  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.
