Global-Local Mixtures: A Unifying Framework

Anindya Bhadra (), Jyotishka Datta (), Nicholas G. Polson () and Brandon T. Willard ()
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Anindya Bhadra: Purdue University
Jyotishka Datta: University of Arkansas
Nicholas G. Polson: The University of Chicago Booth School of Business
Brandon T. Willard: The University of Chicago Booth School of Business

Sankhya A: The Indian Journal of Statistics, 2020, vol. 82, issue 2, No 6, 426-447

Abstract: Abstract 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-Schlömilch 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 Lasso $\sqrt {\text {Lasso}}$ (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.

Keywords: Bayes regularization; Lasso $\sqrt {\text {Lasso}}$; Convolution; Lasso; Logistic; Quantile; Primary 62F15; Secondary 62J07; 62C10 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s13171-019-00191-2

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