Global sensitivity analysis via optimal transport

We examine the construction of variable importance measures for multivariate responses using the theory of optimal transport. We start with the classical optimal transport formulation. We show that the resulting sensitivity indices are well-defined under input dependence, are equal to zero under statistical independence, and are maximal under fully functional dependence. Also, they satisfy a continuity property for information refinements. We show that the new indices encompass Wagner's variance-based sensitivity measures. Moreover, they provide deeper insights into the effect of an input's uncertainty, quantifying its impact on the output mean, variance, and higher-order moments. We then consider the entropic formulation of the optimal transport problem and show that the resulting global sensitivity measures satisfy the same properties, with the exception that, under statistical independence, they are minimal but not necessarily equal to zero. We prove the consistency of a given-data estimation strategy and test the feasibility of algorithmic implementations based on alternative optimal transport solvers. Application to the assemble-to-order simulator reveals a significant difference in the key drivers of uncertainty between the case in which the quantity of interest is profit (univariate) or inventory (multivariate). The new importance measures contribute to meeting the increasing demand for methods that make black-box models more transparent to analysts and decision-makers.