  We develop a novel Bayesian method to select important predictors in
regression models with multiple responses of diverse types. A sparse Gaussian
copula regression model is used to account for the multivariate dependencies
between any combination of discrete and/or continuous responses and their
association with a set of predictors. We utilize the parameter expansion for
data augmentation strategy to construct a Markov chain Monte Carlo algorithm
for the estimation of the parameters and the latent variables of the model.
Based on a centered parametrization of the Gaussian latent variables, we design
a fixed-dimensional proposal distribution to update jointly the latent binary
vectors of important predictors and the corresponding non-zero regression
coefficients. For Gaussian responses and for outcomes that can be modeled as a
dependent version of a Gaussian response, this proposal leads to a
Metropolis-Hastings step that allows an efficient exploration of the
predictors' model space. The proposed strategy is tested on simulated data and
applied to real data sets in which the responses consist of low-intensity
counts, binary, ordinal and continuous variables.
