Faithfulness in Chain Graphs: The Gaussian Case

Abstract

This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that the regular Gaussian distributions that factorize with respect to a chain graph $G$ with $d$ parameters have positive Lebesgue measure with respect to $\mathbb{R}^d$, whereas those that factorize with respect to $G$ but are not faithful to it have zero Lebesgue measure with respect to $\mathbb{R}^d$. This means that, in the measure-theoretic sense described, almost all the regular Gaussian distributions that factorize with respect to $G$ are faithful to it.