Baptiste Ferrere

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition is explicit. It is closely connected to SHAP values, generalized additive models, and orthogonal polynomial expansions, and therefore constitutes a fundamental tool for additive explainability. In the more general and realistic dependent setting, however, obtaining a tractable representation and estimating the decomposition from data remain challenging. In this work, we address this problem for continuous inputs. By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

Functional ANOVA offers a principled framework for interpretability by decomposing a model's prediction into main effects and higher-order interactions. For independent features, this decomposition is well-defined, strongly linked with SHAP values, and serves as a cornerstone of additive explainability. However, the lack of an explicit closed-form expression for general dependent distributions has forced practitioners to rely on costly sampling-based approximations. We completely resolve this limitation for categorical inputs. By bridging functional analysis with the extension of discrete Fourier analysis, we derive a closed-form decomposition without any assumption. Our formulation is computationally very efficient. It seamlessly recovers the classical independent case and extends to arbitrary dependence structures, including distributions with non-rectangular support. Furthermore, leveraging the intrinsic link between SHAP and ANOVA under independence, our framework yields a natural generalization of SHAP values for the general categorical setting.

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

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.