A%20Bayesian%20view%20of%20language%20evolution%20by%20iterated%20learning

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A Bayesian view of language evolution by iterated
learning

• Tom Griffiths
• Brown University

Mike Kalish University of Louisiana
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Linguistic universals

• Human languages are a subset of all logically
  possible communication schemes
• universal properties common to all languages
• (Comrie, 1981 Greenberg, 1963 Hawkins, 1988)
• Two questions
• why do linguistic universals exist?
• why are particular properties universal?

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Possible explanations

• Traditional answer
• linguistic universals reflect innate constraints
  specific to a system for acquiring language
• (e.g., Chomsky, 1965)
• Alternative answer
• linguistic universals emerge as the result of the
  fact that language is learned anew by each
  generation
• (e.g., Briscoe, 1998 Kirby, 2001)

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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

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The information bottleneck(Kirby, 2001)
size indicates compressibility
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Analyzing iterated learning
What are the consequences of iterated learning?
?
Komarova, Niyogi, Nowak (2002) Brighton (2002)
Kirby (2001) Smith, Kirby, Brighton (2003)
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Outline

• Iterated Bayesian learning
• Markov chains
• Convergence results
• Example Emergence of compositionality
• Conclusion

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Outline

• Iterated Bayesian learning
• Markov chains
• Convergence results
• Example Emergence of compositionality
• Conclusion

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Bayesian inference

• Rational procedure for updating beliefs
• Foundation of many learning algorithms
• (e.g., Mackay, 2003)
• Widely used for language learning
• (e.g., Charniak, 1993)

Reverend Thomas Bayes
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Bayes theorem
h hypothesis d data
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Iterated Bayesian learning
p(hd)
p(hd)
p(dh)
p(dh)
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Outline

• Iterated Bayesian learning
• Markov chains
• Convergence results
• Example Emergence of compositionality
• Conclusion

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Markov chains
x
x
x
x
x
x
x
x
Transition matrix P(x(t1)x(t))

• Variables x(t1) independent of history given
  x(t)
• Converges to a stationary distribution under
  easily checked conditions for ergodicity

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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

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Gibbs sampling

• For variables x x1, x2, , xn
• Draw xi(t1) from P(xix-i)
• x-i x1(t1), x2(t1),, xi-1(t1), xi1(t), ,
  xn(t)
• Converges to P(x1, x2, , xn)

(Geman Geman, 1984)
(a.k.a. the heat bath algorithm in statistical
physics)
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Gibbs sampling
(MacKay, 2003)
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Outline

• Iterated Bayesian learning
• Markov chains
• Convergence results
• Example Emergence of compositionality
• Conclusion

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Analyzing iterated learning

• Iterated learning is a Markov chain on (h,d)

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Analyzing iterated learning
p(hd)
p(hd)
p(dh)
p(dh)

• Iterated learning is a Markov chain on (h,d)
• Iterated Bayesian learning is a Gibbs sampler for

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Analytic results

• Iterated Bayesian learning converges to

(geometrically Liu, Wong, Kong, 1995)
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Analytic results

• Iterated Bayesian learning converges to
• Corollaries
• distribution over hypotheses converges to p(h)
• distribution over data converges to p(d)
• the proportion of a population of iterated
  learners with hypothesis h converges to p(h)

(geometrically Liu, Wong, Kong, 1995)
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Outline

• Iterated Bayesian learning
• Markov chains
• Convergence results
• Example Emergence of compositionality
• Conclusion

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A simple language model
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A simple language model
0
1

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

0
1
holistic
0
1
0
1
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Analysis technique

• Compute transition matrix on languages
• Sample Markov chains
• Compare language frequencies with prior
• (can also compute eigenvalues etc.)

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Convergence to priors
? 0.50, ? 0.05, m 3
Chain
Prior
? 0.01, ? 0.05, m 3
Iteration
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The information bottleneck
? 0.50, ? 0.05, m 1
Chain
Prior
? 0.01, ? 0.05, m 3
? 0.50, ? 0.05, m 10
Iteration
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The information bottleneck
Bottleneck affects relative stability of
languages favored by prior
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Outline

• Iterated Bayesian learning
• Markov chains
• Convergence results
• Example Emergence of compositionality
• Conclusion

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Implications for linguistic universals

• Two questions
• why do linguistic universals exist?
• why are particular properties universal?
• Different answers
• existence explained through iterated learning
• universal properties depend on the prior
• Focuses inquiry on the priors of the learners
• languages reflect the biases of human learners

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Extensions and future directions

• Results extend to
• unbounded populations
• continuous time population dynamics
• Iterated learning applies to other knowledge
• religious concepts, social norms, legends
• Provides a method for evaluating priors
• experiments in iterated learning with humans

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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

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Function learning in the lab
Examine iterated learning with different initial
data
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Initial data
Iteration
1 2 3 4
5 6 7 8 9
(Kalish, 2004)
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An example Gaussians

• If we assume
• data, d, is a single real number, x
• hypotheses, h, are means of a Gaussian, ?
• prior, p(?), is Gaussian(?0,?02)
• then p(xn1xn) is Gaussian(?n, ?x2 ?n2)

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?0 0, ?02 1, x0 20 Iterated learning
results in rapid convergence to prior
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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,?02)

y

?
x 1
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y

?
?0 1, ?02 0.1, y0 -1
x 1
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