On the convergence of the Metropolis-Hastings Markov chains

TL;DR

This paper establishes conditions under which Metropolis-Hastings Markov chains converge to the target distribution, demonstrating that proposal density positivity and reversibility ensure convergence in total variation distance.

Contribution

It provides a standalone proof that reversibility combined with kernel positivity guarantees convergence of Metropolis-Hastings chains.

Findings

Positivity of the proposal density ensures convergence.

Reversibility plus kernel positivity imply convergence.

Convergence occurs in total variation distance for any initial density.

Abstract

In this paper we study Markov chains associated with the Metropolis-Hastings algorithm. We consider conditions under which the sequence of the successive densities of such a chain converges to the target density according to the total variation distance for any choice of the initial density. In particular we prove that the positiveness of the proposal density is enough for the chain to converge. The content of this work basically presents a stand alone proof that the reversibility along with the kernel positivity imply the convergence.