Reading dependencies from covariance graphs

Abstract

The covariance graph (aka bi-directed graph) of a probability distribution p is the undirected graph G where two nodes are adjacent iff their corresponding random variables are marginally dependent in p. (It is worth mentioning that our definition of covariance graph is somewhat non-standard. The standard definition states that the lack of an edge between two nodes of G implies that their corresponding random variables are marginally independent in p. This difference in the definition is important in this paper.) In this paper, we present a graphical criterion for reading dependencies from G, under the assumption that p satisfies the graphoid properties as well as weak transitivity and composition. We prove that the graphical criterion is sound and complete in certain sense. We argue that our assumptions are not too restrictive. For instance, all the regular Gaussian probability distributions satisfy them.