  We consider the problem of approximating the product of $n$ expectations with
respect to a common probability distribution $\mu$. Such products routinely
arise in statistics as values of the likelihood in latent variable models.
Motivated by pseudo-marginal Markov chain Monte Carlo schemes, we focus on
unbiased estimators of such products. The standard approach is to sample $N$
particles from $\mu$ and assign each particle to one of the expectations. This
is wasteful and typically requires the number of particles to grow
quadratically with the number of expectations. We propose an alternative
estimator that approximates each expectation using most of the particles while
preserving unbiasedness. We carefully study its properties, showing that in
latent variable contexts the proposed estimator needs only $\mathcal{O}(n)$
particles to match the performance of the standard approach with
$\mathcal{O}(n^{2})$ particles. We demonstrate the procedure on two latent
variable examples from approximate Bayesian computation and single-cell gene
expression analysis, observing computational gains of the order of the number
of expectations, i.e. data points, $n$.
