Bayesian explanations have swept through cognitive science over the past two decades, from intuitive physics and causal learning, to perception, motor control and language. Yet people flounder with even the simplest probability questions. What explains this apparent paradox? How can a supposedly Bayesian brain reason so poorly with probabilities? In this paper, we propose a direct and perhaps unexpected answer: that Bayesian brains need not represent or calculate probabilities at all and are, indeed, poorly adapted to do so. Instead, the brain is a Bayesian sampler. Only with infinite samples does a Bayesian sampler conform to the laws of probability; with finite samples it systematically generates classic probabilistic reasoning errors, including the unpacking effect, base-rate neglect, and the conjunction fallacy.

Rational models of cognition typically consider the abstract computational problems posed by the environment, assuming that people are capable of optimally solving those problems. This differs from more traditional formal models of cognition, which focus on the psychological processes responsible for behavior. A basic challenge for rational models is thus explaining how optimal solutions can be approximated by psychological processes.We outline a general strategy for answering this question, namely to explore the psychological plausibility of approximation algorithms developed in computer science and statistics. In particular, we argue that Monte Carlo methods provide a source of "rational process models" that connect optimal solutions to psychological processes. We support this argument through a detailed example, applying this approach to Anderson's (1990Anderson's ( , 1991 Rational Model of Categorization (RMC), which involves a particularly challenging computational problem. Drawing on a connection between the RMC and ideas from nonparametric Bayesian statistics, we propose two alternative algorithms for approximate inference in this model. The algorithms we consider include Gibbs sampling, a procedure appropriate when all stimuli are presented simultaneously, and particle filters, which sequentially approximate the posterior distribution with a small number of samples that are updated as new data become available. Applying these algorithms to several existing datasets shows that a particle filter with a single particle provides a good description of human inferences. Rational Approximations to Category Learning 3Rational approximations to rational models:Alternative algorithms for category learning Rational models of cognition aim to explain human thought and behavior as an optimal solution to the computational problems that are posed by our environment (Anderson, 1990;Chater & Oaksford, 1999;Marr, 1982; Oaksford & Chater, 1998). This approach has been used to model several aspects of cognition, including memory (Anderson, 1990;Shiffrin & Steyvers, 1997), reasoning (Oaksford & Chater, 1994), generalization (Shepard, 1987;Tenenbaum & Griffiths, 2001a), and causal induction (Anderson, 1990;Griffiths & Tenenbaum, 2005). However, executing optimal solutions to these problems can be extemely computationally expensive, a point that is commonly raised as an argument against the validity of rational models (e.g., Gigerenzer & Todd, 1999;Tversky & Kahneman, 1974). This establishes a basic challenge for advocates of rational models of cognition: identifying psychologically plausible mechanisms that would allow the human mind to approximate optimal performance.The question of how rational models of cognition can be approximated by psychologically plausible mechanisms addresses a fundamental issue in cognitive science: bridging levels of analysis. Rational models provide answers to questions posed at Marr's (1982) computational level -questions about the abstract computational problems involved in cognition. Th...