Normative models of human inductive inference Tom Griffiths Department of Psychology Cognitive Science Program University of California, Berkeley

Perception is optimal Körding & Wolpert (2004)

Cognition is not

Optimality and cognition Can optimal solutions to computational problems shed light on human cognition?

Optimality and cognition Can optimal solutions to computational problems shed light on human cognition? Can we explain aspects of cognition as the result of sensitivity to natural statistics? What kind of representations are extracted from those statistics?

Optimality and cognition Can optimal solutions to computational problems shed light on human cognition? Can we explain aspects of cognition as the result of sensitivity to natural statistics? What kind of representations are extracted from those statistics? Joint work with Josh Tenenbaum

Natural statistics Images of natural scenes sparse coding (Olshausen & Field, 1996) Neural representation

Predicting the future How often is Google News updated? t = time since last update t total = time between updates What should we guess for t total given t?

Reverend Thomas Bayes

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

Bayes’ theorem h: hypothesis d: data

Bayesian inference p(t total |t)  p(t|t total ) p(t total ) posterior probability likelihoodprior

Bayesian inference p(t total |t)  p(t|t total ) p(t total ) p(t total |t)  1/t total p(t total ) assume random sample (0 < t < t total ) posterior probability likelihoodprior

The effects of priors

Evaluating human predictions Different domains with different priors: –a movie has made $60 million [power-law] –your friend quotes from line 17 of a poem [power-law] –you meet a 78 year old man [Gaussian] –a movie has been running for 55 minutes [Gaussian] –a U.S. congressman has served for 11 years [Erlang] Prior distributions derived from actual data Use 5 values of t for each People predict t total

people parametric prior empirical prior Gott’s rule

Predicting the future People produce accurate predictions for the duration and extent of everyday events People are sensitive to the statistics of their environment in making these predictions –form of the prior (power-law or exponential) –distribution given that form (parameters)

Optimality and cognition Can optimal solutions to computational problems shed light on human cognition? Can we explain aspects of cognition as the result of sensitivity to natural statistics? What kind of representations are extracted from those statistics? Joint work with Adam Sanborn

Categories are central to cognition

Sampling from categories Frog distribution P(x|c)

Markov chain Monte Carlo Sample from a target distribution P(x) by constructing Markov chain for which P(x) is the stationary distribution Markov chain converges to its stationary distribution, providing outcomes that can be used similarly to samples

Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings, 1970) Step 1: propose a state (we assume symmetrically) Q(x (t+1) |x (t) ) = Q(x (t) )|x (t+1) ) Step 2: decide whether to accept, with probability Metropolis acceptance function Barker acceptance function

Metropolis-Hastings algorithm p(x)p(x)

p(x)p(x)

p(x)p(x)

A(x (t), x (t+1) ) = 0.5 p(x)p(x)

Metropolis-Hastings algorithm p(x)p(x)

A(x (t), x (t+1) ) = 1 p(x)p(x)

A task Ask subjects which of two alternatives comes from a target category Which animal is a frog?

A Bayesian analysis of the task Assume:

Response probabilities If people probability match to the posterior, response probability is equivalent to the Barker acceptance function for target distribution p(x|c)

Collecting the samples Which is the frog? Trial 1Trial 2Trial 3

Verifying the method

Training Subjects were shown schematic fish of different sizes and trained on whether they came from the ocean (uniform) or a fish farm (Gaussian)

Between-subject conditions

Choice task Subjects judged which of the two fish came from the fish farm (Gaussian) distribution

Examples of subject MCMC chains

Estimates from all subjects Estimated means and standard deviations are significantly different across groups Estimated means are accurate, but standard deviation estimates are high –result could be due to perceptual noise or response gain

Sampling from natural categories Examined distributions for four natural categories: giraffes, horses, cats, and dogs Presented stimuli with nine-parameter stick figures (Olman & Kersten, 2004)

Choice task

Samples from Subject 3 (projected onto plane from LDA)

Mean animals by subject giraffe horse cat dog S1S2S3S4S5S6S7S8

Marginal densities (aggregated across subjects) Giraffes are distinguished by neck length, body height and body tilt Horses are like giraffes, but with shorter bodies and nearly uniform necks Cats have longer tails than dogs

Markov chain Monte Carlo with people Normative models can guide the design of experiments to measure psychological variables Markov chain Monte Carlo (and other methods) can be used to sample from subjective probability distributions –category distributions –prior distributions

Conclusion Optimal solutions to computational problems can shed light on human cognition We can explain aspects of cognition as the result of sensitivity to natural statistics We can use optimality to explore representations extracted from those statistics

Relative volume of categories Minimum Enclosing Hypercube GiraffeHorseCatDog Convex hull content divided by enclosing hypercube content Convex Hull

Discrimination method (Olman & Kersten, 2004)

Parameter space for discrimination Restricted so that most random draws were animal-like

MCMC and discrimination means

Iterated learning (Kirby, 2001) Each learner sees data, forms a hypothesis, produces the data given to the next learner With Bayesian learners, the distribution over hypotheses converges to the prior (Griffiths & Kalish, 2005)

Explaining convergence to the prior PL(h|d)PL(h|d) P P (d|h) PL(h|d)PL(h|d) Intuitively: data acts once, prior many times Formally: iterated learning with Bayesian agents is a Gibbs sampler on P(d,h) (Griffiths & Kalish, in press)

Iterated function learning (Kalish, Griffiths, & Lewandowsky, in press) 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