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Modern statistical learning algorithms are capable of amazing flexibility,but struggle with interpretability. One possible solution is sparsity: makinginference such that many of the parameters are estimated as being identically0, which may be imposed through the use of nonsmooth penalties such as the$\ell_1$ penalty. However, the $\ell_1$ penalty introduces significant biaswhen high sparsity is desired. In this article, we retain the $\ell_1$ penalty,but define learnable penalty weights $\lambda_p$ endowed with hyperpriors. Westart the article by investigating the optimization problem this poses,developing a proximal operator associated with the $\ell_1$ norm. We then studythe theoretical properties of this variable-coefficient $\ell_1$ penalty in thecontext of penalized likelihood. Next, we investigate application of thispenalty to Variational Bayes, developing a model we call the Sparse BayesianLasso which allows for behavior qualitatively like Lasso regression to beapplied to arbitrary variational models. In simulation studies, this gives usthe Uncertainty Quantification and low bias properties of simulation-basedapproaches with an order of magnitude less computation. Finally, we apply ourmethodology to a Bayesian lagged spatiotemporal regression model of internaldisplacement that occurred during the Iraqi Civil War of 2013-2017.

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