Many regularization priors for Bayesian regression assume the regression coefficients are a priori independent. In particular this is the case for standard Bayesian treatments of the lasso and the elastic net. While independence may be reasonable in some data-analytic settings, incorporating dependence in these prior distributions provides greater modeling flexibility. This paper introduces the orthant normal distribution in its general form and shows how it can be used to structure prior dependence in the Bayesian elastic net regression model. An L1-regularized version of Zellner's g prior is introduced as a special case, creating a new link between the literature on penalized optimization and an important class of regression priors. Computation is challenging due to an intractable normalizing constant in the prior. We avoid this issue by modifying slightly a standard prior of convenience for the hyperparameters in such a way to enable simple and fast Gibbs sampling of the posterior distribution. The benefit of including structured prior dependence in the Bayesian elastic net regression model is demonstrated through simulation and a near-infrared spectroscopy data example.

翻译：许多贝叶斯回归的正则化先验假设回归系数在先验分布中是相互独立的。这在标准贝叶斯方法处理lasso和弹性网络时尤为常见。虽然独立性在某些数据分析情境下是合理的，但在这些先验分布中引入依赖性可以提供更大的建模灵活性。本文介绍了正交正态分布的一般形式，并展示了如何利用它在贝叶斯弹性网络回归模型中构建先验依赖性。作为特例，我们引入了Zellner g先验的L1正则化版本，从而在惩罚优化文献与一类重要的回归先验之间建立了新的联系。由于先验分布中存在难以处理的归一化常数，计算面临挑战。我们通过略微修改超参数的标准便利先验来规避这一问题，使得后验分布的吉布斯采样变得简单快速。通过模拟实验和一个近红外光谱数据实例，我们证明了在贝叶斯弹性网络回归模型中引入结构化先验依赖性的优势。