Product-form estimators: exploiting independence to scale up Monte Carlo

TL;DR

This paper introduces product-form Monte Carlo estimators that leverage independence structures to reduce variance and improve efficiency, combining them with existing methods like importance sampling and pseudo-marginal MCMC.

Contribution

The paper develops a new class of unbiased, consistent, and asymptotically normal estimators exploiting independence, and demonstrates their integration with other Monte Carlo techniques for enhanced performance.

Findings

Estimators achieve minimum variance among broad classes.

Merging with importance sampling and MCMC reduces variance.

Numerical examples confirm improved efficiency.

Abstract

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-Hasting. 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.