Comparing the Efficiency of General State Space Reversible MCMC Algorithms

Abstract We prove new results about comparing the efficiency of general state space Markov chain Monte Carlo algorithms that randomly select a possibly different reversible method at each step (previously known only for finite state spaces). We also provide new, simpler, more accessible proofs of key results, and analyse numerous examples. We provide a full proof of the formula for the asymptotic variance for real-valued functionals on phi φ $\varphi$ -irreducible reversible Markov chains, first introduced by Kipnis and Varadhan (1986, Commun. Math. Phys. 104 , 1–19). Given two Markov kernels P and Q with stationary measure pi π $\pi$ , we say that the Markov kernel P efficiency-dominates the Markov kernel Q if the asymptotic variance with respect to P is at most the asymptotic variance with respect to Q for every real-valued functional f element of upper L squared left parenthesis pi right parenthesis f ∈ L 2 ( π ) $f\in L^2(\pi)$ . Assuming only a basic background in functional analysis, we prove that for two reversible Markov kernels P and Q , P efficiency-dominates Q if and only if the operator script upper Q minus script upper P Q − P $\mathcal{Q}-\mathcal{P}$ , where script upper P P $\mathcal{P}$ is the operator on upper L squared left parenthesis pi right parenthesis L 2 ( π ) $L^2(\pi)$ that maps f right arrow from bar integral f left parenthesis y right parenthesis upper P left parenthesis dot comma normal d y right parenthesis