We introduce a class of Monte Carlo estimators that aim to overcome the rapid growth of variance with dimension often observed for standard estimators by exploiting the target’s independence structure. We identify the most basic incarnations of these estimators with a class of generalized U-statistics and thus establish their unbiasedness, consistency, and asymptotic normality. Moreover, we show that they obtain the minimum possible variance amongst a broad class of estimators, and we investigate their computational cost and delineate the settings in which they are most efficient. We exemplify the merger of these estimators with other well known Monte Carlo estimators so as to better adapt the latter to the target’s independence structure and improve their performance. We do this via three simple mergers: one with importance sampling, another with importance sampling squared, and a final one with pseudo-marginal Metropolis–Hastings. In all cases, we show that the resulting estimators are well founded and achieve lower variances than their standard counterparts. Lastly, we illustrate the various variance reductions through several examples.

我们引入了一类蒙特卡罗估计器，旨在通过利用目标的独立结构来克服标准估计器经常观察到的维度方差的快速增长。我们用一类广义 U 统计量识别这些估计量的最基本的化身，从而建立它们的无偏性、一致性和渐近正态性。此外，我们表明他们在广泛的估计器类别中获得了最小可能的方差，我们调查了它们的计算成本并描述了它们最有效的设置。我们举例说明了这些估计器与其他众所周知的蒙特卡罗估计器的合并，以便更好地使后者适应目标的独立结构并提高其性能。我们通过三个简单的合并来做到这一点：一个是重要性抽样，另一个是重要性采样的平方，最后一个是伪边际 Metropolis-Hastings。在所有情况下，我们都表明，由此产生的估计量是有根据的，并且比标准估计量的方差更低。最后，我们通过几个例子来说明各种方差减少。