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On a dualization of graphical Gaussian models. (English) Zbl 0912.62006
Summary: Graphical Gaussian models as defined by T. P. Speed and H. T. Kiiveri [Ann. Stat. 14, 138-150 (1986; Zbl 0589.62033)] present the conditional independence structure of normally distributed variables by a graph. A similar approach was recently motivated by D. R. Cox and N. Wermuth [Stat. Sci. 8, 204-218, 247-283 (1993)] who introduced graphs showing the marginal independence structure. The interpretation of a graph in terms of conditional independence relations is based on the definition of a pairwise, local and global Markov property, respectively, which are equivalent in the normal distribution. Similar definitions can be formulated for the interpretation of graphs in terms of marginal independencies. Their equivalence is proven in the normal distribution. M. Frydenberg [Ann. Stat. 18, No. 2, 790-805 (1990; Zbl 0725.62057)] discusses equivalence statements between the graphical approach and the concept of a cut in exponential families.
In this paper, similar relations are shown for the normal distribution and graphical models for marginal independencies. Parameter estimation in graphical models with marginal independence interpretation is achieved by the dual likelihood concept, which shows interesting relations to results available for maximum likelihood estimation in graphical Gaussian models for conditional independence.

62A01 Foundations and philosophical topics in statistics
05C90 Applications of graph theory