  This paper deals with the Bayesian estimation of high dimensional Gaussian
graphical models. We develop a quasi-Bayesian implementation of the
neighborhood selection method of Meinshausen and Buhlmann (2006) for the
estimation of Gaussian graphical models. The method produces a product-form
quasi-posterior distribution that can be efficiently explored by parallel
computing. We derive a non-asymptotic bound on the contraction rate of the
quasi-posterior distribution. The result shows that the proposed
quasi-posterior distribution contracts towards the true precision matrix at a
rate given by the worst contraction rate of the linear regressions that are
involved in the neighborhood selection. We develop a Markov Chain Monte Carlo
algorithm for approximate computations, following an approach from Atchade
(2015). We illustrate the methodology with a simulation study. The results show
that the proposed method can fit Gaussian graphical models at a scale unmatched
by other Bayesian methods for graphical models.
