A partial graphical model with a structural prior on the direct links between predictors and responses

¹ Univ Angers, CNRS, LAREMA, SFR MATHSTIC, 49000 Angers, France.
² Unité de Bioinformatique, Institut de Cancérologie de l’Ouest, Bd Jacques Monod, 44805 Saint Herblain Cedex, France.
³ SIRIC ILIAD, Nantes, Angers, France.
⁴ CRCINA, INSERM, CNRS, University of Nantes, University of Angers, Health Research Institute-University of Nantes, 8 Quai Moncousu - BP 70721, 44007, Nantes Cedex 1, France.

^* Corresponding author: frederic.proia@univ-angers.fr

Received: 17 September 2020
Accepted: 7 June 2021

Abstract

This paper is devoted to the estimation of a partial graphical model with a structural Bayesian penalization. Precisely, we are interested in the linear regression setting where the estimation is made through the direct links between potentially high-dimensional predictors and multiple responses, since it is known that Gaussian graphical models enable to exhibit direct links only, whereas coefficients in linear regressions contain both direct and indirect relations (due e.g. to strong correlations among the variables). A smooth penalty reflecting a generalized Gaussian Bayesian prior on the covariates is added, either enforcing patterns (like row structures) in the direct links or regulating the joint influence of predictors. We give a theoretical guarantee for our method, taking the form of an upper bound on the estimation error arising with high probability, provided that the model is suitably regularized. Empirical studies on synthetic data and a real dataset are conducted.