Semi-Linear Gaussian Bayesian Networks: Enhancing Flexibility in Node Dependencies via penalized B-Splines

Aitor Oviedo-Madrid, Mabel Morales-Otero, Jose Gonzalez-Gomariz, Ruben Armañanzas

Bayesian networks for continuous variables often rely on restrictive assumptions: linear Gaussian models lack flexibility to capture nonlinear dependencies, while non-parametric approaches provide greater flexibility at a high computational cost. This creates a gap for modeling scenarios where variables are approximately Gaussian but exhibit nonlinear relationships. We introduce Semi-Linear Gaussian Bayesian Networks (SLGBN), a novel class of continuous Bayesian networks designed to bridge this gap, preserving Gaussian conditional distributions while introducing nonlinear conditional mean functions. The proposed framework provides a unified probabilistic framework combining linear Gaussian and nonlinear Gaussian conditional models within the same Bayesian network Nonlinear conditional mean functions are estimated using generalized additive models (GAMs) with penalized B-splines. We present parameter estimation procedures for both types of nodes, as well as a greedy hill-climbing algorithm for learning the network structure under the SLGBN framework. To assess its performance, we compare SLGBN with other continuous Bayesian network models, including semiparametric Bayesian…