Group Inverse-Gamma Gamma Shrinkage for Sparse Regression with Block-Correlated Predictors

TL;DR

This paper introduces the GIGG prior, a new heavy-tailed shrinkage method for sparse regression with grouped, collinear predictors, demonstrating improved estimation accuracy and computational simplicity.

Contribution

The paper proposes the GIGG prior, a novel adaptive shrinkage approach for grouped predictors with collinearity, extending heavy-tailed priors like the horseshoe to structured covariates.

Findings

GIGG prior achieves low mean-squared error in simulations.

Posterior distributions are derived in closed form for efficient computation.

GIGG performs well on real data, revealing associations with liver health.

Abstract

Heavy-tailed continuous shrinkage priors, such as the horseshoe prior, are widely used for sparse estimation problems. However, there is limited work extending these priors to predictors with grouping structures. Of particular interest in this article, is regression coefficient estimation where pockets of high collinearity in the covariate space are contained within known covariate groupings. To assuage variance inflation due to multicollinearity we propose the group inverse-gamma gamma (GIGG) prior, a heavy-tailed prior that can trade-off between local and group shrinkage in a data adaptive fashion. A special case of the GIGG prior is the group horseshoe prior, whose shrinkage profile is correlated within-group such that the regression coefficients marginally have exact horseshoe regularization. We show posterior consistency for regression coefficients in linear regression models and posterior concentration results for mean parameters in sparse normal means models. The full conditional distributions corresponding to GIGG regression can be derived in closed form, leading to straightforward posterior computation. We show that GIGG regression results in low mean-squared error across a wide range of correlation structures and within-group signal densities via simulation. We apply GIGG regression to data from the National Health and Nutrition Examination Survey for associating environmental exposures with liver functionality.