Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods

TL;DR

This paper compares the long-term variance of sample averages in inhomogeneous Markov chains, providing a method to evaluate and improve Markov chain Monte Carlo algorithms, including proposing a new efficient algorithm.

Contribution

It introduces a comparison framework for asymptotic variances of inhomogeneous Markov chains and applies it to enhance data-augmentation MCMC algorithms, including a novel exact method.

Findings

The main result allows direct comparison of asymptotic variances based on lag-one autocovariance ordering.

The new random refreshment algorithm outperforms some existing MCMC algorithms in asymptotic variance.

The proposed algorithm has comparable computational complexity to existing methods.

Abstract

In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different $\pi$-reversible Markov transition kernels $P$ and $Q$. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels $P_i$ and $Q_i$, $i\in\{0,1\}$, as soon as the kernels of each pair $(P_0,P_1)$ and $(Q_0,Q_1)$ can be ordered in the sense of lag-one autocovariance. As an important application, we use this result for comparing different data-augmentation-type Metropolis-Hastings algorithms. In particular, we compare some pseudo-marginal algorithms and propose a novel exact algorithm, referred to as the random refreshment algorithm, which is more efficient, in terms of asymptotic variance, than the Grouped Independence Metropolis-Hastings algorithm and has a computational complexity that does not exceed that of the Monte Carlo Within Metropolis algorithm.