Efficient sampling for Gaussian linear regression with arbitrary priors

This paper develops a computationally efficient posterior sampling algorithm for Bayesian linear regression models with Gaussian errors. Our new approach is motivated by the fact that existing software implementations for Bayesian linear regression do not readily handle problems with large number of observations (hundreds of thousands) and predictors (thousands). Moreover, existing sampling algorithms for popular shrinkage priors are bespoke Gibbs samplers based on case-specific latent variable representations. By contrast, the new algorithm does not rely on case-specific auxiliary variable representations, which allows for rapid prototyping of novel shrinkage priors outside the conditionally Gaussian framework. Specifically, we propose a slice-within-Gibbs sampler based on the elliptical slice sampler of Murray et al. [2010].