Local Shrinkage Rules, Lévy Processes and Regularized Regression

Summary We use Lévy processes to generate joint prior distributions, and therefore penalty functions, for a location parameter β=(β1,…,βp) as p grows large. This generalizes the class of local–global shrinkage rules based on scale mixtures of normals, illuminates new connections between disparate methods and leads to new results for computing posterior means and modes under a wide class of priors. We extend this framework to large-scale regularized regression problems where pn, and we provide comparisons with other methodologies.