Hypothesis generation, sparse categories, and the positive test strategy.

We consider the situation in which a learner must induce the rule that explains an observed set of data but the hypothesis space of possible rules is not explicitly enumerated or identified. The first part of the article demonstrates that as long as hypotheses are sparse (i.e., index less than half of the possible entities in the domain) then a positive test strategy is near optimal. The second part of this article then demonstrates that a preference for sparse hypotheses (a sparsity bias) emerges as a natural consequence of the family resemblance principle; that is, it arises from the requirement that good rules index entities that are more similar to one another than they are to entities that do not satisfy the rule.

Similarity, feature discovery, and the size principle.

In this paper we consider the "size principle" for featural similarity, which states that rare features should be weighted more heavily than common features in people's evaluations of the similarity between two entities. Specifically, it predicts that if a feature is possessed by n objects, the expected weight scales according to a 1/n law. One justification of the size principle emerges from a Bayesian analysis of simple induction problems (Tenenbaum & Griffiths, 2001), and is closely related to work by Shepard (1987) proposing universal laws for inductive generalization. In this article, we (1) show that the size principle can be more generally derived as an expression of a form of representational optimality, and (2) present analyses suggesting that across 11 different data sets in the domains of animals and artifacts, human judgments are in agreement with this law. A number of implications are discussed.