On the Null Distribution of Bayes Factors for Bayesian Linear Regression

• Activity Number: 29
• Type: Contributed
• Date/Time: Sunday, August 9, 2015 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): QUAN Zhou* and Yongtao Guan
• Companies: Baylor College of Medicine and Baylor College of Medicine
• Keywords: Bayes factor ; Bayesian linear regression ; p-value ; normal-inverse-gamma prior ; weighted sum of Chi-squared random variables

Abstract

We study the null distribution of Bayes factors (BF-null) for Bayesian linear regression model. An asymptotically equivalent approximation to 2log(BF-null) is given, which is a weighted sum of Chi-squared random variables with a shifted mean. This claim holds for the full multiple linear regression model under both the Normal-Inverse-Gamma prior and the Zellner's g-prior. Our result has three immediate impacts. First, it allows us to directly compute p-values for Bayes factors without the need of permutation. Our implementation of a recently published method can quickly compute the p-values with high accuracy. Second, it enables the quantification of how prior affect the BF and the relationship between the BF and the p-value. Power of the BF-based p-value and the p-value of likelihood ratio test is compared. Lastly, we propose a new statistic for measuring evidence, permuted Bayes factor, which may have better performance for variable selection in genome-wide association studies. Simulation and analysis of real whole-genome genotype data-sets are provided to help illustrate our theories.