arXiv subjects

Joseph Muré

Publications and source records attributed to Joseph Muré.

7 recordsLinked to original sources

arXiv2026

Exact Functional ANOVA Decomposition for Categorical Inputs Models

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.

stat.ML↗

arXiv2026

Fourier Analysis on the Boolean Hypercube via Hoeffding Functional Decomposition

Fourier analysis on the Boolean hypercube is fundamentally defined as the orthogonal decomposition of the space of pseudo-Boolean functions with respect to the uniform probability measure. In this work, we propose an ANOVA-based generalization of the Fourier decomposition on the Boolean hypercube endowed with any arbitrary probability measure. We provide an \emph{explicit} decomposition basis which generalizes the Walsh-Hadamard (or parity functions) basis under any \emph{arbitrary} probability measure on the Boolean hypercube. We formulate the computation of the entire functional decomposition as a least squares problem and also provide a method to address the classical \emph{curse of dimensionality} challenge. We provide a comprehensive generalization of Fourier analysis on the Boolean hypercube, enabling the handling of non-uniform configuration spaces inherent to real-world machine learning tasks, \textit{e.g.} when dealing with \emph{one-hot encoded} features. Finally, we demonstrate its practical impact in the field of explainable AI, by conducting comparative studies with feature attribution methods such as SHAP or TreeHFD.

stat.ML↗

arXiv2022

Model predictivity assessment: incremental test-set selection and accuracy evaluation

Unbiased assessment of the predictivity of models learnt by supervised machine-learning methods requires knowledge of the learned function over a reserved test set (not used by the learning algorithm). The quality of the assessment depends, naturally, on the properties of the test set and on the error statistic used to estimate the prediction error. In this work we tackle both issues, proposing a new predictivity criterion that carefully weights the individual observed errors to obtain a global error estimate, and using incremental experimental design methods to "optimally" select the test points on which the criterion is computed. Several incremental constructions are studied, including greedy-packing (coffee-house design), support points and kernel herding techniques. Our results show that the incremental and weighted versions of the latter two, based on Maximum Mean Discrepancy concepts, yield superior performance. An industrial test case provided by the historical French electricity supplier (EDF) illustrates the practical relevance of the methodology, indicating that it is an efficient alternative to expensive cross-validation techniques.

math.ST↗

arXiv2020

Propriety of the reference posterior distribution in Gaussian Process modeling

In a seminal article, Berger, De Oliveira and Sansó (2001) compare several objective prior distributions for the parameters of Gaussian Process regression models with isotropic correlation kernel. The reference prior distribution stands out among them insofar as it always leads to a proper posterior. They prove this result for rough correlation kernels - Spherical, Exponential with power $q<2$, Matérn with smoothness $ν<1$. This paper provides a proof for smooth correlation kernels - Exponential with power $q=2$, Matérn with smoothness $ν\geqslant 1$, Rational Quadratic - along with tail rates of the reference prior for these kernels.

math.ST↗

arXiv2018

Optimal compromise between incompatible conditional probability distributions, with application to Objective Bayesian Kriging

Models are often defined through conditional rather than joint distributions, but it can be difficult to check whether the conditional distributions are compatible, i.e. whether there exists a joint probability distribution which generates them. When they are compatible, a Gibbs sampler can be used to sample from this joint distribution. When they are not, the Gibbs sampling algorithm may still be applied, resulting in a "pseudo-Gibbs sampler". We show its stationary probability distribution to be the optimal compromise between the conditional distributions, in the sense that it minimizes a mean squared misfit between them and its own conditional distributions. This allows us to perform Objective Bayesian analysis of correlation parameters in Kriging models by using univariate conditional Jeffreys-rule posterior distributions instead of the widely used multivariate Jeffreys-rule posterior. This strategy makes the full-Bayesian procedure tractable. Numerical examples show it has near-optimal frequentist performance in terms of prediction interval coverage.

math.ST↗

arXiv2018

Trans-Gaussian Kriging in a Bayesian framework : a case study

In the context of Gaussian Process Regression or Kriging, we propose a full-Bayesian solution to deal with hyperparameters of the covariance function. This solution can be extended to the Trans-Gaussian Kriging framework, which makes it possible to deal with spatial data sets that violate assumptions required for Kriging. It is shown to be both elegant and efficient. We propose an application to computer experiments in the field of nuclear safety, where it is necessary to model non-destructive testing procedures based on eddy currents to detect possible wear in steam generator tubes.

stat.AP↗

arXiv2018

A Comprehensive Bayesian Treatment of the Universal Kriging model with Matérn correlation kernels

The Gibbs reference posterior distribution provides an objective full-Bayesian solution to the problem of prediction of a stationary Gaussian process with Matérn anisotropic kernel. A full-Bayesian approach is possible, because the posterior distribution is expressed as the invariant distribution of a uniformly ergodic Markovian kernel for which we give an explicit expression. In this paper, we show that it is appropriate for the Universal Kriging framework, that is when an unknown function is added to the stationary Gaussian process. We give sufficient conditions for the existence and propriety of the Gibbs reference posterior that apply to a wide variety of practical cases and illustrate the method with several examples. Finally, simulations of Gaussian processes suggest that the Gibbs reference posterior has good frequentist properties in terms of coverage of prediction intervals.

math.ST↗