A Closer Look at the Median Probability Model for Bayesian Model Selection

• Activity Number: 111
• Type: Invited
• Date/Time: Monday, August 5, 2013 : 8:30 AM to 10:20 AM
• Sponsor: ASA
• Author(s): Joyee Ghosh*+
• Companies: The University of Iowa
• Keywords: Bayesian variable selection ; Linear regression ; Markov chain Monte Carlo ; Multicollinearity ; Multimodality ; Zellner's g-prior

Abstract

For high-dimensional linear regression, the highest posterior probability model is difficult to estimate, so Bayesian variable selection is often performed with the marginal posterior inclusion probabilities. The use of these can be further motivated by the median probability model (MPM) of Barbieri and Berger. The MPM includes all variables whose posterior marginal inclusion probabilities are greater than or equal to 0.5, and it is the optimal predictive model under certain conditions. In this talk, we first use toy examples with Zellner's g-prior to gain some theoretical insight about the behavior of the posterior distribution over models when the design matrix exhibits high multicollinearity. The results suggest that if there are three or more important, strongly correlated covariates, the MPM could potentially discard all of them. Using simulation studies and real data, we illustrate that several popular priors may be adversely affected by multicollinearity. Our results suggest that a routine examination of the joint inclusion probabilities for correlated covariates can help practitioners to cope with the problem. This is joint work with Andrew Ghattas.