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

Estimating sparse direct effects in multivariate regression with the spike-and-slab LASSO