Efficient birth-death MCMC inference for Gaussian graphical models

We propose a new framework for Bayesian inference of Gaussian graphical models for both the decomposable and non-decomposable case. We employ the birth-death MCMC methodology in order to obtain the correct stationary distribution. In particular, the BDMCMC algorithm updates the graph by adding a new edge in a birth move or by deleting an edge in a death move. The posterior on the precision matrix provides valuable information about stable (sub)parts of the underlying graph. Our BDMCMC algorithm is easy to implement, computationally feasible for large graphs and much faster compared to other MCMC algorithms in this area. Unlike frequentist approaches, this method gives a principled and, in practice, sensible model selection estimation, as we show in a cell signaling example. Finally, we illustrate the method on both artificial and real datasets.