  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 counterexamples to the
LM conjecture, thus completely resolving the problem. Our analysis also
proceeds to give useful insights on graphical Wishart distributions with
implications for Bayesian inference for such models.
