Weak consistency of Markov chain Monte Carlo methods

TL;DR

This paper introduces a new convergence rate for MCMC methods based on diffusion process approximation, providing insights into their behavior and demonstrating the approach with a mixture model and simulations.

Contribution

It presents a novel convergence rate for MCMC methods derived from diffusion process approximation, expanding beyond traditional exact convergence analyses.

Findings

New convergence rate derived from diffusion approximation

Application to a simple mixture model

Numerical simulations illustrating the rate's effect

Abstract

Markov chain Monte Calro methods (MCMC) are commonly used in Bayesian statistics. In the last twenty years, many results have been established for the calculation of the exact convergence rate of MCMC methods. We introduce another rate of convergence for MCMC methods by approximation techniques. This rate can be obtained by the convergence of the Markov chain to a diffusion process. We apply it to a simple mixture model and obtain its convergence rate. Numerical simulations are performed to illustrate the effect of the rate.