Hoeffding decomposition of functions of random dependent variables

Abstract

Hoeffding’s functional decomposition is the cornerstone of many post-hoc interpretability methods. It entails decomposing arbitrary functions of mutually independent random variables as a sum of interactions. Many generalizations to dependent covariables have been proposed throughout the years, which rely on finding a set of suitable projectors. This paper characterizes such projectors under hierarchical orthogonality constraints and mild assumptions on the variable’s probabilistic structure. Our approach is deeply rooted in Hilbert space theory, giving intuitive insights on defining, identifying, and separating interactions from the effects due to the variables’ dependence structure. This new decomposition is then leveraged to define a new functional analysis of variance. Toy cases of functions of bivariate Bernoulli and Gaussian random variables are studied.