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.

翻译：现代统计学习算法具备惊人的灵活性，但在可解释性方面存在困难。稀疏性是一种可能的解决方案：通过使用非光滑罚项（如$\ell_1$罚项），使得推断结果中许多参数被估计为零。然而，当需要高度稀疏性时，$\ell_1$罚项会引入显著偏差。在本文中，我们保留$\ell_1$罚项，但定义具有超先验的可学习罚权重$\lambda_p$。首先，我们研究该问题所蕴含的优化问题，推导出与$\ell_1$范数相关的近端算子。随后，在罚似然框架下研究这种变系数$\ell_1$罚项的理论性质。接着，我们探索将该罚项应用于变分贝叶斯方法，提出一种名为稀疏贝叶斯Lasso的模型，该模型能将类似于Lasso回归的行为应用于任意变分模型。模拟研究表明，该方法在获得基于模拟方法的量化不确定性和低偏差特性的同时，计算量降低了一个数量级。最后，我们将所提方法应用于2013-2017年伊拉克内战期间国内流离失所现象的贝叶斯滞后时空回归模型。