**W**hile I was not involved in any way, or even aware of this research, Anthony Lee, Simone Tiberi, and Giacomo Zanella have an incoming paper in Biometrika, and which was partly written while all three authors were at the University of Warwick. The purpose is to design an efficient manner to approximate the product of n unidimensional expectations (or integrals) all computed against the same reference density. Which is not a real constraint. A neat remark that motivates the method in the paper is that an improved estimator can be connected with the permanent of the n x N matrix A made of the values of the n functions computed at N different simulations from the reference density. And involves N!/ (N-n)! terms rather than N to the power n. Since it is NP-hard to compute, a manageable alternative uses random draws from constrained permutations that are reasonably easy to simulate. Especially since, given that the estimator recycles most of the particles, it requires a much smaller version of N. Essentially N=O(n) with this scenario, instead of O(n²) with the basic Monte Carlo solution, towards a similar variance.

This framework offers many applications in latent variable models, including pseudo-marginal MCMC, of course, but also for ABC since the ABC posterior based on getting each simulated observation close enough from the corresponding actual observation fits this pattern (albeit the dependence on the chosen ordering of the data is an issue that can make the example somewhat artificial).