Sharp inequalities between variance-based dependent sensitivity indices and Shapley effects: upper-bounds

Abstract

For models evaluated at a random set of independent variables, the variance-based Shapley effects range between Sobol' indices, and the corresponding total indices admit derivative-based upper-bounds. Such relationships fail when the inputs are non-independent. This study investigates a general inequality link between the variance-based dependent sensitivity indices, recently introduced by us, and the variance-based Shapley effects for models with dependent inpus. It turns out that Shapley effects range between the main and total dependent sensitivity indices, and such indices are computationally more attractive. Moreover, different upper-bounds of such indices are provided so as to ease the identification of non-relevant inputs in higher dimensions as well as to obtain sometimes practical estimates of total dependent indices. Some of such bounds rely on the traditional gradients, while others rely on generalized sensitivity indices using dependency models.