Comparison of Markov chains via weak Poincaré inequalities with application to pseudo-marginal MCMC
研究用弱庞加莱不等式来界定马尔可夫链收敛到平稳分布的速度，为独立Metropolis-Hastings采样器和伪边际方法等提供了次几何收敛界的简单推导，并分析了近似贝叶斯计算和粒子边际Metropolis-Hastings中的实际应用。
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).