Geometric ergodicity of trans-dimensional Markov chain Monte Carlo algorithms

TL;DR

This paper establishes conditions under which trans-dimensional MCMC algorithms exhibit geometric convergence, extending existing techniques to certain non-reversible chains and applying the results to Bayesian model selection problems.

Contribution

It extends the theory of geometric ergodicity to some non-reversible trans-dimensional MCMC algorithms using an $L^2$ framework and Markov chain decomposition.

Findings

Geometric convergence guaranteed under mild conditions for certain trans-dimensional chains.

Extension of Markov chain decomposition technique to non-reversible chains.

Application to Bayesian models including probit regression, Gaussian mixtures, and autoregression.

Abstract

This article studies the convergence properties of trans-dimensional MCMC algorithms when the total number of models is finite. It is shown that, for reversible and some non-reversible trans-dimensional Markov chains, under mild conditions, geometric convergence is guaranteed if the Markov chains associated with the within-model moves are geometrically ergodic. This result is proved in an $L^2$ framework using the technique of Markov chain decomposition. While the technique was previously developed for reversible chains, this work extends it to the point that it can be applied to some commonly used non-reversible chains. The theory herein is applied to reversible jump algorithms for three Bayesian models: a probit regression with variable selection, a Gaussian mixture model with unknown number of components, and an autoregression with Laplace errors and unknown model order.