Importance sampling for Sobol' indices estimation

TL;DR

This paper introduces an importance sampling method for more efficient estimation of Sobol' indices, enabling variance reduction and distributional sensitivity analysis by reweighting samples from auxiliary distributions.

Contribution

It develops a new importance sampling framework that improves Sobol' indices estimation by deriving optimal sampling distributions and supporting sensitivity analysis.

Findings

Optimal sampling distribution minimizes estimator variance.

Reweighting samples enables efficient Sobol' indices estimation.

Framework facilitates distributional sensitivity analysis.

Abstract

We propose a new importance sampling framework for the estimation and analysis of Sobol' indices. We focus on the estimation of the conditional second-moment quantity underlying these indices, which is the most challenging term to estimate. We show that this quantity, originally defined under a reference input distribution, can be estimated from samples drawn under auxiliary distributions by reweighting the model outputs. We derive the optimal sampling distribution that minimises the asymptotic variance of efficient estimators and demonstrate its impact on estimation. Beyond variance reduction, the framework also supports distributional sensitivity analysis through reverse importance sampling.