Abstract 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.

中文翻译：

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就渐近方差而言，重要性采样校正与可逆 MCMC 的标准平均值

摘要 我们为两个一致马尔可夫链蒙特卡罗 (MCMC) 估计器的渐近方差建立了一个排序标准：一个重要性采样 (IS) 估计器，基于近似可逆链和后续 IS 加权，以及一个标准 MCMC 估计器，基于一个精确的可逆链。本质上，我们通过考虑两个不同的不变概率来放宽 Peskun 类型协方差排序的标准，并获得 IS 的渐近方差与直接 MCMC 的渐近方差的界限，而不是渐近方差的严格排序。简单的例子表明，与 Metropolis-Hastings 和延迟接受 (DA) MCMC 相比，IS 可以具有任意更好或更差的渐近方差。我们的排序意味着，根据（边际）IS 权重的最高值，保证 IS 具有竞争力。我们详细说明了无偏估计量作为辅助变量框架的一部分的标准。我们展示了该标准如何在伪边际 (PM) 和 DA 校正方面暗示 IS 的渐近方差保证，基本上是如果精确和近似似然的比率是有界的。我们还表明，与 PM 和 DA 链相比，IS 链的收敛受无界高方差无偏估计量的影响较小。