Bayesian Variable Selection for Non-Gaussian Responses: A Marginally Calibrated Copula Approach
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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 vector 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 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 Monte Carlo methods. A simulation study shows that when the responses are non-Gaussian the approach selects variables more accurately than contemporary benchmarks. A marketing example illustrates that accounting for even mild deviations from normality can lead to a substantial improvement. To illustrate the full potential of our approach we extend it to spatial variable selection for fMRI data. It allows for voxel-specific marginal calibration of the magnetic resonance signal at over 6,000 voxels, leading to a considerable increase in the quality of the activation maps.
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