  Gaussian graphical models can capture complex dependency structures among
variables. For such models, Bayesian inference is attractive as it provides
principled ways to incorporate prior information and to quantify uncertainty
through the posterior distribution. However, posterior computation under the
conjugate G-Wishart prior distribution on the precision matrix is expensive for
general non-decomposable graphs. We therefore propose a new Markov chain Monte
Carlo (MCMC) method named the G-Wishart weighted proposal algorithm (WWA).
WWA's distinctive features include delayed acceptance MCMC, Gibbs updates for
the precision matrix and an informed proposal distribution on the graph space
that enables embarrassingly parallel computations. Compared to existing
approaches, WWA reduces the frequency of the relatively expensive sampling from
the G-Wishart distribution. This results in faster MCMC convergence, improved
MCMC mixing and reduced computing time. Numerical studies on simulated and real
data show that WWA provides a more efficient tool for posterior inference than
competing state-of-the-art MCMC algorithms.
