Analyzing iterated learning Tom Griffiths Brown University Mike Kalish University of Louisiana.

Analyzing iterated learning

Tom GriffithsBrown University

Mike KalishUniversity of Louisiana

Cultural transmission

• Most knowledge is based on secondhand data

• Some things can only be learned from others– cultural objects transmitted across generations

• Studying the cognitive aspects of cultural transmission provides unique insights…

Iterated learning(Kirby, 2001)

• Each learner sees data, forms a hypothesis, produces the data given to the next learner

• c.f. the playground game “telephone”

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Objects of iterated learning

• It’s not just about languages…

• In the wild:– religious concepts– social norms– myths and legends– causal theories

• In the lab:– functions and categories

Outline

1. Analyzing iterated learning

2. Iterated Bayesian learning

3. Examples

4. Iterated learning with humans

5. Conclusions and open questions

• Variables x(t+1) independent of history given x(t)

• Converges to a stationary distribution under easily checked conditions for ergodicity

x x x x x x x x

Transition matrixT = P(x(t+1)|x(t))

Markov chains

Stationary distributions

• Stationary distribution:

• In matrix form

is the first eigenvector of the matrix T

• Second eigenvalue sets rate of convergence

€

i = P(x(t +1) = i |j

∑ x(t ) = j)π j = Tijπ j

j

∑

€

=Tπ

A Markov chain on hypotheses

• Transition probabilities sum out data

• Stationary distribution and convergence rate from eigenvectors and eigenvalues of Q– can be computed numerically for matrices of

reasonable size, and analytically in some cases

€

Qij = P(hn +1 = i | hn = j) = P(hn +1 = i |d

∑ d) P(d | hn = j)

Infinite populations in continuous time

• “Language dynamical equation”

• “Neutral model” (fj(x) constant)

• Stable equilibrium at first eigenvector of Q

€

dx i

dt= Qij f j (x)

j

∑ x j − φ(x)x i

(Nowak, Komarova, & Niyogi, 2001)

€

dx i

dt= Qij

j

∑ x j − x i

(Komarova & Nowak, 2003)

€

dx

dt= (Q − I)x

Outline



3. Examples



Bayesian inference

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Reverend Thomas Bayes

• Rational procedure for updating beliefs

• Foundation of many learning algorithms

(e.g., Mackay, 2003)

• Widely used for language learning

(e.g., Charniak, 1993)

Bayes’ theorem

€

P(h | d) =P(d | h)P(h)

P(d | ′ h )P( ′ h )′ h ∈H

∑

Posteriorprobability

Likelihood Priorprobability

Sum over space of hypothesesh: hypothesis

d: data

Iterated Bayesian learning







Learners are Bayesian agents

€

PL (h | d) =PP (d | h)P(h)

PP (d | ′ h )P( ′ h )′ h ∈H

∑

Markov chains on h and d

• Markov chain on h has stationary distribution

• Markov chain on d has stationary distribution

€

i = P(h = i) the prior

€

i = PP (d = i | h)h

∑ P(h)the prior

predictivedistribution

Markov chain Monte Carlo

• A strategy for sampling from complex probability distributions

• Key idea: construct a Markov chain which converges to a particular distribution– e.g. Metropolis algorithm– e.g. Gibbs sampling

Gibbs sampling

For variables x = x1, x2, …, xn

Draw xi(t+1) from P(xi|x-i)

x-i = x1(t+1), x2

(t+1),…, xi-1(t+1)

, xi+1(t)

, …, xn(t)

Converges to P(x1, x2, …, xn)

(a.k.a. the heat bath algorithm in statistical physics)

(Geman & Geman, 1984)

Gibbs sampling

(MacKay, 2003)

Iterated learning is a Gibbs sampler

• Iterated Bayesian learning is a sampler for

• Implies:– (h,d) converges to this distribution– converence rates are known

(Liu, Wong, & Kong, 1995)€

p(d,h) = p(d | h) p(h)

Outline



3. Examples



An example: Gaussians

€

μn =xn /σ x

2 + μ0 /σ 02

1/σ x2 +1/σ 0

2

€

σ n2 =

1

1/σ x2 +1/σ 0

2

• If we assume…– data, d, is a single real number, x– hypotheses, h, are means of a Gaussian, μ– prior, p(μ), is Gaussian(μ0,σ0

2)

• …then p(xn+1|xn) is Gaussian(μn, σx2 + σn

2)


• If we assume…– data, d, is a single real number, x– hypotheses, h, are means of a Gaussian, μ– prior, p(μ), is Gaussian(μ0,σ0

2)

• …then p(xn+1|xn) is Gaussian(μn, σx2 + σn

2)

• p(xn|x0) is Gaussian(μ0+cnx0, (σx2 + σ0

2)(1 - c2n))

i.e. geometric convergence to prior

€

c =1

1+ σ x2

σ 02


• p(xn+1|x0) is Gaussian(μ0+cnx0,(σx2 + σ0

2)(1-c2n))

μ0 = 0, σ02 = 1, x0 = 20

Iterated learning results in rapid convergence to prior

An example: Linear regression

• Assume– data, d, are pairs of real numbers (x, y)– hypotheses, h, are functions

• An example: linear regression– hypotheses have slope and pass through origin

– p() is Gaussian(0,σ02)

}x = 1

y

}x = 1

y

0 = 1, σ02 = 0.1, y0 = -1

An example: compositionality

0

1

0 1

0

1

0 1

events utteranceslanguage

x yfunction

“actions”

“agents” “nouns”

“verbs”

compositional

An example: compositionality

• Data: m event-utterance pairs• Hypotheses: languages, with error

0

1

0 1

0

1

0 1compositional

0

1

0 1

0

1

0 1holistic

P(h)

€

α4

€

(1−α )

256

Analysis technique

1. Compute transition matrix on languages

2. Sample Markov chains

3. Compare language frequencies with prior

(can also compute eigenvalues etc.)€

P(hn = i | hn−1 = j) = P(hn = i | d)P(d | hn−1 = j)d

∑

Convergence to priors

α = 0.50, = 0.05, m = 3

α = 0.01, = 0.05, m = 3

Chain Prior

Iteration

Effect of Prior

The information bottleneck

α = 0.50, = 0.05, m = 1

α = 0.01, = 0.05, m = 3

Chain Prior

Iteration

α = 0.50, = 0.05, m = 10

No effect of bottleneck

The information bottleneck

€

Stability ratio = P(hn = i | hn−1 = i)

i∈C

P(hn = i | hn−1 = i)i∈H

Bottleneck affects relative stability of languages favored by prior

Outline



3. Examples



A method for discovering priors

Iterated learning converges to the prior…

…evaluate prior by producing iterated learning

Iterated function learning

• Each learner sees a set of (x,y) pairs

• Makes predictions of y for new x values

• Predictions are data for the next learner

data hypotheses

Function learning in the lab

Stimulus

Response

Slider

Feedback

Examine iterated learning with different initial data

1 2 3 4 5 6 7 8 9

IterationInitialdata

(Kalish, 2004)

Outline



3. Examples



Conclusions and open questions

• Iterated Bayesian learning converges to prior– properties of languages are properties of learners– information bottleneck doesn’t affect equilibrium

• What about other learning algorithms?

• What determines rates of convergence?– amount and structure of input data

• What happens with people?

Analyzing iterated learning Tom Griffiths Brown University Mike Kalish University of Louisiana.

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