Comparison of Markov chains via weak Poincaré inequalities with application to pseudo-marginal MCMC

We investigate the use of a certain class of functional inequalities known
as weak Poincaré inequalities to bound convergence of Markov chains to
equilibrium. We show that this enables the straightforward and transparent derivation of subgeometric convergence bounds for methods such as
the Independent Metropolis–Hastings sampler and pseudo-marginal methods
for intractable likelihoods, the latter being subgeometric in many practical
settings. These results rely on novel quantitative comparison theorems between Markov chains. Associated proofs are simpler than those relying on
drift/minorisation conditions and the tools developed allow us to recover and
further extend known results as particular cases. We are then able to provide
new insights into the practical use of pseudo-marginal algorithms, analyse the
effect of averaging in Approximate Bayesian Computation (ABC) and the use
of products of independent averages and also to study the case of log-normal
weights relevant to particle marginal Metropolis–Hastings (PMMH).