A Double Proposal Normalized Importance Sampling Estimator

Monte Carlo methods are widely used in signal processing for computing integrals of interest. Among Monte Carlo methods, Importance Sampling is a variance reduction technique which consists in sampling from an instrumental distribution and reweighting the samples in order to correct the discrepancy between the target and proposal distributions. When either the target or the proposal distribution is known only up to a constant, the moment of interest can be rewritten as a ratio of two expectations, which can be approximated via self-normalized importance sampling. In this paper we show that it is possible to improve the self-normalized importance sampling estimate by approximating the two expectations in this ratio via two importance distributions. In order to tune them we optimize the variance of the final estimate under a reasonable constraint. Our results are validated via simulations.