Global-Local Mixtures: A Unifying Framework

Global-local mixtures, including Gaussian scale mixtures, have gained prominence in recent times, both as a sparsity inducing prior in p >> n problems as well as default priors for non-linear many-to-one functionals of high-dimensional parameters. Here we propose a unifying framework for global-local scale mixtures using the Cauchy-Schlomilch and Liouville integral transformation identities, and use the framework to build a new Bayesian sparse signal recovery method. This new method is a Bayesian counterpart of the root Lasso (Belloni et al., Biometrika 98, 4, 791-806, 2011) that adapts to unknown error variance. Our framework also characterizes well-known scale mixture distributions including the Laplace density used in Bayesian Lasso, logit and quantile via a single integral identity. Finally, we derive a few convolutions that commonly arise in Bayesian inference and posit a conjecture concerning bridge and uniform correlation mixtures.