Multivariate Bernoulli Hoeffding Decomposition: From Theory to Sensitivity Analysis

Abstract

Understanding the behavior of predictive models with random inputs can be achieved through functional decompositions into sub-models that capture interpretable effects of input groups. Building on recent advances in uncertainty quantification, the existence and uniqueness of a generalized Hoeffding decomposition have been established for correlated input variables, using oblique projections onto suitable functional subspaces. This work focuses on the case of Bernoulli inputs and provides a complete analytical characterization of the decomposition. We show that, in this discrete setting, the associated subspaces are one-dimensional and that the decomposition admits a closed-form representation. One of the main contributions of this study is to generalize the classical Fourier--Walsh--Hadamard decomposition for pseudo-Boolean functions to the correlated case, yielding an oblique version when the underlying distribution is not a product measure, and recovering the standard orthogonal form when independence holds. This explicit structure offers a fully interpretable framework, clarifying the contribution of each input combination and theoretically enabling model reverse engineering. From this formulation, explicit sensitivity measures-such as Sobol' indices and Shapley effects-can be directly derived. Numerical experiments illustrate the practical interest of the approach for decision-support problems involving binary features. The paper concludes with perspectives on extending the methodology to high-dimensional settings and to models involving inputs with finite, non-binary support.