Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:130

Article

Franks, Jordan^1,2  Vihola, Matti¹ 

[1] Univ Jyvaskyla, Dept Math & Stat, POB 35, FI-40014 Jyvaskyla, Finland

[2] Newcastle Univ, Sch Math Stat & Phys, Newcastle Upon Tyne NE1 7RU, Tyne & Wear, England

关键词: Asymptotic variance;    Delayed-acceptance;    Importance sampling;    Markov chain Monte Carlo;    Pseudo-marginal algorithm;    Unbiased estimator;   

DOI  :  10.1016/j.spa.2020.05.006

来源: Elsevier

【 摘 要 】

We establish an ordering criterion for the asymptotic variances of two consistent Markov chain Monte Carlo (MCMC) estimators: an importance sampling (IS) estimator, based on an approximate reversible chain and subsequent IS weighting, and a standard MCMC estimator, based on an exact reversible chain. Essentially, we relax the criterion of the Peskun type covariance ordering by considering two different invariant probabilities, and obtain, in place of a strict ordering of asymptotic variances, a bound of the asymptotic variance of IS by that of the direct MCMC. Simple examples show that IS can have arbitrarily better or worse asymptotic variance than Metropolis-Hastings and delayed-acceptance (DA) MCMC. Our ordering implies that IS is guaranteed to be competitive up to a factor depending on the supremum of the (marginal) IS weight. We elaborate upon the criterion in case of unbiased estimators as part of an auxiliary variable framework. We show how the criterion implies asymptotic variance guarantees for IS in terms of pseudo-marginal (PM) and DA corrections, essentially if the ratio of exact and approximate likelihoods is bounded. We also show that convergence of the IS chain can be less affected by unbounded high-variance unbiased estimators than PM and DA chains. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free