Faithfulness in Chain Graphs: The Gaussian Case

This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that almost all the regular Gaussian distributions that factorize with respect to a chain graph are faithful to it. This result has three important consequences. First, chain graphs are more powerful than undirected graphs and acyclic directed graphs for representing regular Gaussian distributions, as some of these distributions can be represented exactly by the former but not by the latter. Second, the moralization and c-separation criteria for reading independencies from a chain graph are complete, in the sense that they identify all the independencies that can be identified from the chain graph alone. Third, some definitions of equivalence in chain graphs coincide and, thus, they have the same graphical characterization.