Efficient sampling of Gaussian graphical models using conditional Bayes factors

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.