Local Consistency of Markov Chain Monte Carlo Methods

TL;DR

This paper explores the local asymptotic properties of Markov chain Monte Carlo methods, particularly data augmentation, showing that their empirical distributions converge to the posterior as sample size and runtime grow, offering insights for designing efficient algorithms.

Contribution

It introduces the concept of local efficiency for MCMC methods and demonstrates its application to data augmentation, providing a simpler, more general framework for analyzing algorithm performance.

Findings

Empirical distribution of DA converges to the posterior with increasing data and runtime.

Local properties offer a more practical understanding of MCMC behavior than global properties.

Results facilitate the construction of more efficient MCMC algorithms.

Abstract

In this paper, we introduce the notion of efficiency (consistency) and examine some asymptotic properties of Markov chain Monte Carlo methods. We apply these results to the data augmentation (DA) procedure for independent and identically distributed observations. More precisely, we show that if both the sample size and the running time of the DA procedure tend to infinity the empirical distribution of the DA procedure tends to the posterior distribution. This is a local property of the DA procedure, which may be, in some cases, more helpful than the global properties to describe its behavior. The advantages of using the local properties are the simplicity and the generality of the results. The local properties provide useful insight into the problem of how to construct efficient algorithms.