  We propose a new highly flexible and tractable Bayesian approach to undertake
variable selection in non-Gaussian regression models. It uses a copula
decomposition for the joint distribution of observations on the dependent
variable. This allows the marginal distribution of the dependent variable to be
calibrated accurately using a nonparametric or other estimator. The family of
copulas employed are `implicit copulas' that are constructed from existing
hierarchical Bayesian models widely used for variable selection, and we
establish some of their properties. Even though the copulas are
high-dimensional, they can be estimated efficiently and quickly using Markov
chain Monte Carlo (MCMC). A simulation study shows that when the responses are
non-Gaussian the approach selects variables more accurately than contemporary
benchmarks. A real data example in the Web Appendix illustrates that accounting
for even mild deviations from normality can lead to a substantial increase in
accuracy. To illustrate the full potential of our approach we extend it to
spatial variable selection for fMRI. Using real data, we show our method allows
for voxel-specific marginal calibration of the magnetic resonance signal at
over 6,000 voxels, leading to an increase in the quality of the activation
maps.
