  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.
