A Hierarchical Bayesian Framework for Constructing Sparsity-inducing Priors

TL;DR

This paper introduces a hierarchical Bayesian framework that generalizes sparsity-inducing priors, enabling efficient MAP estimation for high-dimensional regression, classification, and graphical models, while incorporating prior information.

Contribution

It provides a novel hierarchical Bayesian approach that unifies and extends existing sparsity-inducing priors and optimization methods, with practical algorithms for various models.

Findings

Effective MAP estimation for linear and logistic regression

Application to sparse precision matrix estimation in Gaussian graphical models

Development of an adaptive group lasso method

Abstract

Variable selection techniques have become increasingly popular amongst statisticians due to an increased number of regression and classification applications involving high-dimensional data where we expect some predictors to be unimportant. In this context, Bayesian variable selection techniques involving Markov chain Monte Carlo exploration of the posterior distribution over models can be prohibitively computationally expensive and so there has been attention paid to quasi-Bayesian approaches such as maximum a posteriori (MAP) estimation using priors that induce sparsity in such estimates. We focus on this latter approach, expanding on the hierarchies proposed to date to provide a Bayesian interpretation and generalization of state-of-the-art penalized optimization approaches and providing simultaneously a natural way to include prior information about parameters within this framework. We give examples of how to use this hierarchy to compute MAP estimates for linear and logistic regression as well as sparse precision-matrix estimates in Gaussian graphical models. In addition, an adaptive group lasso method is derived using the framework.