Hoeffding decomposition of black-box models with dependent inputs

Abstract

Performing an additive decomposition of arbitrary functions of random elements is paramount for global sensitivity analysis and, therefore, the interpretation of black-box models. The well-known seminal work of Hoeffding characterized the summands in such a decomposition in the particular case of mutually independent inputs. Going beyond the framework of independent inputs has been an ongoing challenge in the literature. Existing solutions have so far required constraining assumptions or suffer from a lack of interpretability. In this paper, we generalize Hoeffding's decomposition for dependent inputs under very mild conditions. For that purpose, we propose a novel framework to handle dependencies based on probability theory, functional analysis, and combinatorics. It allows for characterizing two reasonable assumptions on the dependence structure of the inputs: non-perfect functional dependence and non-degenerate stochastic dependence. We then show that any square-integrable, real-valued function of random elements respecting these two assumptions can be uniquely additively decomposed and offer a characterization of the summands using oblique projections. We then introduce and discuss the theoretical properties and practical benefits of the sensitivity indices that ensue from this decomposition. Finally, the decomposition is analytically illustrated on bivariate functions of Bernoulli inputs.