Structuring Dependence in Regression: Spherical Symmetry and Variable Selection

• Activity Number: 230
• Type: Topic Contributed
• Date/Time: Monday, August 5, 2013 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Christopher Hans*+ and Steven MacEachern and Agniva Som
• Companies: Ohio State University and Ohio State University and Ohio State University
• Keywords: objective Bayes ; model uncertainty ; variable selection ; shrinkage ; regularization ; subjective Bayes

Abstract

The most commonly-used priors for linear regression models assume coefficients are a priori independent or induce dependence via the empirical design matrix. While these standard priors may exhibit desirable behavior with respect to targeted inferential goals, we should not expect them to distribute probability throughout the entire space in a way that is consistent with all of one's prior beliefs. Examination reveals that when we focus on the strength of the regression relationship, standard priors place nearly all of their mass in regions of parameter space that are inconsistent with any reasonable prior belief. We describe a new class of priors for linear regression models that remedies these issues. We construct the priors with the strength of the regression relationship as the primitive, highlighting the roles of spherical symmetric and exchangeability. We compare the Bayesian model uncertainty properties of our priors with those of standard priors, highlighting the consequences of inappropriately ignoring prior information when it is indeed available, and highlighting the consequences of unintentionally incorporating strong prior information when it does not exist.