2 set it off.
The role of causal mechanism
knowledge
• Is mechanism knowledge necessary?
– Constraint-based learning using 2 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).
Bayes with correct theory:

Bayes with “noisy sufficient causes” theory:
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
• Extends to more complex cases with hidden
variables, dynamic systems: come to my talk!
Summary
• Causal graphical models provide a language
• Key issues in modeling causal induction:
– what do we mean by causal induction?
– how do knowledge and statistics interact?
• Bayesian approach allows exploration of
Outline
• Morning
– 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)
Property induction
Collaborators
Charles Kemp       Neville Sanjana
Lauren Schmidt     Amy Perfors
Fei Xu             Liz Baraff
Pat Shafto
The Big Question
• How can we generalize new concepts
reliably from just one or a few examples?
– Learning word meanings

“horse”        “horse”        “horse”
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
Gorillas have T4 cells.
Squirrels have T4 cells.
All mammals have T4 cells.

How probable is the the conclusion (target) given
the premises (examples)?
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
Gorillas have T4 cells.         Gorillas have T4 cells.
Squirrels have T4 cells.        Chimps have T4 cells.
All mammals have T4 cells.      All mammals have T4 cells.

More diverse examples               stronger generalization
• 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.
Theory-Based Bayesian Models
• Rational statistical inference (Bayes):
p ( d | h) p ( h)
p(h | d ) 
 p(d | h) p(h)
hH

• 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.
The plan
• Similarity-based models
• Theory-based model
• Bayesian models
– “Empiricist” Bayes
– Theory-based Bayes, with different theories
• Connectionist (PDP) models
– Learning with multiple domain theories
– Learning domain theories
The plan
• Similarity-based models
• Theory-based model
• Bayesian models
– “Empiricist” Bayes
– Theory-based Bayes, with different theories
• Connectionist (PDP) models
– Learning with multiple domain theories
– Learning domain theories
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.
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )           sim(i, X )
imammals

x
x

x

Mammals:
Examples:    x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )           sim(i, X )
imammals

x
x

x

Mammals:
Examples:    x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )           sim(i, X )
imammals

x
x

x

Mammals:
Examples:    x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )           sim(i, X )
imammals

x
x

x

Mammals:
Examples:    x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )             sim(i, X )
imammals

• Sum-Similarity:
sim(i, X )         sim(i, j )          x
x

jX
x

Mammals:
Examples:      x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )            sim(i, X )
imammals

• Max-Similarity:
x
sim( i, X )  max sim( i, j )           x
j X
x

Mammals:
Examples:      x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )            sim(i, X )
imammals

• Max-Similarity:
x
sim( i, X )  max sim( i, j )           x
j X
x

Mammals:
Examples:      x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )            sim(i, X )
imammals

• Max-Similarity:
x
sim( i, X )  max sim( i, j )           x
j X
x

Mammals:
Examples:      x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )            sim(i, X )
imammals

• Max-Similarity:
x
sim( i, X )  max sim( i, j )           x
j X
x

Mammals:
Examples:      x
Similarity-based models
(Osherson et al.)

strength(“all mammals” | X )            sim(i, X )
imammals

• Max-Similarity:
x
sim( i, X )  max sim( i, j )           x
j X
x

Mammals:
Examples:      x
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 . . . .
Data vs. models
Data

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

Max-sim

Sum-sim

Conclusion
kind:   “all mammals”   “horses”   “horses”

Number of
examples:          3             2       1, 2, or 3
Feature rating data
(Osherson and Wilkie)

• People were given 48 animals, 85 features,
each feature.
• E.g., elephant: 'gray' 'hairless' 'toughskin'
'big' 'bulbous' 'longleg'
'tail' 'chewteeth' 'tusks'
'smelly' 'walks' 'slow'
'inactive' 'vegetation' 'grazer'
'oldworld' 'bush' 'jungle'
'ground' 'timid' 'smart'
'group'
?

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,  A  B   A  B  or cosine.
• Generalize based on Max-sim or Sum-sim.
Three data sets
r = 0.77       r = 0.75    r = 0.94

Max-Sim

r = – 0.21      r = 0.63    r = 0.19

Sum-Sim

Conclusion
kind:   “all mammals”   “horses”    “horses”

Number of
examples:          3             2        1, 2, or 3
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.
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
The plan
• Similarity-based models
• Theory-based model
• Bayesian models
– “Empiricist” Bayes
– Theory-based Bayes, with different theories
• Connectionist (PDP) models
– Learning with multiple domain theories
– Learning domain theories
Theory-based induction
• Scientific biology: species generated by an
evolutionary branching process.
– A tree-structured taxonomy of species.

• Taxonomy also central in folkbiology (Atran).
Theory-based induction
Begin by reconstructing intuitive taxonomy
from similarity judgments:

clustering
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.
“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])
“large herbivores”

Cows have property P.                Cows have property P.
Dolphins have property P.            Horses have property P.
Squirrels have property P.           Rhinos have property P.

All mammals have property P.         All mammals have property P.

Strong: 0.76 [max = 0.82])           Weak: 0.17 [min = 0.14]
“all mammals”

Cows have property P.          Seals have property P.
Dolphins have property P.      Dolphins have property P.
Squirrels have property P.     Squirrels have property P.

All mammals have property P.   All mammals have property P.

Strong: 0.76 [max = 0.82]      Weak: 0.30 [min = 0.14]
Taxonomic
distance

Max-sim

Sum-sim

Conclusion
kind:   “all mammals”   “horses”   “horses”

Number of
examples:         3             2       1, 2, or 3
The challenge
• Can we build models with the best of both
– Quantitatively accurate predictions.
– Strong rational basis.

• Will require novel ways of integrating
structured knowledge with statistical inference.
The plan
• Similarity-based models
• Theory-based model
• Bayesian models
– “Empiricist” Bayes
– Theory-based Bayes, with different theories
• Connectionist (PDP) models
– Learning with multiple domain theories
– Learning domain theories
The Bayesian approach
?

Species 1
Species 2                             ?
Species 3                             ?
Species 4                             ?
Species 5                             ?
Species 6                             ?
Species 7                             ?
Species 8                             ?
Species 9
Species 10                            ?

Features       New property
The Bayesian approach
?

Species 1
Species 2                                           ?
Species 3                                           ?
Species 4                                           ?
Species 5                                           ?
Species 6                                           ?
Species 7                                           ?
Species 8                                           ?
Species 9
Species 10                                          ?

Features       Generalization   New property
Hypothesis
The Bayesian approach
?

Species 1
Species 2                                           ?
Species 3                                           ?
Species 4                                           ?
Species 5                                           ?
Species 6                                           ?
Species 7                                           ?
Species 8                                           ?
Species 9
Species 10                                          ?

Features       Generalization   New property
Hypothesis
The Bayesian approach
?

Species 1
Species 2                                           ?
Species 3                                           ?
Species 4                                           ?
Species 5                                           ?
Species 6                                           ?
Species 7                                           ?
Species 8                                           ?
Species 9
Species 10                                          ?

Features       Generalization   New property
Hypothesis
The Bayesian approach
?

Species 1
Species 2                                           ?
Species 3                                           ?
Species 4                                           ?
Species 5                                           ?
Species 6                                           ?
Species 7                                           ?
Species 8                                           ?
Species 9
Species 10                                          ?

Features       Generalization   New property
Hypothesis
The Bayesian approach
?

Species 1
Species 2                                           ?
Species 3                                           ?
Species 4                                           ?
Species 5                                           ?
Species 6                                           ?
Species 7                                           ?
Species 8                                           ?
Species 9
Species 10                                          ?

Features       Generalization   New property
Hypothesis
The Bayesian approach
?

Species 1
Species 2                                           ?
Species 3                                           ?
Species 4                                           ?
Species 5                                           ?
Species 6                                           ?
Species 7                                           ?
Species 8                                           ?
Species 9
Species 10                                          ?

Features       Generalization   New property
Hypothesis
The Bayesian approach
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   New property
Hypothesis
p ( d | h) p ( h)
Bayes’ rule:    p(h | d ) 
 p(d | h) p(h)
hH

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   New property
Hypothesis
Probability that property Q
holds for species x: p(Q( x) | d )                   p(h | d )
h consistent
with Q ( 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    New property
Hypothesis
1   if d is
“Size principle”:          p ( d | h) 
h   consistent
|h| = # of positive                        with h
instances of h
0     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   New property
Hypothesis
The size principle
h1      2    4    6    8 10     h2
“even numbers”     12   14   16   18 20    “multiples of 10”
22   24   26   28 30
32   34   36   38 40
42   44   46   48 50
52   54   56   58 60
62   64   66   68 70
72   74   76   78 80
82   84   86   88 90
92   94   96   98 100
The size principle
h1      2    4    6    8 10     h2
“even numbers”     12   14   16   18 20    “multiples of 10”
22   24   26   28 30
32   34   36   38 40
42   44   46   48 50
52   54   56   58 60
62   64   66   68 70
72   74   76   78 80
82   84   86   88 90
92   94   96   98 100

Data slightly more of a coincidence under h1
The size principle
h1      2    4    6    8 10     h2
“even numbers”     12   14   16   18 20    “multiples of 10”
22   24   26   28 30
32   34   36   38 40
42   44   46   48 50
52   54   56   58 60
62   64   66   68 70
72   74   76   78 80
82   84   86   88 90
92   94   96   98 100

Data much more of a coincidence under h1
Illustrating the size principle
Which argument is stronger?

Grizzly bears have property P.

All mammals have property P.

“Non-monotonicity”
Grizzly bears have property P.
Brown bears have property P.
Polar bears have property P.

All mammals have property P.
Probability that property Q holds
for species x: p(Q( x) | d )   p(h) / h                              p ( h) / h
h consistent              h consistent
with Q ( x ), d              with 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          New property
Hypotheses
Probability that property Q holds
for species x: p(Q( x) | d )   p(h) / h                               p ( h) / h
h consistent                 h consistent
with Q ( x ), d                 with d

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    New property
Hypothesis
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.
“Empiricist”
h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12
Species 1
Bayes:         Species 2
Species 3
(Heit, 1998)   Species 4
Species 5
Species 6
Species 7
Species 8
Species 9
Species 10

p(h) =        1
15
1 2 1 3
15 15 15 15
1 1 1
15 15 15
1 1
15 15
1
15
1
15

h                d
Species 1
Species 2                                                                    ?
Species 3                                                                    ?
Species 4                                                                    ?
Species 5                                                                    ?
Species 6                                                                    ?
Species 7                                                                    ?
Species 8                                                                    ?
Species 9
Species 10                                                                   ?

Features                          Generalization        New property
Hypothesis
Results
r = 0.38     r = 0.16   r = 0.79

“Empiricist”
Bayes

r = 0.77     r = 0.75   r = 0.94

Max-Sim
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=5           N = 100         N = 500
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
to better generalization
from sparse data.
N=5           N=5
“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.
Mutation process generates p(h|T):
– Choose label for root.
– Probability that label mutates
along branch b : 1  e 2 l b
s1 s2 s3 s4 s5 s6 s7 s8 s9 s10
2
l = mutation rate
T       p(h|T)
|b| = length of branch b
h               d
Species 1
Species 2                                                         ?
Species 3                                                         ?
Species 4                                                         ?
Species 5                                                         ?
Species 6                                                         ?
Species 7                                                         ?
Species 8                                                         ?
Species 9
Species 10                                                        ?

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

Features           Generalization   New property
Hypothesis
Samples from the prior
• Labelings that cut the data along fewer
branches are more probable:

>

“monophyletic”            “polyphyletic”
Samples from the prior
• Labelings that cut the data along longer
branches are more probable:

>

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

Features               Generalization   New property
Hypothesis
Results
r = 0.91     r = 0.95   r = 0.91

“Theory-based”
Bayes

r = 0.38     r = 0.16   r = 0.79

“Empiricist”
Bayes

r = 0.77     r = 0.75   r = 0.94

Max-Sim
Grounding in similarity
Reconstruct intuitive taxonomy from
similarity judgments:

clustering
Theory-based
Bayes

Max-sim

Sum-sim

Conclusion
kind:   “all mammals”   “horses”   “horses”

Number of
examples:          3             2       1, 2, or 3
Explaining similarity
• Why does Max-sim fit so well?
– An efficient and accurate approximation to
this Theory-Based Bayesian model.
– Correlation with
Mean r = 0.94
Bayes on three-
premise general
arguments, over 100
simulated trees:
Correlation (r)

– Theorem. Nearest neighbor classification
approximates evolutionary Bayes in the limit of
high mutation rate, if domain is tree-structured.
Alternative feature-based models
• Taxonomic Bayes (strictly taxonomic
hypotheses, with no mutation process)

>

“monophyletic”           “polyphyletic”
Alternative feature-based models
• Taxonomic Bayes (strictly taxonomic
hypotheses, with no mutation process)
• PDP network (Rogers and McClelland)

Species

Features
Results
r = 0.91     r = 0.95   r = 0.91
Bias is
Theory-based                                        just
Bayes                                           right!

r = 0.51     r = 0.53   r = 0.85
Bias is
Taxonomic                                          too
Bayes                                           strong

r = 0.41     r = 0.62   r = 0.71
Bias is
PDP network                                        too
weak
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?
Mutation process generates p(h|T):
– Choose label for root.
– Probability that label mutates
along branch b : 1  e 2 l b
s1 s2 s3 s4 s5 s6 s7 s8 s9 s10
2
l = mutation rate
T       p(h|T)
|b| = length of branch b
h               d
Species 1
Species 2                                                         ?
Species 3                                                         ?
Species 4                                                         ?
Species 5                                                         ?
Species 6                                                         ?
Species 7                                                         ?
Species 8                                                         ?
Species 9
Species 10                                                        ?

Features              Generalization   New property
Hypothesis
Mutation process generates p(h|T):
– Choose label for root.
– Probability that label mutates
along branch b :
l                s1 s2 s3 s4 s5 s6 s7 s8 s9 s10

l = mutation rate
T       p(h|T)
|b| = length of branch b
h               d
Species 1
Species 2                                                         ?
Species 3                                                         ?
Species 4                                                         ?
Species 5                                                         ?
Species 6                                                         ?
Species 7                                                         ?
Species 8                                                         ?
Species 9
Species 10                                                        ?

Features              Generalization   New property
Hypothesis
Bayes
(taxonomy+
Premise typicality effect (Rips,
mutation)
1975; Osherson et al., 1990):

Bayes                       Strong:
(taxonomy+
Horses have property P.
Occam)
All mammals have property P.

Max-sim
Weak:
Seals have property P.
Conclusion                     All mammals have property P.
kind:   “all mammals”

Number of
examples:          1
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.
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
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.
The plan
• Similarity-based models
• Theory-based model
• Bayesian models
– “Empiricist” Bayes
– Theory-based Bayes, with different theories
• Connectionist (PDP) models
– Learning with multiple domain theories
– Learning domain theories
Property type
Generic “essence”

Theory Structure
Taxonomic Tree

Lion
Cheetah
Hyena
Giraffe
Gazelle
Gorilla
Monkey

Lion
Cheetah
Hyena
Giraffe             ...
Gazelle
Gorilla
Monkey
Property type
Generic “essence”       Size-related                  Food-carried

Theory Structure
Taxonomic Tree          Dimensional                   Directed Acyclic
Network
Giraffe
Lion                                                          Giraffe
Cheetah            Lion                      Lion
Hyena             Gorilla
Gazelle
Giraffe            Hyena
Hyena          Cheetah
Gazelle           Gazelle                                     Monkey
Gorilla           Cheetah
Monkey
Monkey                                      Gorilla

Lion
Cheetah
Hyena
Giraffe             ...                  ...                            ...
Gazelle
Gorilla
Monkey
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
One-dimensional predicates

Bayes
(taxonomy+
mutation)

Max-sim

Bayes
(1D model)
Disease    Food web model fits (Shafto et al.)

r = 0.77                  r = 0.82
Property

r = -0.35                 r = -0.05

Mammals                Island
Disease
Taxonomic tree model fits (Shafto et al.)

r = -0.12            r = 0.16
Property

r = 0.81             r = 0.62

Mammals               Island
The plan
• Similarity-based models
• Theory-based model
• Bayesian models
– “Empiricist” Bayes
– Theory-based Bayes, with different theories
• Connectionist (PDP) models
– Learning with multiple domain theories
– Learning domain theories
Theory      • Species organized in taxonomic tree structure
• Feature i generated by mutation process with rate li
p(S|T)
F9
F8
Domain                                    F7 F11                 F14
F13
Structure                           F12
F6
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
Data           Species 7
Species 8
Species 9
Species 10

l10 high ~ weight low
Theory      • Species organized in taxonomic tree structure
• Feature i generated by mutation process with rate li
p(S|T)
F9
F8
Domain                                              F7 F11                       F14
F13
Structure                                     F12
F6
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
Data           Species 7
Species 8
Species 9
Species 10

Species X    ?   ?   ?   ? ?    ?    ?     ?     ?      ?     ?   ?     ?
Theory      • Species organized in taxonomic tree structure
• Feature i generated by mutation process with rate li
p(S|T)
F9
F8
Domain                                    F7 F11                 F14
F13
Structure                           F12
F6
F14   F10
F3      F1      F2      F4   F5
F10        F10
S3 S4 S1 S2 S9 S10 S5 S6 S7 S8

p(D|S)          SX
Species 1
Species 2
Species 3
Species 4
Species 5
Species 6
Data           Species 7
Species 8
Species 9
Species 10

Species X
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).
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.
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: p( D | T )   p( D | S , T ) p(S | T )
S
 max p( D | S , T ) p( S | T )
S
T0:
• No organizational structure
for species.
• Features distributed
independently over species.

F1
F2
F3            F1              F2
F2 F5 F2          F6 F2           F4
F1   F4 F7 F4 F1       F7 F4 F2 F1 F8
F2   F6 F8 F7 F5       F8 F5 F3 F6 F9
F5   F7 F10 F9 F7      F9 F12 F6 F8 F10
F8   F9 F12 F12 F13    F10 F13 F11 F9 F11
F9   F14 F13 F14 F14   F13 F14 F13 F12 F14
S1   S2   S3 S4   S5 S6    S7   S8 S9 S10

Species 1
Species 2
Species 3
Species 4
Data               Species 5
Species 6
Species 7
Species 8
Species 9
Species 10

Features
T0:
• No organizational structure
for species.
• Features distributed
independently over species.

F1 F3 F3
F1    F6 F7 F7                      F2    F2
F6    F7 F8 F8              F5 F5 F6      F6
F7    F8 F9 F9              F9 F9 F7      F7
F8    F9 F11 F11 F4    F4   F10 F10 F8    F8
F9    F10 F12 F12 F8   F8   F13 F13 F9    F9
F11   F11 F14 F14 F9   F9   F14 F14 F11   F11
S1 S2     S3 S4   S5 S6     S7   S8 S9 S10

Species 1
Species 2
Species 3
Species 4
Data               Species 5
Species 6
Species 7
Species 8
Species 9
Species 10

Features
T0:                                                         T1:
• No organizational structure                               • Species organized in
for species.                                                taxonomic tree structure.
• Features distributed                                      • Features distributed via
independently over species.                                 stochastic mutation process.

F9

F1 F3 F3                                                                   F8
F1    F6 F7 F7                      F2    F2                           F7 F11               F14
F6    F7 F8 F8              F5 F5 F6      F6
F7    F8 F9 F9                                                                              F13
F9 F9 F7      F7                                F6
F8    F9 F11 F11 F4    F4   F10 F10 F8    F8                     F12                  F14   F10
F9    F10 F12 F12 F8   F8   F13 F13 F9    F9                     F3      F1      F2   F4    F5
F11   F11 F14 F14 F9   F9   F14 F14 F11   F11                              F10
S1 S2     S3 S4   S5 S6     S7   S8 S9 S10                      S3 S4 S1 S2 S9 S10 S5 S6 S7 S8

Species 1
Species 2
Species 3
Species 4
Data               Species 5
Species 6
Species 7
Species 8
Species 9
Species 10

Features
T0: p(Data|T1) ~ 1.83 x 10-41                               T1: p(Data|T2) ~ 2.42 x 10-32
• No organizational structure                               • Species organized in
for species.                                                taxonomic tree structure.
• Features distributed                                      • Features distributed via
independently over species.                                 stochastic mutation process.

F9

F1 F3 F3                                                                   F8
F1    F6 F7 F7                      F2    F2                           F7 F11               F14
F6    F7 F8 F8              F5 F5 F6      F6
F7    F8 F9 F9                                                                              F13
F9 F9 F7      F7                                F6
F8    F9 F11 F11 F4    F4   F10 F10 F8    F8                     F12                  F14   F10
F9    F10 F12 F12 F8   F8   F13 F13 F9    F9                     F3      F1      F2   F4    F5
F11   F11 F14 F14 F9   F9   F14 F14 F11   F11                              F10
S1 S2     S3 S4   S5 S6     S7   S8 S9 S10                      S3 S4 S1 S2 S9 S10 S5 S6 S7 S8

Species 1
Species 2
Species 3
Species 4
Data               Species 5
Species 6
Species 7
Species 8
Species 9
Species 10

Features
T0:                                                      T1:
• No organizational structure                            • Species organized in
for species.                                             taxonomic tree structure.
• Features distributed                                   • Features distributed via
independently over species.                              stochastic mutation process.

F1                                                               F2
F2                                                         F4                        F1
F5 F7 F13
F3            F1              F2                           F14              F8 F9
F12
F2 F5 F2          F6 F2           F4                     F9
F13 F10
F1   F4 F7 F4 F1       F7 F4 F2 F1 F8                         F7       F11 F13
F2   F6 F8 F7 F5       F8 F5 F3 F6 F9                                                   F10        F8
F13 F10 F11
F5   F7 F10 F9 F7      F9 F12 F6 F8 F10                                              F12 F7        F3
F8   F9 F12 F12 F13    F10 F13 F11 F9 F11                       F12 F12   F9 F6   F5
F9   F14 F13 F14 F14   F13 F14 F13 F12 F14                   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
Data               Species 5
Species 6
Species 7
Species 8
Species 9
Species 10

Features
T0: p(Data|T1) ~ 2.29 x 10-42                            T1: p(Data|T2) ~ 4.38 x 10-53
• No organizational structure                            • Species organized in
for species.                                             taxonomic tree structure.
• Features distributed                                   • Features distributed via
independently over species.                              stochastic mutation process.

F1                                                               F2
F2                                                         F4                        F1
F5 F7 F13
F3            F1              F2                           F14              F8 F9
F12
F2 F5 F2          F6 F2           F4                     F9
F13 F10
F1   F4 F7 F4 F1       F7 F4 F2 F1 F8                         F7       F11 F13
F2   F6 F8 F7 F5       F8 F5 F3 F6 F9                                                   F10        F8
F13 F10 F11
F5   F7 F10 F9 F7      F9 F12 F6 F8 F10                                              F12 F7        F3
F8   F9 F12 F12 F13    F10 F13 F11 F9 F11                       F12 F12   F9 F6   F5
F9   F14 F13 F14 F14   F13 F14 F13 F12 F14                   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
Data               Species 5
Species 6
Species 7
Species 8
Species 9
Species 10

Features
Empirical tests
• Synthetic data: 32 objects, 120 features
– tree-structured generative model
– linear chain generative model
– unconstrained (independent features).
• Real data
– Animal feature judgments: 48 species, 85
features.
– US Supreme Court decisions, 1981-1985: 9
people, 637 cases.
Results   Preferred
Model
Null
Tree
Linear
Tree
Linear
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?
Outline
• Morning
– 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)
Structure and statistics
• Statistical language modeling
– topic models

• Relational categorization
– attributes and relations
Structure and statistics
• Statistical language modeling
– topic models

• Relational categorization
– attributes and relations
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
Generative models for language

latent structure

observed data
Generative models for language

meaning

sentences
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)
Generating a document

q        distribution over topics

z     z    z     topic assignments

w     w    w       observed words
w     P(w|z = 1) = f (1)   w    P(w|z = 2) = f (2)
HEART              0.2     HEART             0.0
LOVE               0.2     LOVE              0.0
SOUL               0.2     SOUL              0.0
TEARS              0.2     TEARS             0.0
JOY                0.2     JOY               0.0
SCIENTIFIC         0.0     SCIENTIFIC        0.2
KNOWLEDGE          0.0     KNOWLEDGE         0.2
WORK               0.0     WORK              0.2
RESEARCH           0.0     RESEARCH          0.2
MATHEMATICS        0.0     MATHEMATICS       0.2
topic 1                     topic 2
Choose mixture weights for each document, generate “bag of words”
q = {P(z = 1), P(z = 2)}
MATHEMATICS KNOWLEDGE RESEARCH WORK MATHEMATICS
{0, 1}                 RESEARCH WORK SCIENTIFIC MATHEMATICS WORK

SCIENTIFIC KNOWLEDGE MATHEMATICS SCIENTIFIC
{0.25, 0.75}                   HEART LOVE TEARS KNOWLEDGE HEART

MATHEMATICS HEART RESEARCH LOVE MATHEMATICS
{0.5, 0.5}                    WORK TEARS SOUL KNOWLEDGE HEART

{0.75, 0.25}                    WORK JOY SOUL TEARS MATHEMATICS
TEARS LOVE LOVE LOVE SOUL

{1, 0}               TEARS LOVE JOY SOUL LOVE TEARS SOUL SOUL TEARS JOY
A selection of topics (from 500)
THEORY          SPACE         ART     STUDENTS      BRAIN      CURRENT       NATURE       THIRD
SCIENTISTS       EARTH         PAINT    TEACHER       NERVE    ELECTRICITY      WORLD        FIRST
EXPERIMENT         MOON         ARTIST    STUDENT      SENSE      ELECTRIC       HUMAN      SECOND
OBSERVATIONS       PLANET      PAINTING   TEACHERS     SENSES       CIRCUIT    PHILOSOPHY     THREE
SCIENTIFIC      ROCKET        PAINTED   TEACHING        ARE          IS         MORAL      FOURTH
EXPERIMENTS         MARS        ARTISTS      CLASS    NERVOUS     ELECTRICAL   KNOWLEDGE       FOUR
HYPOTHESIS         ORBIT      MUSEUM    CLASSROOM     NERVES      VOLTAGE      THOUGHT       GRADE
EXPLAIN    ASTRONAUTS        WORK       SCHOOL      BODY         FLOW        REASON        TWO
SCIENTIST        FIRST     PAINTINGS   LEARNING      SMELL      BATTERY        SENSE       FIFTH
OBSERVED     SPACECRAFT        STYLE       PUPILS     TASTE        WIRE           OUR     SEVENTH
EXPLANATION       JUPITER     PICTURES    CONTENT      TOUCH         WIRES        TRUTH       SIXTH
BASED       SATELLITE       WORKS  INSTRUCTION   MESSAGES      SWITCH       NATURAL     EIGHTH
OBSERVATION     SATELLITES        OWN      TAUGHT     IMPULSES    CONNECTED     EXISTENCE      HALF
IDEA     ATMOSPHERE    SCULPTURE       GROUP       CORD     ELECTRONS        BEING      SEVEN
EVIDENCE     SPACESHIP       PAINTER     GRADE      ORGANS     RESISTANCE        LIFE        SIX
THEORIES       SURFACE         ARTS     SHOULD      SPINAL       POWER          MIND      SINGLE
BELIEVED     SCIENTISTS   BEAUTIFUL     GRADES       FIBERS   CONDUCTORS     ARISTOTLE     NINTH
DISCOVERED    ASTRONAUT       DESIGNS     CLASSES    SENSORY      CIRCUITS     BELIEVED        END
OBSERVE        SATURN      PORTRAIT        PUPIL      PAIN        TUBE      EXPERIENCE     TENTH
FACTS          MILES      PAINTERS      GIVEN         IS      NEGATIVE      REALITY    ANOTHER
A selection of topics (from 500)
DISEASE                 MIND       STORY     FIELD     SCIENCE      BALL         JOB
WATER
BACTERIA                WORLD     STORIES  MAGNETIC      STUDY       GAME        WORK
FISH
DISEASES               DREAM        TELL   MAGNET    SCIENTISTS     TEAM        JOBS
SEA
GERMS                 DREAMS   CHARACTER    WIRE    SCIENTIFIC FOOTBALL       CAREER
SWIM
FEVER   SWIMMING    THOUGHT CHARACTERS    NEEDLE  KNOWLEDGE BASEBALL EXPERIENCE
CAUSE      POOL   IMAGINATION  AUTHOR    CURRENT      WORK      PLAYERS EMPLOYMENT
CAUSED                MOMENT       READ       COIL   RESEARCH       PLAY   OPPORTUNITIES
LIKE
SPREAD              THOUGHTS       TOLD     POLES   CHEMISTRY      FIELD     WORKING
SHELL
VIRUSES                  OWN     SETTING      IRON  TECHNOLOGY PLAYER         TRAINING
SHARK
INFECTION                REAL       TALES   COMPASS      MANY    BASKETBALL     SKILLS
TANK
VIRUS                  LIFE       PLOT     LINES  MATHEMATICS COACH         CAREERS
SHELLS
MICROORGANISMS SHARKS      IMAGINE    TELLING     CORE     BIOLOGY     PLAYED     POSITIONS
PERSON                 SENSE     SHORT    ELECTRIC     FIELD     PLAYING       FIND
DIVING
DIRECTION    PHYSICS       HIT      POSITION
INFECTIOUS  DOLPHINS CONSCIOUSNESS FICTION
COMMON                STRANGE     ACTION    FORCE   LABORATORY     TENNIS       FIELD
SWAM
CAUSING                FEELING      TRUE   MAGNETS     STUDIES     TEAMS    OCCUPATIONS
LONG
SMALLPOX                WHOLE      EVENTS       BE      WORLD       GAMES      REQUIRE
SEAL
BODY                  BEING      TELLS  MAGNETISM SCIENTIST      SPORTS   OPPORTUNITY
DIVE
INFECTIONS                MIGHT       TALE      POLE   STUDYING        BAT        EARN
DOLPHIN
CERTAIN                 HOPE      NOVEL    INDUCED    SCIENCES     TERRY        ABLE
UNDERWATER
A selection of topics (from 500)
DISEASE                 MIND       STORY     FIELD     SCIENCE      BALL          JOB
WATER
BACTERIA                WORLD     STORIES  MAGNETIC      STUDY       GAME        WORK
FISH
DISEASES               DREAM        TELL   MAGNET    SCIENTISTS     TEAM         JOBS
SEA
GERMS                 DREAMS   CHARACTER    WIRE    SCIENTIFIC FOOTBALL       CAREER
SWIM
FEVER   SWIMMING    THOUGHT CHARACTERS    NEEDLE  KNOWLEDGE BASEBALL EXPERIENCE
CAUSE      POOL   IMAGINATION  AUTHOR    CURRENT      WORK      PLAYERS EMPLOYMENT
CAUSED                MOMENT       READ       COIL   RESEARCH       PLAY   OPPORTUNITIES
LIKE
SPREAD              THOUGHTS       TOLD     POLES   CHEMISTRY      FIELD     WORKING
SHELL
VIRUSES                  OWN     SETTING      IRON  TECHNOLOGY PLAYER         TRAINING
SHARK
INFECTION                REAL       TALES   COMPASS      MANY    BASKETBALL     SKILLS
TANK
VIRUS                  LIFE       PLOT     LINES  MATHEMATICS COACH         CAREERS
SHELLS
MICROORGANISMS SHARKS      IMAGINE    TELLING     CORE     BIOLOGY     PLAYED     POSITIONS
PERSON                 SENSE     SHORT    ELECTRIC     FIELD     PLAYING        FIND
DIVING
DIRECTION    PHYSICS       HIT      POSITION
INFECTIOUS  DOLPHINS CONSCIOUSNESS FICTION
COMMON                STRANGE     ACTION    FORCE   LABORATORY     TENNIS       FIELD
SWAM
CAUSING                FEELING      TRUE   MAGNETS     STUDIES     TEAMS    OCCUPATIONS
LONG
SMALLPOX                WHOLE      EVENTS       BE      WORLD       GAMES      REQUIRE
SEAL
BODY                  BEING      TELLS  MAGNETISM SCIENTIST      SPORTS   OPPORTUNITY
DIVE
INFECTIONS                MIGHT       TALE      POLE   STUDYING        BAT        EARN
DOLPHIN
CERTAIN                 HOPE      NOVEL    INDUCED    SCIENCES     TERRY        ABLE
UNDERWATER
Learning topic hiearchies

(Blei, Griffiths, Jordan, & Tenenbaum, 2004)
Syntax and semantics from statistics
Factorization of language based on   semantics: probabilistic topics
statistical dependency patterns:
q
long-range, document specific,
dependencies                      z          z      z

w          w      w

short-range dependencies constant
across all documents                x          x      x

syntax: probabilistic regular grammar

(Griffiths, Steyvers, Blei, & Tenenbaum, submitted)
x=2

OF      0.6
x=1                         0.8          FOR     0.3
BETWEEN 0.1
z = 1 0.4              z = 2 0.6
HEART   0.2         SCIENTIFIC      0.2
LOVE    0.2         KNOWLEDGE       0.2         0.7
SOUL    0.2         WORK            0.2
0.3         0.1
TEARS   0.2         RESEARCH        0.2
JOY     0.2         MATHEMATICS     0.2
0.2
x=3

THE  0.6
0.9          A    0.3
MANY 0.1
x=2

OF      0.6
x=1                         0.8          FOR     0.3
BETWEEN 0.1
z = 1 0.4              z = 2 0.6
HEART   0.2         SCIENTIFIC      0.2
LOVE    0.2         KNOWLEDGE       0.2         0.7
SOUL    0.2         WORK            0.2
0.3         0.1
TEARS   0.2         RESEARCH        0.2
JOY     0.2         MATHEMATICS     0.2
0.2
x=3

THE  0.6
0.9          A    0.3
MANY 0.1

THE ………………………………
x=2

OF      0.6
x=1                         0.8          FOR     0.3
BETWEEN 0.1
z = 1 0.4              z = 2 0.6
HEART   0.2         SCIENTIFIC      0.2
LOVE    0.2         KNOWLEDGE       0.2         0.7
SOUL    0.2         WORK            0.2
0.3         0.1
TEARS   0.2         RESEARCH        0.2
JOY     0.2         MATHEMATICS     0.2
0.2
x=3

THE  0.6
0.9          A    0.3
MANY 0.1

THE LOVE……………………
x=2

OF      0.6
x=1                         0.8          FOR     0.3
BETWEEN 0.1
z = 1 0.4              z = 2 0.6
HEART   0.2         SCIENTIFIC      0.2
LOVE    0.2         KNOWLEDGE       0.2         0.7
SOUL    0.2         WORK            0.2
0.3         0.1
TEARS   0.2         RESEARCH        0.2
JOY     0.2         MATHEMATICS     0.2
0.2
x=3

THE  0.6
0.9          A    0.3
MANY 0.1

THE LOVE OF………………
x=2

OF      0.6
x=1                         0.8          FOR     0.3
BETWEEN 0.1
z = 1 0.4              z = 2 0.6
HEART   0.2         SCIENTIFIC      0.2
LOVE    0.2         KNOWLEDGE       0.2         0.7
SOUL    0.2         WORK            0.2
0.3         0.1
TEARS   0.2         RESEARCH        0.2
JOY     0.2         MATHEMATICS     0.2
0.2
x=3

THE  0.6
0.9          A    0.3
MANY 0.1

THE LOVE OF RESEARCH ……
Semantic categories
FOOD       MAP        DOCTOR       BOOK      GOLD      BEHAVIOR      CELLS     PLANTS
FOODS     NORTH         PATIENT     BOOKS     IRON         SELF        CELL      PLANT
BODY     EARTH         HEALTH    READING    SILVER    INDIVIDUAL ORGANISMS     LEAVES
NUTRIENTS   SOUTH        HOSPITAL INFORMATION COPPER   PERSONALITY     ALGAE       SEEDS
DIET      POLE        MEDICAL    LIBRARY    METAL      RESPONSE   BACTERIA       SOIL
FAT      MAPS          CARE      REPORT   METALS        SOCIAL  MICROSCOPE     ROOTS
SUGAR    EQUATOR       PATIENTS      PAGE    STEEL     EMOTIONAL   MEMBRANE    FLOWERS
ENERGY      WEST          NURSE      TITLE     CLAY      LEARNING   ORGANISM      WATER
MILK      LINES       DOCTORS    SUBJECT     LEAD       FEELINGS     FOOD       FOOD
EATING     EAST       MEDICINE      PAGES    ADAM   PSYCHOLOGISTS    LIVING      GREEN
FRUITS  AUSTRALIA      NURSING      GUIDE      ORE    INDIVIDUALS     FUNGI       SEED
VEGETABLES   GLOBE      TREATMENT     WORDS   ALUMINUM PSYCHOLOGICAL     MOLD       STEMS
WEIGHT    POLES         NURSES    MATERIAL  MINERAL   EXPERIENCES MATERIALS     FLOWER
FATS   HEMISPHERE    PHYSICIAN    ARTICLE    MINE   ENVIRONMENT    NUCLEUS       STEM
NEEDS    LATITUDE     HOSPITALS   ARTICLES   STONE        HUMAN      CELLED       LEAF
CARBOHYDRATES PLACES            DR       WORD   MINERALS     RESPONSES STRUCTURES   ANIMALS
VITAMINS    LAND           SICK      FACTS      POT      BEHAVIORS  MATERIAL      ROOT
CALORIES    WORLD       ASSISTANT    AUTHOR   MINING      ATTITUDES STRUCTURE     POLLEN
PROTEIN   COMPASS     EMERGENCY   REFERENCE  MINERS   PSYCHOLOGY      GREEN     GROWING
MINERALS  CONTINENTS    PRACTICE       NOTE      TIN        PERSON     MOLDS       GROW
Syntactic categories
SAID     THE       MORE         ON        GOOD        ONE         HE         BE
ASKED      HIS       SUCH        AT       SMALL      SOME         YOU       MAKE
THOUGHT    THEIR       LESS       INTO       NEW       MANY        THEY        GET
TOLD     YOUR       MUCH       FROM    IMPORTANT      TWO           I       HAVE
SAYS     HER      KNOWN        WITH      GREAT       EACH        SHE         GO
MEANS       ITS       JUST    THROUGH      LITTLE       ALL        WE        TAKE
CALLED      MY      BETTER       OVER      LARGE      MOST          IT         DO
CRIED     OUR      RATHER     AROUND         *        ANY       PEOPLE      FIND
SHOWS      THIS    GREATER    AGAINST        BIG      THREE     EVERYONE      USE
ANSWERED    THESE     HIGHER     ACROSS       LONG       THIS      OTHERS       SEE
TELLS       A      LARGER       UPON       HIGH      EVERY    SCIENTISTS    HELP
REPLIED     AN      LONGER     TOWARD    DIFFERENT   SEVERAL    SOMEONE      KEEP
SHOUTED     THAT     FASTER      UNDER     SPECIAL      FOUR       WHO        GIVE
EXPLAINED    NEW     EXACTLY      ALONG        OLD       FIVE      NOBODY      LOOK
LAUGHED    THOSE    SMALLER       NEAR     STRONG       BOTH        ONE       COME
MEANT     EACH    SOMETHING    BEHIND      YOUNG        TEN    SOMETHING     WORK
WROTE       MR      BIGGER        OFF     COMMON        SIX      ANYONE      MOVE
SHOWED      ANY       FEWER      ABOVE      WHITE      MUCH     EVERYBODY      LIVE
BELIEVED    MRS      LOWER       DOWN       SINGLE    TWENTY       SOME        EAT
WHISPERED     ALL    ALMOST      BEFORE     CERTAIN     EIGHT       THEN      BECOME
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
Structure and statistics
• Statistical language modeling
– topic models

• Relational categorization
– attributes and relations
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)
Attributes and relations
Data                        Model
objects
attributes

P(X) = ik z P(xik|zi) i P(zi)
X
mixture model (c.f. Anderson, 1990)

objects
objects

Y           P(Y) = ij z P(yij|zi) i P(zi)
stochastic blockmodel
Stochastic blockmodels
• For any pair of objects, (i,j), probability of
relation is determined by classes, (zi, zj)
To type j
l11   l12     l13        Each entity has a type = Z
From
type i   l21   l22     l23   L
l31   l32     l33
P(Z,L|Y)  P(Y|Z,LP(Z)P(L

• Allows types of objects and class
probabilities to be learned from data
Stochastic blockmodels

A       B
A       B

C               D       C

A       B   C       A B C       D
A
A                   B
B                   C
C                   D
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
Categorizing words

BAND        TIE       SEW       WASH
INSTRUMENT     COAT    MATERIAL     LIQUID
BLOW      SHOES      WOOL     BATHROOM
HORN       ROPE      YARN        SINK
FLUTE     LEATHER     WEAR      CLEANER
BRASS       SHOE       TEAR      STAIN
GUITAR       HAT       FRAY      DRAIN
PIANO      PANTS      JEANS      DISHES
TUBA     WEDDING    COTTON       TUB
TRUMPET     STRING    CARPET      SCRUB
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
Categorizing actors

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

Germany  UK        British comedy          Italian          US Westerns
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
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

Taking stock
Bayesian models of inductive learning
• Inductive leaps can be explained with
hierarchical Theory-based Bayesian models:

Domain Theory
Probabilistic
Bayesian
Generative      Structural Hypotheses
Model                                   inference

Data
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 ...
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.
Bayesian models of inductive learning
• Inductive leaps can be explained with
hierarchical Theory-based Bayesian models.
• Challenges:
– Theories are hard.
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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