  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.
