Asymptotic Performance of Bayesian Variable Selection Based on G-Priors in Normal Linear Regression Models

• Activity Number: 130
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Minerva Mukhopadhyay*+
• Companies: Indian Statistical Institute
• Keywords: $g$ prior ; loss efficiency ; model selection consistency ; variable selection

Abstract

We consider Bayesian variable selection in normal linear regression models based on Zellner's $g$-prior -- a very popular and conventional choice of prior for such problems. We study theoretical properties of this method when the sample size $n$ grows and consider both the cases when the number of regressors is fixed and when it grows with $n$. Conditions are found under which this method has desirable asymptotic properties. We first consider the situation where the true model is not in the model space and prove that the method is consistent in an appropriate sense and ``loss efficient" (in the sense of Li, K.C., 1987 and Shao, J., 1997). We then consider the case when the true model is in the model space and prove that the posterior probability of the true model goes to one as $n$ goes to infinity. ``Loss efficiency" is also proved in this situation. We give explicit conditions on the rate of growth of $g$ possibly depending on that of $p$ as $n$ grows for our results to hold and this helps in making recommendations for the choice of $g$.

(a joint work with Tapas Samanta and Arijit Chakrabarti)