  A hierarchical Bayesian approach that permits simultaneous inference for the
regression coefficient matrix and the error precision (inverse covariance)
matrix in the multivariate linear model is proposed. Assuming a natural
ordering of the elements of the response, the precision matrix is
reparameterized so it can be estimated with univariate-response linear
regression techniques. A novel generalized bridge regression prior that
accommodates both sparse and dense settings and is competitive with alternative
methods for univariate-response regression is proposed and used in this
framework. Two component-wise Markov chain Monte Carlo algorithms are developed
for sampling, including a data augmentation algorithm based on a scale mixture
of normals representation. Numerical examples demonstrate that the proposed
method is competitive with comparable joint mean-covariance models,
particularly in estimation of the precision matrix. The method is also used to
estimate the 253 by 253 precision matrices of two classes of spectra extracted
from images taken by the Hubble Space Telescope. Some interesting structural
patterns in the estimates are discussed.
