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 knownas weak Poincaré inequalities to bound convergence of Markov chains toequilibrium. We show that this enables the straightforward and transparent derivation of subgeometric convergence bounds for methods such asthe Independent Metropolis–Hastings sampler and pseudo-marginal methodsfor intractable likelihoods, the latter being subgeometric in many practicalsettings. These results rely on novel quantitative comparison theorems between Markov chains. Associated proofs are simpler than those relying ondrift/minorisation conditions and the tools developed allow us to recover andfurther extend known results as particular cases. We are then able to providenew insights into the practical use of pseudo-marginal algorithms, analyse theeffect of averaging in Approximate Bayesian Computation (ABC) and the useof products of independent averages and also to study the case of log-normalweights relevant to particle marginal Metropolis–Hastings (PMMH)