Importance sampling and independent Metropolis-Hastings (IMH) are among the fundamental building blocks of Monte Carlo methods. Both require a proposal distribution that globally approximates the target distribution. The Radon-Nikodym derivative of the target distribution relative to the proposal is called the weight function. Under the weak assumption that the weight is unbounded but has a number of finite moments under the proposal distribution, we obtain new results on the approximation error of importance sampling and of the particle independent Metropolis-Hastings algorithm (PIMH), which includes IMH as a special case. For IMH and PIMH, we show that the common random numbers coupling is maximal. Using that coupling we derive bounds on the total variation distance of a PIMH chain to the target distribution. The bounds are sharp with respect to the number of particles and the number of iterations. Our results allow a formal comparison of the finite-time biases of importance sampling and IMH. We further consider bias removal techniques using couplings of PIMH, and provide conditions under which the resulting unbiased estimators have finite moments. We compare the asymptotic efficiency of regular and unbiased importance sampling estimators as the number of particles goes to infinity.

翻译：重要性采样与独立Metropolis-Hastings（IMH）算法是蒙特卡罗方法的基础组成部分。两者均需要一个在全局范围内逼近目标分布的提议分布。目标分布相对于提议分布的Radon-Nikodym导数称为权重函数。在权重函数无界但在提议分布下具有若干有限矩的弱假设下，我们获得了关于重要性采样及粒子独立Metropolis-Hastings算法（PIMH，其包含IMH作为特例）近似误差的新结果。对于IMH与PIMH，我们证明其公共随机数耦合是极大耦合。利用该耦合，我们推导出PIMH链与目标分布之间全变差距离的上界。这些上界在粒子数与迭代次数方面是尖锐的。我们的结果允许对重要性采样与IMH的有限时间偏差进行形式化比较。我们进一步研究了基于PIMH耦合的偏差消除技术，并给出了使所得无偏估计量具有有限矩的条件。我们比较了当粒子数趋于无穷时，常规与无偏重要性采样估计量的渐近效率。