  The multivariate regression interpretation of the Gaussian chain graph model
simultaneously parametrizes (i) the direct effects of $p$ predictors on $q$
outcomes and (ii) the residual partial covariances between pairs of outcomes.
We introduce a new method for fitting sparse Gaussian chain graph models with
spike-and-slab LASSO (SSL) priors. We develop an Expectation Conditional
Maximization algorithm to obtain sparse estimates of the $p \times q$ matrix of
direct effects and the $q \times q$ residual precision matrix. Our algorithm
iteratively solves a sequence of penalized maximum likelihood problems with
self-adaptive penalties that gradually filter out negligible regression
coefficients and partial covariances. Because it adaptively penalizes
individual model parameters, our method is seen to outperform fixed-penalty
competitors on simulated data. We establish the posterior contraction rate for
our model, buttressing our method's excellent empirical performance with strong
theoretical guarantees. Using our method, we estimated the direct effects of
diet and residence type on the composition of the gut microbiome of elderly
adults.
