Abstract
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 X-u is conditionally independent of X-v given X-S. The opposite direction need not be true, that is, X-u perpendicular to X-v vertical bar X-S need not imply S is a node separator of u and v. When it does, the relation X-u perpendicular to X-v vertical bar X-S 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 X-i perpendicular to X-j vertical bar X-S. We study two classes of separable Gaussian graphical models, namely, weakly K-separable and strongly K- separable Gaussian graphical models. Using the above test for faithfulness, we introduce algorithms to learn the topologies of weakly K-separable and strongly K-separable Gaussian graphical models with Omega (K log p) sample complexity. For strongly K-separable Gaussian graphical models, we additionally provide a method with error bounds for learning the off-diagonal precision matrix entries.