Proyecto De Investigacion

Markov Chain Monte Carlo with People

Adam N. Sanborn Psychological and Brain Sciences Indiana University Bloomington, IN 47045 asanborn@indiana.edu

Thomas L. Griffiths Department of Psychology University of California Berkeley, CA 94720 tom griffiths@berkeley.edu

Abstract
Many formal models of cognition implicitly use subjective probability distributions to capture the assumptions ofhuman learners. Most applications of these models determine these distributions indirectly. We propose a method for directly determining the assumptions of human learners by sampling from subjective probability distributions. Using a correspondence between a model of human choice and Markov chain Monte Carlo (MCMC), we describe a method for sampling from the distributions over objects that peopleassociate with different categories. In our task, subjects choose whether to accept or reject a proposed change to an object. The task is constructed so that these decisions follow an MCMC acceptance rule, defining a Markov chain for which the stationary distribution is the category distribution. We test this procedure for both artificial categories acquired in the laboratory, and natural categoriesacquired from experience.

1

Introduction

Determining the assumptions that guide human learning and inference is one of the central goals of cognitive science. Subjective probability distributions are used to model the degrees of belief that learners assign to hypotheses in many domains, including categorization, decision making, and memory [1, 2, 3, 4]. If the knowledge of learners can bemodeled in this way, then exploring this knowledge becomes a matter of asking questions about the nature of their associated probability distributions. A common way to learn about a probability distribution is to draw samples from it. In the machine learning and statistics literature, drawing samples from probability distributions is a major area of research, and is often done using Markov chainMonte Carlo (MCMC) algorithms. In this paper, we describe a method for directly obtaining information about subjective probability distributions, by having people act as elements of an MCMC algorithm. Our approach is to design a task that will allow us to sample from a particular subjective probability distribution. Much research has been devoted to relating the magnitude of psychological responsesto choice probabilities, resulting in mathematical models of these tasks. We point out an equivalence between a model of human choice behavior and an MCMC acceptance function, and use this equivalence to develop a method for obtaining samples from a subjective distribution. In this way we can use the power of MCMC algorithms to explore the knowledge of human learners. The plan of the paper is asfollows. In Section 2, we describe MCMC in general and the Metropolis method and Barker acceptance function in particular. Section 3 describes the experimental task we use to connect human judgments to MCMC. In Section 4, we present an experiment showing that this method can be used to recover trained category distributions from human judgments. Section 5 gives a demonstration of our MCMC methodapplied to recovering natural categories of animal shape. Section 6 summarizes the results and discusses some implications. 1

2 Markov chain Monte Carlo
Models of physical phenomena used by scientists are often expressed in terms of complex probability distributions over different events. Generating samples from these distributions can be an efficient way to determine their properties,indicating which events are assigned high probabilities and providing a way to approximate various statistics of interest. Often, the distributions used in these models are difficult to sample from, being defined over large state spaces or having unknown normalization constants. Consequently, a great deal of research has been devoted to developing sophisticated Monte Carlo algorithms that can be used to...