  We review criteria for comparing the efficiency of Markov chain Monte Carlo
(MCMC) methods with respect to the asymptotic variance of estimates of
expectations of functions of state, and show how such criteria can justify ways
of combining improvements to MCMC methods. We say that a chain on a finite
state space with transition matrix $P$ efficiency-dominates one with transition
matrix $Q$ if for every function of state it has lower (or equal) asymptotic
variance. We give elementary proofs of some previous results regarding
efficiency dominance, leading to a self-contained demonstration that a
reversible chain with transition matrix $P$ efficiency-dominates a reversible
chain with transition matrix $Q$ if and only if none of the eigenvalues of
$Q-P$ are negative. This allows us to conclude that modifying a reversible MCMC
method to improve its efficiency will also improve the efficiency of a method
that randomly chooses either this or some other reversible method, and to
conclude that improving the efficiency of a reversible update for one component
of state (as in Gibbs sampling) will improve the overall efficiency of a
reversible method that combines this and other updates. It also explains how
antithetic MCMC can be more efficient than i.i.d. sampling. We also establish
conditions that can guarantee that a method is not efficiency-dominated by any
other method.
