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Title: Undirected, indirected and regression graph models for categorical data in a common framework Authors:  Alberto Roverato - University of Bologna (Italy) [presenting]
Luca La Rocca - University of Modena and Reggio Emilia (Italy)
Abstract: The problem of specifying a suitable parameterization for graphical models for categorical data is considered. We focus on three of the most relevant families of graphical models, that is, undirected, bidirected and regression graph models. In this respect, we first give some general properties concerning conditional independence and Moebius inversion. Next, we exploit these basic results to provide a unified approach to the parameterization of the three classes of models. The parameterizations are derived by applying, in the three cases, the same Moebius inversion formula to obtain a log-linear expansion of certain probabilities. In the undirected case this procedure leads to the usual corner-constrained parameterization of the class of log-linear models for contingency tables. This is the standard parameterization of undirected graph models, and we show that some well-known properties of this parameterization, such as the connection between vanishing terms and independence relationships, as well as the capability of defining context specific independences, follow directly from the constructing procedure. In this way, the former properties automatically hold true also for other parameterizations based on the same constructing procedure, and we exploit this feature to present the theory of the three classes of models in a common framework.