Generalizing Generalization1 Generalization, Similarity, and Bayesian Inference

Shepard's theoretical analysis of generalization, originally formulated only for the ideal case of encountering a single consequential stimulus that can be represented as a point in a continuous metric space, is here recast in a more general Bayesian framework. This formulation naturally extends to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure. Our framework also subsumes a version of Tversky's set-theoretic models of similarity, which are conventionally thought of as the primary alternative to Shepard's approach. This uniication not only allows us to draw deep parallels between the set-theoretic and spatial approaches, but also to signiicantly advance the explanatory power of set-theoretic models. Long abstract Shepard has argued that a universal law should govern generalization across diierent domains of perception and cognition, as well as across organisms from diierent species or even diierent planets. Starting with some basic assumptions about natural kinds, he derived an exponential decay function as the form of the universal generalization gradient, which accords strikingly well with a wide range of empirical data. However, his original formulation applied only to the ideal case of generalization from a single encountered stimulus to a single novel stimulus, and for stimuli that can be represented as points in a continuous metric psychological space. Here we recast Shepard's theory in a more general Bayesian framework and show how this naturally extends his approach to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure. Our framework also subsumes a version of Tversky's set-theoretic models of similarity, which is conventionally thought of as the primary alternative to Shepard's continuous metric space model of similarity and generalization. This uniication allows us not only to draw deep parallels between the set-theoretic and spatial approaches, but also to signiicantly advance the explanatory power of set-theoretic models.