  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.
