Faithfulness in Chain Graphs: The Gaussian Case

Previously, it has been proven that for any undirected graph there exists a regular Gaussian distribution that is faithful to it (Lněnička & Matúš, 2007, Corollary 3). A stronger result has been proven for acyclic directed graphs: In certain measure-theoretic sense, almost all the regular Gaussian distributions that factorize with respect to an acyclic directed graph are faithful to it (Spirtes et al., 1993, Theorem 3.2). Therefore, this paper extends the latter result to chain graphs. It is worth mentioning that we have recently proved in (Peña, 2009) a result analogous to the one in this paper but for strictly positive discrete probability distributions with arbitrary prescribed sample space. It is also worth noticing that a result analogous to the one in this paper has been proven in (Levitz et al., 2001, Theorem 6.1) under the