An empirical $G$-Wishart prior for sparse high-dimensional Gaussian graphical models

In Gaussian graphical models, the zero entries in the precision matrix determine the dependence structure, so estimating that sparse precision matrix and, thereby, learning this underlying structure, is an important and challenging problem. We propose an empirical version of the $G$-Wishart prior for sparse precision matrices, where the prior mode is informed by the data in a suitable way. Paired with a prior on the graph structure, a marginal posterior distribution for the same is obtained that takes the form of a ratio of two $G$-Wishart normalizing constants. We show that this ratio can be easily and accurately computed using a Laplace approximation, which leads to fast and efficient posterior sampling even in high-dimensions. Numerical results demonstrate the proposed method's superior performance, in terms of speed and accuracy, across a variety of settings, and theoretical support is provided in the form of a posterior concentration rate theorem.