  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 through partial correlations (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. An existing result for model selection
consistency is reformulated to stick to our sparse and group-sparse settings,
providing a theoretical guarantee under some technical assumptions. Gibbs
samplers are developed and a simulation study shows the efficiency of our
models which give very competitive results, especially in terms of support
recovery. To conclude, a real dataset is investigated.
