Comparing the Efficiency of General State Space Reversible MCMC Algorithms

We review and provide new proofs of results used to compare the efficiency of estimates generated by reversible MCMC algorithms on a general state space. We provide a full proof of the formula for the asymptotic variance for real-valued functionals on $\phi$-irreducible reversible Markov chains, first introduced by Kipnis and Varadhan. Given two Markov kernels $P$ and $Q$ with stationary measure $\pi$, 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 $f\in L^2(\pi)$. Assuming only a basic background in functional analysis, we prove that for two $\phi$-irreducible reversible Markov kernels $P$ and $Q$, $P$ efficiency dominates $Q$ if and only if the operator $Q-P$, where $P$ is the operator on $L^2(\pi)$ that maps $f\mapsto\int f(y)P(\cdot,dy)$ and similarly for $Q$, is positive on $L^2(\pi)$, i.e. $\langle f,(Q-P)f\rangle\geq0$ for every $f\in L^2(\pi)$. We use this result to show that reversible antithetic kernels are more efficient than i.i.d. sampling, and that efficiency dominance is a partial ordering on $\phi$-irreducible reversible Markov kernels. We also provide a proof based on that of Tierney that Peskun dominance is a sufficient condition for efficiency dominance for reversible kernels. Using these results, we show that Markov kernels formed by randomly selecting other "component" Markov kernels will always efficiency dominate another Markov kernel formed in this way, as long as the component kernels of the former efficiency dominate those of the latter. These results on the efficiency dominance of combining component kernels generalises the results on the efficiency dominance of combined chains introduced by Neal and Rosenthal from finite state spaces to general state spaces.