Group Inverse-Gamma Gamma Shrinkage for Sparse Linear Models with Block-Correlated Regressors

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 explicitly incorporate multivariate shrinkage for regressors with grouping structures. Of particular interest in this article, is regression coefficient estimation where pockets of high collinearity in the regressor space are contained within known regressor 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 dependent within-group such that the regression coefficients marginally have exact horseshoe regularization. We establish posterior consistency and posterior concentration results for regression coefficients in linear models and 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.