Exploring cultural transmission by iterated learning Tom Griffiths Brown University Mike Kalish 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) 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 P L (h|d): probability of inferring hypothesis h from data d P P (d|h): probability of generating data d from hypothesis h PL(h|d)PL(h|d) P P (d|h) PL(h|d)PL(h|d)

Analyzing iterated learning What are the consequences of iterated learning? Simulations Analytic results Complex algorithms Simple algorithms Komarova, Niyogi, & Nowak (2002) Brighton (2002) Kirby (2001) Smith, Kirby, & Brighton (2003) ?

Bayesian inference Reverend Thomas Bayes Rational procedure for updating beliefs Foundation of many learning algorithms Widely used for language learning

Bayes’ theorem Posterior probability LikelihoodPrior probability Sum over space of hypotheses h: hypothesis d: data

Iterated Bayesian learning Learners are Bayesian agents PL(h|d)PL(h|d) P P (d|h) PL(h|d)PL(h|d)

Variables x (t+1) independent of history given x (t) Converges to a stationary distribution under easily checked conditions for ergodicity xx x xx x x x Transition matrix T = 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

Analyzing iterated learning d0d0 h1h1 d1d1 h2h2 PL(h|d)PL(h|d) PP(d|h)PP(d|h) PL(h|d)PL(h|d) d2d2 h3h3 PP(d|h)PP(d|h) PL(h|d)PL(h|d)  d P P (d|h)P L (h|d) h1h1 h2h2 h3h3 A Markov chain on hypotheses d0d0 d1d1  h P L (h|d) P P (d|h) d2d2 A Markov chain on data P L (h|d) P P (d|h) h 1,d 1 h 2,d 2 h 3,d 3 A Markov chain on hypothesis-data pairs

Stationary distributions Markov chain on h converges to the prior, p(h) Markov chain on d converges to the “prior predictive distribution” Markov chain on (h,d) is a Gibbs sampler for

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) Each learner sees a set of (x,y) pairs Makes predictions of y for new x values Predictions are data for the next learner datahypotheses

Function learning experiments Stimulus Response Slider Feedback Examine iterated learning with different initial data

Iteration Initial data

Iterated concept learning (heavy lifting by Brian Christian) Each learner sees examples from a species Identifies species of four amoebae Iterated learning is run within-subjects data hypotheses

Two positive examples data (d) hypotheses (h)

Bayesian model (Tenenbaum, 1999; Tenenbaum & Griffiths, 2001) d: 2 amoebae h: set of 4 amoebae m: # of amoebae in the set d (= 2) |h|: # of amoebae in the set h (= 4) Posterior is renormalized prior What is the prior?

Classes of concepts (Shepard, Hovland, & Jenkins, 1958) Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 shape size color

Experiment design (for each subject) Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 6 iterated learning chains 6 independent learning “chains”

Estimating the prior data (d) hypotheses (h)

Estimating the prior Class 1 Class 2 Class 3 Class 4 Class 5 Class Prior r = Bayesian model Human subjects

Two positive examples (n = 20) Probability Iteration Probability Iteration Human learners Bayesian model

Two positive examples (n = 20) Probability Bayesian model Human learners

Three positive examples data (d) hypotheses (h)

Three positive examples (n = 20) Probability Iteration Probability Iteration Human learners Bayesian model

Three positive examples (n = 20) Bayesian model Human learners

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:

Discovering the biases of models EXAM (Delosh, Busemeyer, & McDaniel, 1997):

Discovering the biases of models POLE (Kalish, Lewandowsky, & Kruschke, 2004):