Latin hypercube sampling to reduce the variance in Monte Carlo integration: a practical guide and an application to gravitational-wave astronomy

Abstract

In applied statistics, likelihoods involving high-dimensional integrals are often approximated via Monte Carlo summation. Controlling the resulting numerical variance is crucial for reliable inference. We provide a practical introduction to Latin hypercube sampling (LHS) and its extension based on orthogonal arrays (OA-LHS) as strategies to reduce this variance by stratifying the sampling distribution. LHS asymptotically removes variance contributions from individual coordinates, while strength-2 OA-LHS also removes those from pairwise interactions, yielding orders-of-magnitude improvements over plain Monte Carlo estimators for suitable integrands. As a case study, we investigate their application to the selection function entering hierarchical inference of gravitational-wave populations. At approximately one million injections, we find only a marginal reduction in variance, with low statistical significance. Nevertheless, the computational cost of constructing these designs is negligible relative to evaluating gravitational-wave injections, making even marginal gains worth considering for future applications. This practical introduction to LHS and OA-LHS offers a starting point for other applications in physics and astronomy involving computationally expensive Monte Carlo calculations.