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 all
ows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels $P_i$ and $Q_i$, $iin{0,1}$, 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.
