ON A DUALIZATION OF GRAPHICAL GAUSSIAN MODELS

Graphical Gaussian models as defined by Speed & Kiiveri (1986) present the conditional independence structure of normally distributed variab les by a graph. A similar approach was recently motivated by Cox & Wer muth (1993) who introduced graphs showing the marginal independence st ructure, The interpretation of a graph in terms of conditional indepen dence relations is based on the definition of a pairwise, local and gl obal Markov property respectively, which are equivalent in the normal distribution, Similar definitions can be formulated for the interpreta tion of graphs in terms of marginal independencies. Their equivalence is proven in the normal distribution. Frydenberg (1990a) discusses equ ivalence statements between the graphical approach and the concept of a cut in exponential families (Barndorff-Nielsen, 1978). In this paper , similar relations are shown for the normal distribution and graphica l models for marginal independencies. Parameter estimation in graphica l 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 mode ls for conditional independence.