  This article presents an approach to Bayesian semiparametric inference for
Gaussian multivariate response regression. We are motivated by various small
and medium dimensional problems from the physical and social sciences. The
statistical challenges revolve around dealing with the unknown mean and
variance functions and in particular, the correlation matrix. To tackle these
problems, we have developed priors over the smooth functions and a Markov chain
Monte Carlo algorithm for inference and model selection. Specifically,
Dirichlet process mixtures of Gaussian distributions are used as the basis for
a cluster-inducing prior over the elements of the correlation matrix. The
smooth, multidimensional means and variances are represented using radial basis
function expansions. The complexity of the model, in terms of variable
selection and smoothness, is then controlled by spike-slab priors. A simulation
study is presented, demonstrating performance as the response dimension
increases. Finally, the model is fit to a number of real world datasets. An R
package, scripts for replicating synthetic and real data examples, and a
detailed description of the MCMC sampler are available in the supplementary
materials online.
