Bayesian Modeling with Prior Information on the Strength of Regression

• Activity Number: 598
• Type: Topic Contributed
• Date/Time: Thursday, August 2, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Agniva Som*+ and Chris Hans and Steven MacEachern
• Companies: The Ohio State University and The Ohio State University and The Ohio State University
• Keywords: Population R-squared ; Hyperspherical coordinates ; Model Selection ; Markov Chain Monte Carlo ; Generalized Inverted Beta distribution

Abstract

The most commonly-used prior distributions for Bayesian regression models typically assume that coefficients are a priori independent or induce dependence via the empirical design matrix. While these standard priors (and recently-refined versions of them) may exhibit desirable behavior with respect to targeted inferential goals, we should not expect them to distribute probability throughout the entire parameter space in a way that is consistent with all of our prior beliefs. We describe a new class of priors that directly incorporates information about the strength of the linear regression relationship. We compare the Bayesian model uncertainty properties of our priors with those of standard priors, highlighting the consequences of inappropriately ignoring prior information when available, and those of unintentionally incorporating strong prior information when it does not exist. We describe MCMC algorithms that scale well with model size and require minimal storage by using a fixed-dimensional parameterization across models of different sizes. We discuss several strategies for improving MCMC output-based estimation using the structure of the posterior.