标题： 贝叶斯$l_0$-正则化最小二乘法 显示英文标题

标题： Bayesian $l_0$-regularized Least Squares

摘要： 贝叶斯$l_0$-正则化最小二乘法是一种针对高维预测变量的变量选择技术。 挑战在于通过搜索模型空间（即所有可能的预测变量组合）来优化非凸目标函数。 尖峰-滑动（又称伯努利-高斯）先验是贝叶斯变量选择的标准方法，但存在计算速度和可扩展性的问题。 单次最佳替换（SBR）提供了一种快速且可扩展的替代方案。 我们建立了贝叶斯正则化与近端更新之间的联系，这表明找到后验模式与使用不同正则化先验的后验均值是等价的。 这使我们可以利用SBR找到尖峰-滑动估计量。 为了说明我们的方法，我们提供了模拟证据以及基于SBR与直接使用尖峰-滑动先验进行后验采样的统计性质和计算效率的真实数据实例。 最后，我们提出了未来研究的方向。 显示英文摘要

摘要： Bayesian $l_0$-regularized least squares is a variable selection technique for high dimensional predictors. The challenge is optimizing a non-convex objective function via search over model space consisting of all possible predictor combinations. Spike-and-slab (a.k.a. Bernoulli-Gaussian) priors are the gold standard for Bayesian variable selection, with a caveat of computational speed and scalability. Single Best Replacement (SBR) provides a fast scalable alternative. We provide a link between Bayesian regularization and proximal updating, which provides an equivalence between finding a posterior mode and a posterior mean with a different regularization prior. This allows us to use SBR to find the spike-and-slab estimator. To illustrate our methodology, we provide simulation evidence and a real data example on the statistical properties and computational efficiency of SBR versus direct posterior sampling using spike-and-slab priors. Finally, we conclude with directions for future research.