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Title: Group inverse-gamma gamma shrinkage for sparse regression with application to correlated environmental exposure data Authors:  Jonathan Boss - University of Michigan (United States) [presenting]
Jyotishka Datta - Virginia Polytechnic Institute and State University (United States)
Xin Wang - University of Michigan (United States)
Sung Kyun Park - University of Michigan (United States)
Jian Kang - University of Michigan (United States)
Bhramar Mukherjee - University of Michigan (United States)
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 incorporate bi-level shrinkage for predictors with grouping structures explicitly. Regression coefficient estimation is particularly interesting, 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.