Ergodicity of Markov chain Monte Carlo with reversible proposal

TL;DR

This paper proves that the mixed preconditioned Crank-Nicolson (MpCN) algorithm exhibits geometric ergodicity for heavy-tailed distributions, enhancing understanding of its convergence properties in Markov Chain Monte Carlo methods.

Contribution

It demonstrates that MpCN, under certain transformations, behaves as a random-walk Metropolis, ensuring geometric ergodicity even with heavy-tailed targets, unlike typical sub-geometric results.

Findings

MpCN is geometrically ergodic for heavy-tailed distributions.

Under transformation, MpCN acts as a random-walk Metropolis.

The analysis extends ergodicity results to a broader class of distributions.

Abstract

We describe ergodic properties of some Metropolis-Hastings (MH) algorithms for heavy-tailed target distributions. The analysis usually falls into sub-geometric ergodicity framework but we prove that the mixed preconditioned Crank-Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that the MpCN algorithm becomes a random-walk Metropolis algorithm under suitable transformation.