Bayesian precision matrix estimation for graphical Gaussian models with edge and vertex symmetries

TL;DR

This paper develops a Bayesian method for estimating the precision matrix in graphical Gaussian models with symmetries, providing explicit formulas and simulation validation for various graph structures.

Contribution

It introduces a conjugate prior for symmetric precision matrices and derives explicit posterior mean estimates for specific graph classes.

Findings

Explicit analytic expressions for the expected precision matrix in certain graph models

Validation of Bayesian estimates through simulations on graphs with up to thirty vertices

Comparison of Bayesian estimates with exact prior means in special cases

Abstract

Graphical Gaussian models with edge and vertex symmetries were introduced by \citet{HojLaur:2008} who also gave an algorithm to compute the maximum likelihood estimate of the precision matrix for such models. In this paper, we take a Bayesian approach to the estimation of the precision matrix. We consider only those models where the symmetry constraints are imposed on the precision matrix and which thus form a natural exponential family with the precision matrix as the canonical parameter. We first identify the Diaconis-Ylvisaker conjugate prior for these models and develop a scheme to sample from the prior and posterior distributions. We thus obtain estimates of the posterior mean of the precision matrix. Second, in order to verify the precision of our estimate, we derive the explicit analytic expression of the expected value of the precision matrix when the graph underlying our model is a tree, a complete graph on three vertices and a decomposable graph on four vertices with various symmetries. In those cases, we compare our estimates with the exact value of the mean of the prior distribution. We also verify the accuracy of our estimates of the posterior mean on simulated data for graphs with up to thirty vertices and various symmetries.