Testing Unfaithful Gaussian Graphical Models

The global Markov property for Gaussian graphical models ensures graph separation implies conditional independence. Specifically if a node set S graph separates nodes u and v then Xu is conditionally independent of Xv given XS . The opposite direction need not be true, that is, Xu ⊥ Xv | XS need not imply S is a node separator of u and v. When it does, the relation Xu ⊥ Xv | XS is called faithful. In this paper we provide a characterization of faithful relations and then provide an algorithm to test faithfulness based only on knowledge of other conditional relations of the form Xi ⊥ Xj | XS .