High-dimensional reliability-oriented Shapley effect estimation with Normalizing Flows

2026-06-25

AI Summary

This work addresses the challenge of accurately and efficiently estimating reliability-oriented Shapley effects in the presence of high-dimensional, correlated input variables. The authors propose a novel approach that leverages only failure samples by reconstructing conditional marginal densities to derive a closed-form expression for reliability-oriented Sobol indices. To enable precise modeling of high-dimensional densities, the method integrates normalizing flows, facilitating efficient computation of Shapley effects. Additionally, an error assessment mechanism tailored to failure samples is introduced to quantify estimation uncertainty. Numerical experiments demonstrate that the proposed framework reliably estimates Shapley effects under high-dimensional dependence structures and provides robust uncertainty quantification.

Abstract

This article presents a new estimation scheme for the reliability-oriented Shapley effects when there is a large number of correlated input variables in the model, using a unique sample of failure points. To do so, we first propose a new writing of the reliability-oriented closed Sobol indices involving the marginal densities conditionally to the failure, which may be high-dimensional. Then, we propose to estimate these densities with the available failing samples using Normalizing Flows, powerful tools from generative modeling that enable the estimation of complex high-dimensional densities. In addition, we provide an error estimation procedure relying on the same sample of failing points, which constitutes a new contribution for the estimation of target Shapley effects. Finally, we illustrate our methodology on numerical use-cases, discuss insightful features of our approach and provide prospects for the future.

Problem

• Shapley effect
• reliability
• high-dimensional
• correlated inputs
• failure samples

Innovation

• Shapley effects
• Normalizing Flows
• reliability analysis
• high-dimensional density estimation
• failure samples