Importance sampling for Sobol' indices estimation

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.