Approximate Bayesian estimation in large coloured graphical Gaussian models

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 article, 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 of our estimate of the precision matrix to its true value, in the Frobenius norm, compares well with the rates in the current literature for the maximum likelihood estimate. The Canadian Journal of Statistics 46: 176-203; 2018 (c) 2017 Statistical Society of Canada