Importance sampling for Sobol' indices estimation

TL;DR

This paper introduces a new importance sampling framework for estimating Sobol' indices, focusing on the challenging conditional second-moment η_u = $\mathbb{E}[m^2(X_u)]$ and showing how sampling under an auxiliary distribution q, via Z_u, can reduce estimator variance. It derives the optimal sequential construction of q (first the conditional, then the marginal) and demonstrates substantial variance reductions, including a theoretical zero-variance case in a deterministic setting. The framework also enables distributional sensitivity analysis through reverse importance sampling, allowing reuse of a single dataset to evaluate Sobol' indices under perturbed input distributions, illustrated on a g-function benchmark and a real physical IGLOO2D model. Together, these results enhance efficiency in given-data Sobol' estimation and broaden sensitivity analysis capabilities to distributional changes with minimal additional model evaluations.

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.

Figures (5)

• Figure 1: Left: optimal sampling density $q^*$. Right: relative dispersion of the estimator decreases as we move from $p$ to the optimal $q^*$.
• Figure 2: Asymptotic variance $\sigma_{\mathrm{opt}}^2(\alpha, \beta)$ over the Beta parameter grid (symmetric case) displayed on a logarithmic scale.
• Figure 3: Samples generated via rejection sampling following $q^*$.
• Figure 4: Dispersion of $\widehat{\eta}_{\{1,2\}}$ under $p$ and the learned $\widehat{q}_\theta$.
• Figure 5: Effect of marginal distribution changes on $\eta_4$: each subplot shows how $\eta_4$ evolves as the Beta parameters of one input $X_j$ are varied, while others are kept uniform. All curves are computed from the same baseline sample ($N = 2500$).

Theorems & Definitions (4)

• Lemma 3.1
• Proposition 3.2
• Lemma 3.3
• Proof 1