Relation between the Rational Model and the Context Model of Categorization

A formal proof is pnnided that Anderson's {1990) rational model of categorization generalizes the Medin and Schaffer {1978) context model. According to the context model, people represent categ<>ries by storing individual exemplars in memoiy. According to the rational mode!, people represent categories in terms of midtiple exemplarclusters or prototypes. In both models, a multiplicative rule is used to compute the similarity of an item to the underlying category representations. In certain special cases, each multiple prototype in the rational model corresponds to an individual exemplar, and in these cases the rational model reduces to the context model. Preliminary quantitative comparisons between the models are illustrated to test whether the multiple-prototype view adds significant explanatory power over the pure exemplar view. Anderson (1990) recently proposed an important new model of categorization known as the rational modei The purpose of the present note is to consider formal relations between the rational model and Medin and Schaffer's (1978) well-known context model of categorization. The main point is to prove that a slightly modified version of Anderson's (1990) rational model generalizes the context model. Relations between the rational model and the fuzzy logical model of perception (Massaro, 1987; Massaro & Friedman, 1990; Oden & Massaro, 1978) are also discussed. The proof that the rational model generalizes the context model has important implications for future research strategies aimed at comparing these two models. For example, Anderson (1990) demonstrated that the rational model could account for numerous important categoAddress correspondence to: Robert Nosofsky, Department of Psychology. Indiana University, Bloomington, IN 47405; e-mail: nosofskyf". iics-indiana-etiu. rization phenomena, such as prototype effects, effects of similarity to specific training exemplars, sensitivity to correlated features, and probability matching behavior. However, the context model has been demonstrated previously to account for all these major phenomena (Busemeyer, Dewey, & Medin, 1984; Estes, 1986; Medin & Schaffer, 1978; Medin, Altom, Edelson, & Freko, 1982; Nosofsky, 1988a, 1990). In light of the finding that the rational model generalizes the context model, the rational model's success in the aforementioned domains is unsurprising: By necessity, a more general model must account for data at least as well as its special cases. The critical issue is whether the extra free parameters accorded the rational model lead to clear improvements in explanatory power over its special case, the context model. Otherwise, on grounds of parsimony, the simpler context model is to be preferred. I illustrate some preliminary explorations of this issue in this note. AN INTUITIVE DESCRIPTION OF THE RATIONAL MODEL, THE CONTEXT MODEL, AND THEIR FORMAL RELATION According to the context model, people store the individual training exemplars of categories in memory, and make classification decisions on the basis of the summed similarity of items to the exemplars of the alternative categories. Overall similarity between exemplars is computed by using a multiplicativecombination rule, in which similarities along individual component dimensions are multiplied together. In Anderson's (1990) rational model, individual exemplars become grouped together into clusters during the learning process. The probability that a particular exemplar Is grouped into a cluster depends on how similar the exemplar is to the cluster's central tendency, and on the prior probability that items are grouped into the cluster. This prior probability is determined jointly by the current size of each cluster and the value of a coupling parameter, which is a free parameter in the model. The model also has mechanisms for computing the probability with which membership in each cluster signals a given category label. According to the model, when classification decisions are to be made, the observer computes the similarity of the item to the central tendency of each cluster, and sums these similarities weighted by the respective category-label probabilities of each cluster. Similarity to the central tendency of each cluster is computed by using a multiplicative-combination rule that is isomorphic to the one used in the context model. The key intuition is that when the value of the coupling parameter is zero, each individual exemplar forms its own cluster, as in the context model. In this special case, the rational model reduces to the context model. (Technically, as explained later, it is a slightly modified version of the rational model that reduces to the context model when the coupling parameter is set at zero.) The rational model also generalizes the context model by allowing for similarities between the category labels that are assigned to stimuli. However, the major conceptual generalization that is involved is the clustering idea that results from the action of the coupling parameter. The clustering idea is interesting, because it provides an elegant formalization of the hypothesis that people store multiple prototypes in memory. In much previous work, Medin and his associates (Medin, Dewey, & Murphy, 1983; Medin & Schaffer. 1978; Medin & Smith, 1981) and Nosofsky (1987, 1988b, 1991b) demonstrated that the exemplar-based context model consistently outperforms prototype models in quantitatively predicting classification performance. A natural idea is that instead of literally storing 416 Copyrighi '", 1991 Arnefican Psyi^hological Society VOL. 2, NO. 6, NOVEMBER 1991 PSYCHOLOGICAL SCIENCE each individual exemplar as a unique memory trace, and instead of simply storing a single category prototype, the cognitive system might do something intermediate, namely store multiple prototypes to represent a category. What was lacking in previous work, however, was a principled model for predicting which multiple prototypes are formed. The rational model provides such an account, in that the multiple clusters act as multiple prototypes. Another special case of the rational model is also of interest. When the value of the coupling parameter is unity, and there is zero similarity among category labels, exemplars with a given category label are all grouped together in one grand cluster. In other words, the clusters that are formed correspond to the experimentally defined categories. In this case, the rational model takes the form of a multiplicative-similarity prototype model (Estes, 1986; Golden & Rumelhart, 1989; Medin, Altom, & Murphy, 1984; Nosofsky, in press), important examples of which are Massaro's (1987) fuzzy logical model of perception and Gluck and Bower's (1988) (nonconfigural) adaptive network models. Details are provided by Nosofsky (1991a, in press).