The Gaussian chain graph model simultaneously parametrizes (i) the direct
effects of p predictors on q correlated outcomes and (ii) the residual
partial covariance between pair 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×q matrix of direct effects and the q×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 model parameters, our method is
seen to outperform fixed-penalty competitors on simulated data. We establish
the posterior concentration rate for our model, buttressing our method's
excellent empirical performance with strong theoretical guarantees. We use our
method to reanalyze a dataset from a study of the effects of diet and residence
type on the composition of the gut microbiome of elderly adults