近端经验贝叶斯用于带后验决策支持的稀疏回归
Proximal Empirical Bayes for Sparse Regression with Posterior Decision Support
- The University of Sheffield（谢菲尔德大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种基于凸惩罚和对数凹先验的近端经验贝叶斯框架，用于高斯稀疏回归，通过分别校准观测尺度和收缩参数、利用众数估计系数、近端采样提供后验不确定性，并采用阈值和激活概率做出稀疏决策，实验验证了其准确性和效率。
AI中文摘要：
稀疏回归需要估计以及关于保留哪些效应的决策。贝叶斯收缩为这一决策提供了不确定性，但更丰富的先验层次结构可能使校准和后验计算变得困难。我们开发了一个基于凸惩罚和对数凹先验的高斯稀疏回归的计算高效的经验贝叶斯框架。观测尺度和全局收缩参数分别校准，众数汇总了联合后验的信息并提供系数估计，而近端采样器提供后验不确定性。后验尺度幅度阈值和激活概率随后将这些输出转换为稀疏决策。相同的近端结构在整个过程中重复使用，使优化和采样成本低廉，并允许扩展到具有可处理近端映射的其他凸惩罚。我们还开发了仿射信息下的经验贝叶斯校准，区分硬齐次约束和软非齐次仿射信息，并在强仿射信息产生低秩刚性时引入几何感知的后验预处理。合成实验表明，当样本量超过预测变量数量时，恢复准确；当预测变量数量超过观测值时，表现出保守的弱信号行为；几何感知缩放显著提高了蒙特卡洛效率。在糖尿病数据集上，后验不确定性与已有的贝叶斯分析非常一致，而最终决策提供了更稀疏的实际摘要。
英文摘要：
In this work, we revisit estimation and variable selection in sparse Gaussian regression, possibly under affine constraints. We adopt a middle-ground approach between optimisation and fully hierarchical Bayesian methods, preserving much of the computational efficiency of the former while adding posterior uncertainty quantification. In particular, we develop an empirical Bayes framework that separates shrinkage calibration from posterior-informed selection while using the same log-concave model throughout. The resulting workflow is modular: different methods can be used for noise-variance estimation, shrinkage calibration and posterior sampling. At the same time, these tasks are linked, and numerical choices made at one stage can propagate through the workflow and ultimately affect the decision about which variables to retain. We investigate in detail one instance of the framework based on a Laplace prior, with the amount of shrinkage determined by stochastic approximation proximal gradient. The proximal machinery used for this calibration also supports posterior simulation. We further show how the empirical Bayes calibration should be modified when homogeneous constraints reduce the number of free parameters. Nonhomogeneous affine information is instead incorporated softly; when this information is strong, posterior sampling can become poorly conditioned, and we develop a preconditioning strategy to improve mixing and posterior exploration. Synthetic experiments examine how scale calibration, posterior run length and sampler conditioning affect estimation, uncertainty quantification and the final sparse decision. On a diabetes dataset, posterior uncertainty agrees closely with established Bayesian analyses, while the final decision provides a sparser practical summary.
