  This paper is devoted to the estimation of a partial graphical model with a
structural Bayesian penalization. Precisely, we are interested in the linear
regression setting where the estimation is made through the direct links
between potentially high-dimensional predictors and multiple responses, since
it is known that Gaussian graphical models enable to exhibit direct links only,
whereas coefficients in linear regressions contain both direct and indirect
relations (due \textit{e.g.} to strong correlations among the variables). A
smooth penalty reflecting a generalized Gaussian Bayesian prior on the
covariates is added, either enforcing patterns (like row structures) in the
direct links or regulating the joint influence of predictors. We give a
theoretical guarantee for our method, taking the form of an upper bound on the
estimation error arising with high probability, provided that the model is
suitably regularized. Empirical studies on synthetic data and a real dataset
are conducted.
