Models of categorization make different representational assumptions, with categories being represented by prototypes, sets of exemplars, and everything in between. Rational models of categorization justify these representational assumptions in terms of different schemes for estimating probability distributions. However, they do not answer the question of which scheme should be used in representing a given category. We show that existing rational models of categorization are special cases of a statistical model called the hierarchical Dirichlet process, which can be used to automatically infer a representation of the appropriate complexity for a given category. Keywords: rational analysis, categorization, Dirichlet process Rational models of cognition aim to explain human behavior as an optimal solution to the computational problems posed by our environment (Anderson, 1990) . Examining these computational problems provides a deeper understanding of the assumptions behind successful models of human cognition, and can lead to new models. In this paper, we pursue a rational analysis of category learning: inferring the structure of categories from a set of stimuli labeled as belonging to those categories. The knowledge acquired through this process can ultimately be used to make decisions about how to categorize new stimuli. Existing rational analyses of category learning (Anderson, 1990; Ashby & Alfonso-Reese, 1995; Rosseel, 2002) agree that the computational problem involved is one of density estimation: determining the probability distributions over stimuli associated with different category labels. Viewing category learning as density estimation helps to clarify the assumptions behind the two main classes of psychological models: exemplar models and prototype models. Exemplar models assume that a category is represented by a set of stored exemplars, and categorization involves comparing new stimuli to the set of exemplars in each category (e.g., Medin & Schaffer, 1978; Nosofsky, 1986) . Prototype models assume that a category is associated with a single prototype and categorization involves comparing new stimuli to these prototypes (e.g., Reed, 1972) . These approaches to category learning correspond to different strategies for density estimation, being nonparametric and parametric density estimation respectively (Ashby & Alfonso-Reese, 1995) . Despite providing insight into the assumptions behind models of categorization, existing rational analyses of category learning leave a number of questions open. In particular, many categorization experiments have explored whether people represent categories with exemplars or prototypes. One desideratum for a rational account of category learning might be that it can indicate when a learner should choose to use one of these forms of representation over the other. The greater flexibility of nonparametric density estimation has motivated the claim that exemplar models are to be preferred as rational models of category learning (Nosofsky, 1998) . However, nonparametric and parametric methods have different advantages and disadvantages: the greater flexibility of nonparametric methods comes at a cost of requiring more data to estimate a distribution. The decision as to which representation scheme to use should be determined by the stimuli presented to the learner, and existing rational analyses do not indicate how this decision should be made (although a similar argument is made by Briscoe & Feldman, 2006) . The question of how to represent categories is complicated by the fact that prototype and exemplar models are not the only options. A number of models have recently explored possibilities between these extremes, representing categories using clusters of several exemplars (Anderson, 1990; Vanpaemel, Storms, & Ons, 2005; Rosseel, 2002; Love, Medin, & Gureckis, 2004 ). The range of representations possible in these models emphasizes the importance of being able to identify an appropriate representation for a category from the stimuli themselves: with more options for the representation of categories, it becomes more important to be able to say which option a learner should choose. Our goal in this paper is to build on previous rational analyses of category learning to provide not just a unifying framework which can be used to understand the assumptions behind existing models of categorization, but a unifying model of which these models are special cases. This model goes beyond previous unifying models of category learning (e.g., Rosseel, 2002; Vanpaemel et al., 2005) by providing a rational solution to the question of which representation should be chosen based purely on the structure of a category. These results are achieved by identifying connections between models of human category learning and ideas from nonparametric Bayesian statistics. In particular, we show that all of the models mentioned above can be viewed as variants of a stochastic
REF unify exemplar and prototype categorization through the hierarchical Dirichlet process to model the transition from prototypes to exemplars as more data is collected.
Unifying Rational Models of Categorization via the Hierarchical Dirichlet Process
Mathematics