COMPARISON OF MARKOV CHAINS VIA WEAK POINCAR? INEQUALITIES WITH APPLICATION TO PSEUDO-MARGINAL MCMC

• 成果类型: Article
• 署名作者: Andrieu, Christophe; Lee, Anthony; Power, Sam; Wang, Andi Q.
• 署名单位: University of Bristol
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2241
• 发表日期: 2022
• 页码: 3592-3618
• 关键词: CONVERGENCE-RATES; averages

摘要

We investigate the use of a certain class of functional inequalities known as weak Poincare inequalities to bound convergence of Markov chains to equilibrium. We show that this enables the straightforward and transpar-ent 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 be-tween 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).