  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.
