Local degeneracy of Markov chain Monte Carlo methods^∗

Graduate School of Engineering Science, Osaka University, Machikaneyama-cho 1-3, Toyonaka-si, 560-0043 Osaka, Japan
kamatani@sigmath.es.osaka-u.ac.jp

Received: 19 January 2012
Revised: 24 May 2013

Abstract

We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes the performances of MCMC procedures are poor and there are great importance for the study of such behavior. In this paper we call degeneracy for a particular type of poor performances. We show some equivalent conditions for degeneracy. As an application, we consider the cumulative probit model. It is well known that the natural data augmentation (DA) procedure does not work well for this model and the so-called parameter-expanded data augmentation (PX-DA) procedure is considered to be a remedy for it. In the sense of degeneracy, the PX-DA procedure is better than the DA procedure. However, when the number of categories is large, both procedures are degenerate and so the PX-DA procedure may not provide good estimate for the posterior distribution.