# Bayesian Variable Selection for Non-Gaussian Responses: A Marginally Calibrated Copula Approach

## TL;DR

This paper introduces a flexible Bayesian method for variable selection in non-Gaussian regression models using copula decomposition, enabling accurate marginal calibration and efficient estimation with improved variable selection and spatial analysis in fMRI data.

## Contribution

It develops a novel copula-based Bayesian variable selection approach that is highly flexible, computationally efficient, and applicable to spatial data, improving accuracy over existing methods.

## Key findings

- Outperforms benchmarks in non-Gaussian response variable selection.
- Enables voxel-specific calibration in fMRI, enhancing activation map quality.
- Efficient MCMC estimation for high-dimensional copula models.

## Abstract

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.