Bayesian variable selection for non-Gaussian responses: a marginally calibrated copula approach

Area of Science:

• Statistics
• Machine Learning
• Computational Neuroscience

Background:

• Standard regression models often assume Gaussian (normal) distributions, limiting their applicability to non-normal data.
• Variable selection is crucial for identifying relevant predictors in complex models.
• Existing methods may struggle with non-Gaussian data, potentially leading to inaccurate variable selection.

Purpose of the Study:

• To develop a flexible and tractable Bayesian approach for variable selection in non-Gaussian regression models.
• To enable accurate marginal distribution calibration for dependent variables.
• To extend the method for spatial variable selection in functional Magnetic Resonance Imaging (fMRI).

Main Methods:

• Utilizes copula decomposition for the joint distribution of observations.
• Employs "implicit copulas" derived from hierarchical Bayesian models.
• Leverages Markov chain Monte Carlo (MCMC) for efficient estimation of high-dimensional copulas.

Main Results:

• The proposed method demonstrates superior variable selection accuracy compared to benchmarks for non-Gaussian responses.
• Accounting for deviations from normality significantly increases model accuracy.
• The approach successfully performs voxel-specific marginal calibration in fMRI data, enhancing activation map quality.

Conclusions:

• The novel Bayesian copula approach offers a flexible and accurate solution for variable selection in non-Gaussian regression.
• It provides substantial improvements in accuracy, particularly when data deviates from normality.
• The method has significant potential for applications in neuroimaging, such as fMRI analysis.