The general applicability and ease of use of the pseudo-marginal Metropolis--Hastings (PMMH) algorithm, and particle Metropolis--Hastings in particular, makes it a popular method for inference on discretely observed Markovian stochastic processes. The performance of these algorithms and, in the case of particle Metropolis--Hastings, the trade off between improved mixing through increased accuracy of the estimator and the computational cost were investigated independently in two papers, both published in 2015. Each suggested choosing the number of particles so that the variance of the logarithm of the estimator of the posterior at a fixed sensible parameter value is approximately 1. This advice has been widely and successfully adopted. We provide new, remarkably simple upper and lower bounds on the asymptotic variance of PMMH algorithms. The bounds explain how blindly following the 2015 advice can hide serious issues with the algorithm and they strongly suggest an alternative criterion. In most situations our guidelines and those from 2015 closely coincide; however, when the two differ it is safer to follow the new guidance. An extension of one of our bounds shows how the use of correlated proposals can fundamentally shift the properties of pseudo-marginal algorithms, so that asymptotic variances that were infinite under the PMMH kernel become finite.

翻译：伪边际Metropolis-Hastings（PMMH）算法（尤其是粒子Metropolis-Hastings算法）因其普适性与易用性，已成为离散观测马尔可夫随机过程推断的常用方法。2015年发表的两篇独立论文分别研究了此类算法的性能，并针对粒子Metropolis-Hastings算法，探讨了估计器精度提升带来的混合性改善与计算成本之间的权衡关系。两篇论文均建议：应选择粒子数量使得在固定合理参数值处后验估计量对数值的方差约为1。该建议已被广泛采纳且成效显著。本文提出了PMMH算法渐近方差的新上界与下界，这些界限具有惊人的简洁性。这些界限揭示了盲目遵循2015年建议可能掩盖的算法严重问题，并强烈指向一种替代准则。在多数情况下，本文准则与2015年准则高度吻合；但当两者出现分歧时，遵循新准则更为可靠。通过对其中一条界限的扩展研究，我们发现相关提案的使用能够根本性改变伪边际算法的性质：原本在PMMH核下为无穷大的渐近方差可转变为有限值。