Wishart distributions for decomposable graphs

TL;DR

This paper introduces two new families of Wishart distributions tailored for decomposable Gaussian graphical models, extending existing distributions and providing flexible priors with desirable conjugacy and Markov properties.

Contribution

It constructs Type I and II Wishart distributions on cones associated with decomposable graphs, generalizing hyper Wishart distributions and establishing their properties.

Findings

Type I and II Wisharts generalize hyper Wishart distributions.

Inverse Type II Wishart is a conjugate prior with multidimensional shape parameters.

Distributions exhibit properties similar to hyper and hyper inverse Wisharts.

Abstract

When considering a graphical Gaussian model ${\mathcal{N}}_G$ Markov with respect to a decomposable graph $G$, the parameter space of interest for the precision parameter is the cone $P_G$ of positive definite matrices with fixed zeros corresponding to the missing edges of $G$. The parameter space for the scale parameter of ${\mathcal{N}}_G$ is the cone $Q_G$, dual to $P_G$, of incomplete matrices with submatrices corresponding to the cliques of $G$ being positive definite. In this paper we construct on the cones $Q_G$ and $P_G$ two families of Wishart distributions, namely the Type I and Type II Wisharts. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272--1317]. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the Type I Wishart is weak hyper Markov. Moreover, the inverse Type II Wishart as a conjugate family presents the advantage of having a multidimensional shape parameter, thus offering flexibility for the choice of a prior.