  We consider sparse Bayesian estimation in the classical multivariate linear
regression model with $p$ regressors and $q$ response variables. In univariate
Bayesian linear regression with a single response $y$, shrinkage priors which
can be expressed as scale mixtures of normal densities are popular for
obtaining sparse estimates of the coefficients. In this paper, we extend the
use of these priors to the multivariate case to estimate a $p \times q$
coefficients matrix $\mathbf{B}$. We derive sufficient conditions for posterior
consistency under the Bayesian multivariate linear regression framework and
prove that our method achieves posterior consistency even when $p>n$ and even
when $p$ grows at nearly exponential rate with the sample size. We derive an
efficient Gibbs sampling algorithm and provide the implementation in a
comprehensive R package called MBSP. Finally, we demonstrate through
simulations and data analysis that our model has excellent finite sample
performance.
