The Letac-Massam conjecture and existence of high dimensional Bayes estimators for Graphical Models

TL;DR

This paper resolves the longstanding Letac-Massam conjecture regarding the parameter domains of graphical Wishart distributions, which are crucial for Bayesian inference in high-dimensional Gaussian graphical models.

Contribution

The paper proves the conjecture's falsehood by providing counterexamples and develops new probabilistic theory to analyze graphical Wishart distributions in high-dimensional settings.

Findings

Counterexamples to the Letac-Massam conjecture

Insights into the existence of Bayesian estimators for graphical models

Enhanced understanding of high-dimensional Wishart distributions

Abstract

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.