In principle, the Bayesian approach to model selection is straightforward. Prior probability distributions are used to describe the uncertainty surrounding all unknowns. After observing the data, the posterior distribution provides a coherent post data summary of the remaining uncertainty which is relevant for model selection. However, the practical implementation of this approach often requires carefully tailored priors and novel posterior calculation methods. In this article, we illustrate some of the fundamental practical issues that arise for two different model selection problems: the variable selection problem for the linear model and the CART model selection problem. Hugh Chipman is Associate Professor of Statistics, Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada; email: hachipman@uwaterloo.ca. Edward I. George is Professor of Statistics, Department of Statistics, The Wharton School of the University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104-6302, U.S.A; email: edgeorge@wharton.upenn.edu. Robert E. McCulloch is Professor of Statistics, Graduate School of Business, University of Chicago, 1101 East 58th Street, Chicago, IL, U.S.A; email: Robert.McCulloch@gsb.uchicago.edu. This work was supported by NSF grant DMS-98.03756 and Texas ARP grant 003658.690.

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