发表机构
KTH Royal Institute of Technology; Competence Centre for Advanced BioProduction by Continuous Processing, AdBIOPRO（皇家理工学院; 连续加工先进生物生产能力中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对线性回归中经验贝叶斯估计对超参数扰动敏感的问题，提出一种广义贝叶斯估计器，保持相同XMSE且局部鲁棒性更强，计算复杂度相当或更优。
AI 中文摘要
贝叶斯估计在统计学、决策理论、信号处理、机器学习和系统辨识中得到了广泛研究与应用。在其变体中，经验贝叶斯（EB）估计因其良好的估计性能和计算可行性而备受关注。然而，EB估计器对超参数的直接插入式依赖可能使其对超参数扰动局部敏感。本文考虑线性回归模型，并聚焦于采用边际最大似然超参数估计器的EB估计器。为简洁起见，该估计器简称为EB估计器。给定一族EB加权函数，构造了一个广义贝叶斯估计器，其具有与对应EB估计器相同的超额均方误差（XMSE）。此处，XMSE是所关注估计器与最大似然估计器之间均方误差差异的二阶渐近度量。此外，EB估计器被证明对超参数扰动至多一阶敏感，而所构造的贝叶斯估计器至多二阶敏感，因此局部鲁棒性更强。还分析了这两种估计器的计算复杂度。在某些情况下，所构造的贝叶斯估计器在计算上可与EB估计器相当，甚至更高效。这些理论结果得到了数值模拟的进一步支持。
英文摘要
Bayes estimation has been extensively studied and widely used in statistics, decision theory, signal processing, machine learning, and system identification. Among its variants, empirical Bayes (EB) estimation has attracted considerable attention due to its favorable estimation performance and computational tractability. However, the direct plug-in dependence of an EB estimator on hyperparameters can make it locally sensitive to hyper-parameter perturbations. This paper considers the linear regression model and focuses on the EB estimator by employing the marginal maximum likelihood hyper-parameter estimator. For conciseness, this estimator is simply referred to as the EB estimator. Given a family of EB weighting functions, a generalized Bayes estimator is constructed with the same excess mean squared error (XMSE) as the corresponding EB estimator. Here, the XMSE is a second-order asymptotic measure of the mean squared error difference between the estimator of interest and the maximum likelihood estimator. Furthermore, the EB estimator is shown to be at most firstorder sensitive to hyper-parameter perturbations, whereas the constructed Bayes estimator is at most second-order sensitive, making it locally more robust. The computational complexities of these two estimators are also analyzed. In some cases, the constructed Bayes estimator can be computationally comparable to, or more efficient than, the EB estimator. These theoretical results are further supported by numerical simulations.
