Bayesian Variable Selection as a Complete Graph Model

• Activity Number: 352
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 10:30 AM to 12:20 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Zaili Fang*+ and Inyoung Kim
• Companies: Virginia Tech and Virginia Tech
• Keywords: Additive models ; Bayesian variable selection ; Complete graph ; Ising model

Abstract

In recent years, Ising prior has been introduced to Bayesian variable selections such as SSVS. However, in this report we show that Bayesian variable selection itself can be considered as a Ising model. Based on the linear regression model advocated by Kuo and Mallick (1998), we derive the posterior distribution of the binary random vector $\boldsymbol\gamma$, which demonstrates the form of the Ising model. Therefore, we consider the $p$ dimension binary random variable $\boldsymbol\gamma$ of the Bayesian variable selection problem as a class of stochastic processes on a finite random graph model $G=(V,E)$. The advantage of our method is that we have only one tuning parameter. It can be understood as a parameter that play the role as the temperature parameter of Ising model in statistical physics. With this consideration, algorithm similar to Extended Ensemble Monte Carlo algorithms (Iba, 2001) has been developed which is more efficient than direct sampling. We also extend the linear regression model to additive regression models by using Lancaster and \v{S}alkauskas basis for natural cubic spline (Chib and Greenberg, 2010). The methods are illustrated with simulated and real data.