## 1. Introduction

The ability to make sensible inductive inferences is one of the most important capabilities of an intelligent entity. The capacity to go beyond the data and make generalizations that can hold for future observations and events is extremely useful, and it is of interest not only to psychologists but also to philosophers (e.g., Goodman, 1955) and researchers with an interest in formal theories of learning (e.g., Solomonoff, 1964). From a psychological perspective, experimental research dating back to Pavlov (1927) demonstrates the tendency of organisms to generalize from one stimulus to another, with learned contingencies being applied to novel but similar stimuli. Critically, in many cases these generalizations do not involve a failure of discrimination. Stimulus generalization is better characterized as a form of inductive inference than of perceptual failure, and indeed the two have a somewhat different formal character (Ennis, 1988). As Shepard (1987, p. 1322) notes, ‘‘we generalize from one situation to another not because we cannot tell the difference between the two situations but because we judge that they are likely to belong to a set of situations having the same consequence.’’

One of the best-known analyses of inductive generalization is Shepard's (1987) exponential law, which emerges from a Bayesian analysis of an idealized single-point generalization problem. In this problem, the learner is presented with a single item known to belong to some target category and the learner is asked to judge the probability that a novel item belongs to the same category. Shepard's analysis correctly predicts the empirical tendency for these generalization probabilities to decay exponentially as a function of distance in a psychological space (this decay function is called a *generalization gradient*). The exponential generalization function is treated as a basic building block for a number of successful theories of categorization and concept learning (e.g., Kruschke, 1992; Love, Medin, & Gureckis, 2004; Nosofsky, 1984; Tenenbaum & Griffiths, 2001a) that seek to explain how people learn a category from multiple known category members. The analysis by Tenenbaum and Griffiths (2001a), in particular, is notable for adopting much the same probabilistic formalism as Shepard's original approach, while extending it to handle multiple observations and cases where spatial representations may not be appropriate (see also Russell, 1986). In a related line of work, other researchers have examined much the same issue using ‘‘property induction’’ problems, leading to the development of the similarity-coverage model (Osherson, Smith, Wilkie, Lopez, & Shafir, 1990), as well as feature-based connectionist models (Sloman, 1993) and a range of other Bayesian approaches (Heit, 1998; Kemp & Tenenbaum, 2009; Sanjana & Tenenbaum, 2003).

In this article we investigate the implicit ‘‘sampling’’ models underlying inductive generalizations. We begin by discussing the ideas behind ‘‘strong sampling’’ and ‘‘weak sampling’’ (Shepard, 1987; Tenenbaum & Griffiths, 2001a) and by developing an extension to the Bayesian generalization model that incorporates both of these as special cases of a more general family of sampling schemes. We then present two experiments designed to test whether people's generalizations are consistent with the model, and more specifically, to allow us to determine what sampling assumptions are involved. These experiments are designed so that we are able to look both at the overall tendencies that people display but also so that we can infer the sampling models used by each individual participant. Our main findings are that there are clear individual differences in the mixture between strong and weak sampling used by different people, and that these mixtures are sensitive to the patterns of observed data. We conclude with a discussion of the psychological meaning of mixing strong and weak sampling, and of possible extensions of our modeling approach to richer problems of inductive generalization.