  Bayesian estimation of Gaussian graphical models has proven to be challenging
because the conjugate prior distribution on the Gaussian precision matrix, the
G-Wishart distribution, has a doubly intractable partition function. Recent
developments provide a direct way to sample from the G-Wishart distribution,
which allows for more efficient algorithms for model selection than previously
possible. Still, estimating Gaussian graphical models with more than a handful
of variables remains a nearly infeasible task. Here, we propose two novel
algorithms that use the direct sampler to more efficiently approximate the
posterior distribution of the Gaussian graphical model. The first algorithm
uses conditional Bayes factors to compare models in a Metropolis-Hastings
framework. The second algorithm is based on a continuous time Markov process.
We show that both algorithms are substantially faster than state-of-the-art
alternatives. Finally, we show how the algorithms may be used to simultaneously
estimate both structural and functional connectivity between subcortical brain
regions using resting-state fMRI.
