The posterior covariance matrix W defined by the log-likelihood of each observation plays important roles both in the sensitivity analysis and frequencist evaluation of the Bayesian estimators. This study is focused on the matrix W and its principal space; we term the latter as an essential subspace. A key tool for treating frequencist properties is the recently proposed Bayesian infinitesimal jackknife approximation (Giordano and Broderick (2023)). The matrix W can be interpreted as a reproducing kernel and is denoted as W-kernel. Using W-kernel, the essential subspace is expressed as a principal space given by the kernel principal component analysis. A relation to the Fisher kernel and neural tangent kernel is established, which elucidates the connection to the classical asymptotic theory. We also discuss a type of Bayesian-frequencist duality, which is naturally appeared from the kernel framework. Finally, two applications are discussed: the selection of a representative set of observations and dimensional reduction in the approximate bootstrap. In the former, incomplete Cholesky decomposition is introduced as an efficient method for computing the essential subspace. In the latter, different implementations of the approximate bootstrap for posterior means are compared.

翻译：由各观测值对数似然定义的后验协方差矩阵W在贝叶斯估计量的敏感性分析与频率主义评估中均扮演重要角色。本研究聚焦于矩阵W及其主空间——后者被称为本质子空间。处理频率性质的关键工具是近期提出的贝叶斯无穷小刀切近似（Giordano and Broderick (2023)）。矩阵W可解释为再生核，记作W-核。利用W-核，本质子空间可表示为通过核主成分分析得到的主空间。本文建立了与Fisher核及神经正切核的关联，从而阐明其与经典渐近理论的联系。我们还讨论了一种贝叶斯-频率主义对偶性，该对偶性自然产生于核框架。最后探讨了两个应用：代表性观测集的选取与近似自助法中的降维。前者引入不完全Cholesky分解作为计算本质子空间的有效方法；后者比较了后验均值近似自助法的不同实现方式。