Approximate Bayesian estimation in large coloured graphical Gaussian models

TL;DR

This paper develops a Bayesian approach for estimating the precision matrix in large coloured graphical Gaussian models, incorporating symmetry constraints and analyzing its asymptotic properties under different regimes.

Contribution

It introduces a Bayesian estimation method for large coloured graphical Gaussian models with symmetry constraints and studies its asymptotic behavior, extending previous frequentist approaches.

Findings

Bayesian estimate converges at rates comparable to MLE in fixed p regime.

Asymptotic consistency established under double asymptotic regime.

Method performs well when local model parameters are bounded.

Abstract

Distributed estimation methods have recently been used to compute the maximum likelihood estimate of the precision matrix for large graphical Gaussian models. Our aim, in this paper, is to give a Bayesian estimate of the precision matrix for large graphical Gaussian models with, additionally, symmetry constraints imposed by an underlying graph which is coloured. We take the sample posterior mean of the precision matrix as our estimate. We study its asymptotic behaviour under the regular asymptotic regime when the number of variables p is fixed and under the double asymptotic regime when both p and n grow to infinity. We show in particular, that when the number of parameters of the local models is uniformly bounded, the standard convergence rate we obtain for the asymptotic consistency, in the Frobenius norm, of our estimate of the precision matrix compares well with the rates in the current literature for the maximum likelihood estimate.