Sparse Bayesian Lasso via a Variable-Coefficient $\ell_1$ Penalty

TL;DR

This paper introduces a Sparse Bayesian Lasso with learnable penalty weights and hyperpriors, enhancing interpretability and reducing bias in sparse models, applicable to variational Bayesian methods and demonstrated on real-world displacement data.

Contribution

It develops a variable-coefficient $ ext{L}_1$ penalty with hyperpriors, integrating it into Variational Bayes for improved sparsity and uncertainty quantification.

Findings

Achieves low bias and uncertainty quantification in simulations.

Reduces computational cost by an order of magnitude.

Successfully applied to spatiotemporal displacement modeling.

Abstract

Modern statistical learning algorithms are capable of amazing flexibility, but struggle with interpretability. One possible solution is sparsity: making inference such that many of the parameters are estimated as being identically 0, which may be imposed through the use of nonsmooth penalties such as the $\ell_1$ penalty. However, the $\ell_1$ penalty introduces significant bias when high sparsity is desired. In this article, we retain the $\ell_1$ penalty, but define learnable penalty weights $\lambda_p$ endowed with hyperpriors. We start the article by investigating the optimization problem this poses, developing a proximal operator associated with the $\ell_1$ norm. We then study the theoretical properties of this variable-coefficient $\ell_1$ penalty in the context of penalized likelihood. Next, we investigate application of this penalty to Variational Bayes, developing a model we call the Sparse Bayesian Lasso which allows for behavior qualitatively like Lasso regression to be applied to arbitrary variational models. In simulation studies, this gives us the Uncertainty Quantification and low bias properties of simulation-based approaches with an order of magnitude less computation. Finally, we apply our methodology to a Bayesian lagged spatiotemporal regression model of internal displacement that occurred during the Iraqi Civil War of 2013-2017.