  We establish an ordering criterion for the asymptotic variances of two
consistent Markov chain Monte Carlo (MCMC) estimators: an importance sampling
(IS) estimator, based on an approximate reversible chain and subsequent IS
weighting, and a standard MCMC estimator, based on an exact reversible chain.
Essentially, we relax the criterion of the Peskun type covariance ordering by
considering two different invariant probabilities, and obtain, in place of a
strict ordering of asymptotic variances, a bound of the asymptotic variance of
IS by that of the direct MCMC. Simple examples show that IS can have
arbitrarily better or worse asymptotic variance than Metropolis-Hastings and
delayed-acceptance (DA) MCMC. Our ordering implies that IS is guaranteed to be
competitive up to a factor depending on the supremum of the (marginal) IS
weight. We elaborate upon the criterion in case of unbiased estimators as part
of an auxiliary variable framework. We show how the criterion implies
asymptotic variance guarantees for IS in terms of pseudo-marginal (PM) and DA
corrections, essentially if the ratio of exact and approximate likelihoods is
bounded. We also show that convergence of the IS chain can be less affected by
unbounded high-variance unbiased estimators than PM and DA chains.
