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

We introduce a novel class of Monte Carlo estimators for product-form targets that aim to overcome the rapid growth of variance with dimension often observed for standard estimators. We establish their unbiasedness, consistency, and asymptotic normality. We show that they achieve lower variances than their conventional counterparts given the same number of samples drawn from the target, investigate the gap in variance via several examples, and identify the situations in which the difference is most, and least, pronounced. We also study the estimators' computational cost and investigate the settings in which they are most efficient. We illustrate their utility beyond the product-form setting by giving two simple extensions (one to targets that are mixtures of product-form distributions and another to targets that are absolutely continuous with respect to a product-form distribution) and conclude by discussing further possible uses.