Florian Maire et al., COMPARISON OF ASYMPTOTIC VARIANCES OF INHOMOGENEOUS MARKOV CHAINS WITH APPLICATION TO MARKOV CHAIN MONTE CARLO METHODS, Annals of statistics , 42(4), 2014, pp. 1483-1510
In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different .-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 Pi and Qi, i . {0,1}, as soon as the kernels of each pair (P., P.) and (Q., Q.) 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 pseudomarginal 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.