Variance Analysis of Multiple Importance Sampling Schemes

TL;DR

This paper extends variance analysis of multiple importance sampling schemes to more than two proposals, providing new proofs for variance relations and guiding practitioners in choosing optimal schemes.

Contribution

It generalizes previous variance analysis results for MIS schemes beyond two proposals, offering novel proofs and broader applicability.

Findings

Balanced heuristic estimator often has lower variance

Variance relations depend on the number of proposals

Guidelines for selecting MIS schemes based on variance

Abstract

Multiple importance sampling (MIS) is an increasingly used methodology where several proposal densities are used to approximate integrals, generally involving target probability density functions. The use of several proposals allows for a large variety of sampling and weighting schemes. Then, the practitioner must choose a given scheme, i.e., sampling mechanism and weighting function. A variance analysis has been proposed in Elvira et al (2019, Statistical Science 34, 129-155), showing the superiority of the balanced heuristic estimator with respect to other competing schemes in some scenarios. However, some of their results are valid only for two proposals. In this paper, we extend and generalize these results, providing novel proofs that allow to determine the variance relations among MIS schemes.