The $f$-Sensitivity Index

TL;DR

This paper introduces a versatile $f$-sensitivity index for global sensitivity analysis, applicable to dependent and independent inputs, unifying various existing measures and providing new theoretical insights and estimation methods.

Contribution

It proposes a general $f$-divergence-based sensitivity index that encompasses existing measures and introduces new theoretical properties and estimation techniques.

Findings

The $f$-sensitivity index generalizes many existing sensitivity measures.

New inequalities and properties of the $f$-sensitivity index are established.

Three approximation methods for estimating the index are proposed and compared.

Abstract

This article presents a general multivariate $f$-sensitivity index, rooted in the $f$-divergence between the unconditional and conditional probability measures of a stochastic response, for global sensitivity analysis. Unlike the variance-based Sobol index, the $f$-sensitivity index is applicable to random input following dependent as well as independent probability distributions. Since the class of $f$-divergences supports a wide variety of divergence or distance measures, a plethora of $f$-sensitivity indices are possible, affording diverse choices to sensitivity analysis. Commonly used sensitivity indices or measures, such as mutual information, squared-loss mutual information, and Borgonovo's importance measure, are shown to be special cases of the proposed sensitivity index. New theoretical results, revealing fundamental properties of the $f$-sensitivity index and establishing important inequalities, are presented. Three new approximate methods, depending on how the probability densities of a stochastic response are determined, are proposed to estimate the sensitivity index. Four numerical examples, including a computationally intensive stochastic boundary-value problem, illustrate these methods and explain when one method is more relevant than the others.