A Direct Approach to Understanding Posterior Consistency of Bayesian Regression Problems

Abstract

Previous approaches to establishing posterior consistency of Bayesian regression problems have used general theorems that involve verifying sufficient conditions for posterior consistency. In this article, we consider a direct approach by computing the posterior density explicitly and evaluating its asymptotic behavior. For this purpose, we deal with a sample size dependent prior based on a truncated regression function with increasing sample size, and evaluate the asymptotic properties of the resulting posterior. Based on a concept called posterior density consistency, we attempt to understand posterior consistency. As an application, we illustrate that the posterior density of an orthogonal semiparametric regression model is consistent.

Keywords:

• Nonparametric regression
• Orthogonality
• Posterior density consistency
• Quadratic form
• Sample size dependent prior

Mathematics Subject Classification:

• Primary 62C10
• Secondary 62G08, 62G20

Acknowledgments

The authors are very grateful to the two referees for constructive comments. Research of Seongbaek Yi was supported by the Pukyong National University Research Foundation in 2006(PS-2006-011). Research of Taeryon Choi was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0063906).