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

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.