Inference of multiple high-dimensional networks with the Graphical Horseshoe prior

We develop a novel full-Bayesian approach for multiple correlated precision matrices, called multiple Graphical Horseshoe (mGHS). The proposed approach relies on a novel multivariate shrinkage prior based on the Horseshoe prior that borrows strength and shares sparsity patterns across groups, improving posterior edge selection when the precision matrices are similar. On the other hand, there is no loss of performance when the groups are independent. Moreover, mGHS provides a similarity matrix estimate, useful for understanding network similarities across groups. We implement an efficient Metropolis-within-Gibbs for posterior inference; specifically, local variance parameters are updated via a novel and efficient modified rejection sampling algorithm that samples from a three-parameter Gamma distribution. The method scales well with respect to the number of variables and provides one of the fastest full-Bayesian approaches for the estimation of multiple precision matrices. Finally, edge selection is performed with a novel approach based on model cuts. We empirically demonstrate that mGHS outperforms competing approaches through both simulation studies and the application to a bike-sharing and a genomic dataset.