Journal of Mathematical Psychology 44, 92107 (2000) Bayesian Model Selection and Model Averaging Larry Wasserman Carnegie Mellon University This paper reviews the Bayesian approach to model selection and model averaging. In this review, I emphasize objective Bayesian methods based on noninformative priors. I will also discuss implementation details, approxima- tions, and relationships to other methods.  2000 Academic Press Key Words: AIC; Bayes factors; BIC; consistency; default Bayes methods; Markov chain Monte Carlo. 1. INTRODUCTION Suppose we are analyzing data and we believe that the data arise from one of a set of possible models M 1 , ..., M k . In this paper, a model will refer to a set of prob- ability distributions. For example, suppose the data consist of a normally dis- tributed outcome Y and a covariate X and that two possibilities are entertained. The first possibility is that Y is unrelated to X and the second possibility is that Y is linearly related to X. Then M 1 consists of the distributions for which YtN( +, _ 2 ) and M 2 consists of the distributions for which YtN( ; 0 +; 1 X, { 2 ). This is a simple example with only two models. There could be many models under consideration and each could be very complicated. Of course, the models are often based on sub- ject matter theories. A referee suggested the following example. If we are studying response times, then under a serial process structure we could be led to a gamma distribution while under a diffusion model of decision processes we might be led to an inverse Gaussian distribution. Thus M 1 consists of the gamma distributions and M 2 consists of inverse Gaussian distributions and we have been led to the models by way of two substantive theories. Model selection refers to the problem of using the data to select one model from the list of candidate models M 1 , ..., M k . Model averaging refers to the process of estimating some quantity under each model M j and then averaging the estimates according to how likely each model is. For example, we could use model M j doi:10.1006jmps.1999.1278, available online at http:www.idealibrary.com on 92 0022-249600 35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved. I am grateful to the organizers of the workshop for inviting me to speak and for financial support. I also thank Fulvio De Santis, Rob Kass, Richard Golden, Herbie Lee, Michael Browne, Thomas Wallsten, In Jae Myung, Malcolm Forster, and Isabella Verdinelli and an anonymous referee for provid- ing feedback on an earlier version of this paper. Research was supported by NIH Grant RO1-CA54852 and NSF Grants DMS-9303557 and DMS-9357646. Correspondence and reprint requests should be sent to Larry Wasserman, Department of Statistics, Bakerhall, Carnegie Mellon University, Pittsburgh, PA 15213.