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 the convergence of Markov chains to equilibrium. We show that this enables the straightforward and transparent derivation of subgeometric convergence bounds for several ‘pseudo-marginal’ sampling algorithms which are popular for carrying out Bayesian inference in the setting of intractable likelihoods, which are necessarily 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 / minorization conditions, and the tools developed allow us to recover and further extend known results as particular cases. As a consequence of our results, we are then able to provide new insights into the practical use of pseudo-marginal algorithms, analysing the effect of averaging in Approximate Bayesian Computation (ABC), the use of products of independent averages, and the complexity trade-offs which arise for particle marginal Metropolis-Hastings (PMMH).
(https://arxiv.org/abs/2112.05605, joint work with Christophe Andrieu, Anthony Lee, and Andi Q. Wang)