  We propose a Bayesian procedure for simultaneous variable and covariance
selection using continuous spike-and-slab priors in multivariate linear
regression models where q possibly correlated responses are regressed onto p
predictors. Rather than relying on a stochastic search through the
high-dimensional model space, we develop an ECM algorithm similar to the EMVS
procedure of Rockova & George (2014) targeting modal estimates of the matrix of
regression coefficients and residual precision matrix. Varying the scale of the
continuous spike densities facilitates dynamic posterior exploration and allows
us to filter out negligible regression coefficients and partial covariances
gradually. Our method is seen to substantially outperform regularization
competitors on simulated data. We demonstrate our method with a re-examination
of data from a recent observational study of the effect of playing high school
football on several later-life cognition, psychological, and socio-economic
outcomes.
