The posterior covariance matrix W defined by the log-likelihood of each observation plays important roles both in the sensitivity analysis and frequencist's evaluation of the Bayesian estimators. This study focused on the matrix W and its principal space; we term the latter as an essential subspace. First, it is shown that they appear in various statistical settings, such as the evaluation of the posterior sensitivity, assessment of the frequencist's uncertainty from posterior samples, and stochastic expansion of the loss; a key tool to treat frequencist's properties is the recently proposed Bayesian infinitesimal jackknife approximation (Giordano and Broderick (2023)). In the following part, we show that the matrix W can be interpreted as a reproducing kernel; it is named as W-kernel. Using the W-kernel, the essential subspace is expressed as a principal space given by the kernel PCA. A relation to the Fisher kernel and neural tangent kernel is established, which elucidates the connection to the classical asymptotic theory; it also leads to a sort of Bayesian-frequencist's duality. Finally, two applications, selection of a representative set of observations and dimensional reduction in the approximate bootstrap, are discussed. In the former, incomplete Cholesky decomposition is introduced as an efficient method to compute the essential subspace. In the latter, different implementations of the approximate bootstrap for posterior means are compared.

翻译：由每个观测的对数似然定义的后验协方差矩阵W在贝叶斯估计的敏感性分析和频域评估中均发挥重要作用。本研究聚焦于矩阵W及其主空间；我们将后者称为"本质子空间"。首先，本文证明该矩阵及其主空间出现在多种统计场景中，例如后验敏感性评估、基于后验样本的频域不确定性评估以及损失的随机展开；处理频域特性的关键工具是最近提出的贝叶斯无穷小刀切逼近（Giordano and Broderick, 2023）。在后续部分，我们证明矩阵W可解释为再生核，并将其命名为W核。通过W核，本质子空间可表示为基于核主成分分析（kernel PCA）得到的主空间。本文建立了W核与Fisher核及神经正切核的关联，阐明了与经典渐近理论的关系，同时揭示了贝叶斯-频域对偶性的某种形式。最后，本文讨论了两种应用场景：代表性观测集合的选取以及近似自助法中的维度约简。针对前者，引入不完全Cholesky分解作为计算本质子空间的高效方法；针对后者，比较了后验均值近似自助法的不同实现方式。