Bayesian Generalized Horseshoe Estimation of Generalized Linear Models

Abstract

Bayesian global-local shrinkage estimation with the generalized horseshoe prior represents the state-of-the-art for Gaussian regression models. The extension to non-Gaussian data, such as binary or Student-t regression, is usually done by exploiting a scale-mixture-of-normals approach. However, many standard distributions, such as the gamma and the Poisson, do not admit such a representation. We contribute two extensions to global-local shrinkage methodology. The first is an adaption of recent auxiliary gradient based-sampling schemes to the global-local shrinkage framework, which yields simple algorithms for sampling from generalized linear models. We also introduce two new samplers for the hyperparameters in the generalized horseshoe model, one based on an inverse-gamma mixture of inverse-gamma distributions, and the second a rejection sampler. Results show that these new samplers are highly competitive with the no U-turn sampler for small numbers of predictors, and potentially perform better for larger numbers of predictors. Results for hyperparameter sampling show our new inverse-gamma inverse-gamma based sampling scheme outperforms the standard sampler based on a gamma mixture of gamma distributions.