Gaussian Covariance faithful Markov Trees

Abstract

A covariance graph is an undirected graph associated with a multivariate probability distribution of a given random vector where each vertex represents each of the different components of the random vector and where the absence of an edge between any pair of variables implies marginal independence between these two variables. Covariance graph models have recently received much attention in the literature and constitute a sub-family of graphical models. Though they are conceptually simple to understand, they are considerably more difficult to analyze. Under some suitable assumption on the probability distribution, covariance graph models can also be used to represent more complex conditional independence relationships between subsets of variables. When the covariance graph captures or reflects all the conditional independence statements present in the probability distribution the latter is said to be faithful to its covariance graph - though no such prior guarantee exists. Despite the increasingly widespread use of these two types of graphical models, to date no deep probabilistic analysis of this class of models, in terms of the faithfulness assumption, is available. Such an analysis is crucial in understanding the ability of the graph, a discrete object, to fully capture the salient features of the probability distribution it aims to describe. In this paper we demonstrate that multivariate Gaussian distributions that have trees as covariance graphs are necessarily faithful. The method of proof is original as it uses an entirely new approach and in the process yields a technique that is novel to the field of graphical models.