Transcript Inductive inference in perception and cognition

Exploring cultural transmission
by iterated learning
Tom Griffiths
Mike Kalish
Brown University
University of Louisiana
With thanks to: Anu Asnaani, Brian Christian, and Alana Firl
Cultural transmission
• Most knowledge is based on secondhand data
• Some things can only be learned from others
– cultural knowledge transmitted across generations
• What are the consequences of learners learning
from other learners?
Iterated learning
(Kirby, 2001)
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Each learner sees data, forms a hypothesis,
produces the data given to the next learner
Objects of iterated learning
• Knowledge communicated through data
• Examples:
–
–
–
–
–
religious concepts
social norms
myths and legends
causal theories
language
Analyzing iterated learning
PL(h|d)
PL(h|d)
PP(d|h)
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PP(d|h)
PL(h|d): probability of inferring hypothesis h from data d
PP(d|h): probability of generating data d from hypothesis h
Analyzing iterated learning
What are the consequences of iterated learning?
Complex
algorithms
Analytic results
?
Kirby (2001)
Simulations
Smith, Kirby, &
Brighton (2003)
Simple
algorithms
Komarova, Niyogi,
& Nowak (2002)
Brighton (2002)
Bayesian inference
• Rational procedure
for updating beliefs
• Foundation of many
learning algorithms
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• Widely used for
language learning
Reverend Thomas Bayes

Bayes’ theorem
Posterior
probability
Likelihood
Prior
probability
P(d | h)P(h)
P(h | d) 
 P(d | h)P(h)
h  H
h: hypothesis
d: data
Sum over space
of hypotheses
Iterated Bayesian learning
PL(h|d)
PL(h|d)
PP(d|h)
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QuickTime™ and a
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PP(d|h)
QuickTime™ and a
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Learners are Bayesian agents
PP (d | h)P(h)
PL (h | d) 
 PP (d | h)P(h)
h  H
Markov chains
x
x
x
x
x
x
x
x
Transition matrix
T = P(x(t+1)|x(t))
• Variables x(t+1) independent of history given x(t)
• Converges to a stationary distribution under
easily checked conditions for ergodicity
Stationary distributions
• Stationary distribution:
 i  P(x(t 1)  i | x(t )  j) j  Tij j
j
j
• In matrix form

  T
•  is the first eigenvector of the matrix T
• Second eigenvalue
sets rate of convergence

Analyzing iterated learning
d0
PL(h|d)
h1
PP(d|h)
d1
PL(h|d)
h2
PP(d|h)
d2
PL(h|d)
h3
A Markov chain on hypotheses
h1
d PP(d|h)PL(h|d)
h2
d PP(d|h)PL(h|d)
h3
A Markov chain on data
d0
h PL(h|d) PP(d|h)
d1
h PL(h|d) PP(d|h)
d2
h PL(h|d) PP
A Markov chain on hypothesis-data pairs
h1,d1
PL(h|d) PP(d|h)
h2 ,d2
PL(h|d) PP(d|h)
h3 ,d3
Stationary distributions
• Markov chain on h converges to the prior, p(h)
• Markov chain on d converges to the “prior
predictive distribution”
p(d)   p(d | h) p(h)
h
• Markov chain on (h,d) is a Gibbs sampler for

p(d,h)  p(d | h) p(h)
Implications
• The probability that the nth learner entertains
the hypothesis h approaches p(h) as n  
• Convergence to the prior occurs regardless of:
– the properties of the hypotheses themselves
– the amount or structure of the data transmitted
• The consequences of iterated learning are
determined entirely by the biases of the learners
Identifying inductive biases
• Many problems in cognitive science can be
formulated as problems of induction
– learning languages, concepts, and causal relations
• Such problems are not solvable without bias
(e.g., Goodman, 1955; Kearns & Vazirani, 1994; Vapnik, 1995)
• What biases guide human inductive inferences?
If iterated learning converges to the prior, then it
may provide a method for investigating biases
Serial reproduction
(Bartlett, 1932)
• Participants see stimuli,
then reproduce them
from memory
• Reproductions of one
participant are stimuli
for the next
• Stimuli were interesting,
rather than controlled
– e.g., “War of the Ghosts”
Iterated function learning
(heavy lifting by Mike Kalish)
data
hypotheses
• Each learner sees a set of (x,y) pairs
• Makes predictions of y for new x values
• Predictions are data for the next learner
Function learning experiments
Stimulus
Feedback
Response
Slider
Examine iterated learning with different initial data
Initial
data
Iteration
1
2
3
4
5
6
7
8
9
Iterated concept learning
(heavy lifting by Brian Christian)
data
hypotheses
• Each learner sees examples from a species
• Identifies species of four amoebae
• Iterated learning is run within-subjects
Two positive examples
data (d)
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hypotheses (h)
Bayesian model
(Tenenbaum, 1999; Tenenbaum & Griffiths, 2001)
P(d | h)P(h)
P(h | d) 
 P(d | h)P(h)
d: 2 amoebae
h: set of 4 amoebae
h  H
m: # of amoebae in
1/ h m
dh
the set d (= 2)
P(d | h)  
|h|: # of amoebae in
otherwise
 0
the set h (= 4)
P(h)
Posterior is renormalized prior
P(h | d) 
 P(h)
What is the prior?
h'|d h'
Classes of concepts
(Shepard, Hovland, & Jenkins, 1958)
color
size
shape
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Experiment design (for each subject)
6
iterated
learning
chains
6
independent
learning
“chains”
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Estimating the prior
hypotheses (h)
data (d)
Estimating the prior
Human subjects
Bayesian model
Prior
Class 1
Class 2
0.861
0.087
Class 3
0.009
Class 4
0.002
Class 5
0.013
Class 6
0.028
r = 0.952
Two positive examples
(n = 20)
Human learners
Probability
Probability
Bayesian model
Iteration
Iteration
Two positive examples
(n = 20)
Probability
Human learners
Bayesian model
Three positive examples
data (d)
hypotheses (h)
Three positive examples
(n = 20)
Human learners
Probability
Probability
Bayesian model
Iteration
Iteration
Three positive examples
(n = 20)
Human learners
Bayesian model
Conclusions
• Consequences of iterated learning with Bayesian
learners determined by the biases of the learners
• Consistent results are obtained with human learners
• Provides an explanation for cultural universals…
– universal properties are probable under the prior
– a direct connection between mind and culture
• …and a novel method for evaluating the inductive
biases that guide human learning
Discovering the biases of models
Generic neural network:
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Discovering the biases of models
EXAM (Delosh, Busemeyer, & McDaniel, 1997):
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Discovering the biases of models
POLE (Kalish, Lewandowsky, & Kruschke, 2004):
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