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.

翻译：先前的马蹄木等重尾连续不断缩缩的前科,被广泛用于稀释的估计问题。然而,将这些前科扩大到有分组结构的预测器的工作有限。对于这一条特别感兴趣的是,在已知的共变组合中包含共变空间中高共线分数小块的回归系数估计。为了缓解多曲线性造成的多重差异性通货膨胀,我们建议先前的小组反伽马伽马(GIGGG)的反伽马(GIGG)的完全有条件分布可以以封闭的形式产生,从而导致直接的后部计算。以前GIGGG的特例是先前的集团马蹄木,其缩缩缩影特征与集团内部相关,因此回归系数在组内稍稍稍有马蹄定型。我们显示了线回归模型中的回归系数和后部浓度结果的一致性。与GGGGG回归完全有条件的分布可以以封闭的形式产生,从而导致直接的后部计算。我们表明,GIGGG回归的结果是低度差差差差差差错,从广泛的相关结构结构中采用GIGHER的模型,并采用内部检索数据。