Group for Research in Decision Analysis

The Hyper Dirichlet Revisited: A Characterization

Hélène Massam

We give a characterization of the hyper Dirichlet distribution hyper Markov with respect to a decomposable graph \(G\) (or equivalently a moral directed acyclic graph). For \(X=(X_1,\ldots,X_d)\) following the hyper Dirichlet distribution, our characterization is through the so-called "local and global independence properties" for a carefully designed family of orders of the variables \(X_1,\ldots,X_d\). The hyper Dirichlet for general directed acyclic graphs was derived from a characterization of the Dirichlet distribution given by Geiger and Heckerman (1997). This characterization of the Dirichlet for \(X=(X_1,\ldots,X_d)\) is obtained through a functional equation derived from the local and global independence properties for two different orders of the variables. These two orders are seemingly chosen haphazardly but, as our results show, this is not so. Our results generalize those of Geiger and Heckerman (1997) and are given without the assumption of existence of a positive density for \(X\).