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In recent years, a variety of useful extensions of the Wishart have been proposed in the literature for the purposes of studying Markov random fields/graphical models. In particular, generalizations of the Wishart, referred to as Type I and Type II Wishart distributions, have been introduced by Letac and Massam (\emph{Annals of Statistics} 2006) and play important roles in both frequentist and Bayesian inference for Gaussian graphical models. These distributions have been especially useful in high-dimensional settings due to the flexibility offered by their multiple shape parameters. The domain of In this paper we resolve a long-standing conjecture of Letac and Massam (LM) concerning the domains of the multi-parameters of graphical Wishart type distributions. This conjecture, posed in \emph{Annals of Statistics}, also relates fundamentally to the existence of Bayes estimators corresponding to these high dimensional priors. To achieve our goal, we first develop novel theory in the context of probabilistic analysis of graphical models. Using these tools, and a recently introduced class of Wishart distributions for directed acyclic graph (DAG) models, we proceed to give counterexample

Author

Ben-David, Emanuel; Rajaratnam, Bala

Published

2014

Language

EN