标题： 具有 horseshoe 类先验的鲁棒贝叶斯高维变量选择和推断 显示英文标题

标题： Robust Bayesian high-dimensional variable selection and inference with the horseshoe family of priors

摘要： 频率稳健变量选择在高维回归中已被广泛研究。尽管取得了成功，开发相应的统计推断程序仍然是一个具有挑战性的任务。最近，从贝叶斯视角解决这一挑战受到了广泛关注。在文献中，能够诱导精确稀疏性的两组尖峰-平板先验已被证明在稳健稀疏线性模型中能产生有效的推断。然而，另一个重要的稀疏先验类别，即马蹄族先验，包括马蹄先验、马蹄+先验和正则化马蹄先验，到目前为止尚未在稳健高维回归中进行过研究。它们在存在重尾模型误差时的变量选择和统计推断表现尚不明确。在本文中，我们通过利用马蹄族先验以及高效的吉布斯采样方案来构建稳健的贝叶斯层次模型来解决这个问题。我们表明，与采用其他采样策略（如切片采样）的竞争方法相比，我们的方法在变量选择、贝叶斯估计和统计推断方面表现出更优的性能。特别是，我们的数值研究表明，即使没有施加精确稀疏性，单组马蹄先验在稳健高维线性回归模型下仍能产生有效的贝叶斯可信区间。所提出的方法和替代方法在真实数据上的应用进一步说明了所提方法的优势。 显示英文摘要

摘要： Frequentist robust variable selection has been extensively investigated in high-dimensional regression. Despite success, developing the corresponding statistical inference procedures remains a challenging task. Recently, tackling this challenge from a Bayesian perspective has received much attention. In literature, the two-group spike-and-slab priors that can induce exact sparsity have been demonstrated to yield valid inference in robust sparse linear models. Nevertheless, another important category of sparse priors, the horseshoe family of priors, including horseshoe, horseshoe+, and regularized horseshoe priors, has not yet been examined in robust high-dimensional regression by far. Their performance in variable selection and especially statistical inference in the presence of heavy-tailed model errors is not well understood. In this paper, we address the question by developing robust Bayesian hierarchical models utilizing the horseshoe family of priors along with an efficient Gibbs sampling scheme. We show that compared with competing methods with alternative sampling strategies such as slice sampling, our proposals lead to superior performance in variable selection, Bayesian estimation and statistical inference. In particular, our numeric studies indicate that even without imposing exact sparsity, the one-group horseshoe priors can still yield valid Bayesian credible intervals under robust high-dimensional linear regression models. Applications of the proposed and alternative methods on real data further illustrates the advantage of the proposed methods.