Graphical Answers to Questions about Likelihood Inference for Gaussian Covariance Models

In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph (alternatively graphs with dashed edges have been used for this purpose). Bi-directed graphs are special instances of ancestral graphs, which are mixed graphs with undirected, directed, and bi-directed edges. In this paper, we show how simplicial sets and the newly define d orientable edges can be used to construct a maximal ancestral graph that is Markov equivalent to a given bi-directed graph, i.e. the independence models associated with the two graphs coincide, and such that the number of arrowheads is minimal. Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi-directed edges. This construction yields an immediate check whether the original bi-directed graph is Markov equivalent to a directed acyclic graph (Bayesian network) or an undirected graph (Markov random field). Moreover, the ancestral graph construction a llows for computationally more efficient maximum likelihood fitting of covariance graph mod els, i.e. Gaussian bi-directed graph models. In particular, we give a necessary and sufficie nt graphical criterion for determining when an entry of the maximum likelihood estimate of the covariance matrix must equal its empirical counterpart.

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