Abstract
Graphical models are useful for characterizing conditional and marginal independence structures in high‐dimensional distributions. An important class of graphical models is covariance graph models, where the nodes of a graph represent different components of a random vector, and the absence of an edge between any pair of variables implies marginal independence. Covariance graph models also 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 in general this is not guaranteed. Faithfulness however is crucial, for instance, in model selection procedures that proceed by testing conditional independences. Hence, an analysis of the faithfulness assumption is important 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.
Highlights
Markov random fields and graphical models are widely used to represent conditional independences in a given multivariate probability distribution see 1–5, to name just a few
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 in general this is not guaranteed
For any triplet A, B, S of subsets of V pairwise disjoint, if S separates A and B in the concentration graph G associated with P, the random vector XA Xv, v ∈ A is independent of XB Xv, v ∈ B given XS Xv, v ∈ S
Summary
Markov random fields and graphical models are widely used to represent conditional independences in a given multivariate probability distribution see 1–5 , to name just a few. For any triplet A, B, S of subsets of V pairwise disjoint, if S separates A and B in the concentration graph G associated with P , the random vector XA Xv, v ∈ A is independent of XB Xv, v ∈ B given XS Xv, v ∈ S This latter property is called concentration global Markov property and is formally defined as. The associated covariance graph G is fully able to capture all the conditional independences present in the multivariate distribution P This result can be considered as a dual of a previous probabilistic result proved by Becker et al 13 for concentration graphs that demonstrates that Gaussian distributions having concentration trees i.e., the concentration graph is a tree are necessarily concentration faithful to its concentration graph implying that property 1.7 is satisfied.
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