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

摘要

In this paper, we study the asymptotic variance of sample path averages forinhomogeneous Markov chains that evolve alternatingly according to twodifferent $\pi$-reversible Markov transition kernels $P$ and $Q$. Morespecifically, our main result allows us to compare directly the asymptoticvariances 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 animportant application, we use this result for comparing differentdata-augmentation-type Metropolis-Hastings algorithms. In particular, wecompare some pseudo-marginal algorithms and propose a novel exact algorithm,referred to as the random refreshment algorithm, which is more efficient, interms of asymptotic variance, than the Grouped Independence Metropolis-Hastingsalgorithm and has a computational complexity that does not exceed that of theMonte Carlo Within Metropolis algorithm.