A Bayesian approach for partial Gaussian graphical models with sparsity

We explore various Bayesian approaches to estimate partial Gaussian graphical models. Our hierarchical structures enable to deal with single-output as well as multiple-output linear regressions, in small or high dimension, enforcing either no sparsity, sparsity, group sparsity or even sparse-group sparsity for a bi-level selection in the direct links between predictors and responses, thanks to spike-and-slab priors corresponding to each setting. Adaptative and global shrinkages are also incorporated in the Bayesian modeling of the direct links. Gibbs samplers are developed and a simulation study shows the efficiency of our models which regularly give better results than the usual Lasso-type procedures, especially in terms of support recovery. To conclude, a real dataset is investigated.