From a Conditional Lindley's Paradox to Poly-Hyper-G Priors

• Activity Number: 432
• Type: Invited
• Date/Time: Wednesday, August 6, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Christopher Hans*+ and Agniva Som and Steven N. MacEachern
• Companies: Ohio State University and Ohio State University and Ohio State University
• Keywords: Bayesian regression ; variable selection ; model uncertainty ; prior distribution ; shrinkage

Abstract

Mixtures of g-priors have gained traction as a default choice of prior in Bayesian regression settings. The motivation for these priors, exemplified by the hyper-g prior of Liang et al. (2008), usually focuses on properties of model comparison and variable selection. Standard mixtures of g-priors mix over a single, common scale parameter that shrinks all regression coefficients in the same manner. In this paper we focus on the effect of this mono-shrinkage and show that the hyper-g prior suffers from a "conditional Lindley's paradox" that results in undesirable performance when one (or several) coefficients are large and others are small. We propose identifying groups of coefficients within which mono-shrinkage seems appropriate and employing independent hyper-g priors across these blocks. We investigate the properties of these poly-hyper-g priors vis-a-vis the hyper-g prior and provide conditions under which the conditional Lindley's paradox can be resolved. We demonstrate the practical effectiveness of our methods by performing an analysis of a dataset and highlighting the qualitative differences in inference and prediction achieved in comparison with other common priors.