In the field of high-dimensional Bayesian statistics, a plethora of methodologies have been developed, including various prior distributions that result in parameter sparsity. However, such priors exhibit limitations in handling the spectral eigenvector structure of data, rendering estimations less effective for analyzing the over-parameterized models (high-dimensional linear models that do not assume sparsity) developed in recent years. This study introduces a Bayesian approach that employs a prior distribution dependent on the eigenvectors of data covariance matrices without inducing parameter sparsity. We also provide contraction rates of the derived posterior estimation and develop a truncated Gaussian approximation of the posterior distribution. The former demonstrates the efficiency of posterior estimation, whereas the latter facilitates the uncertainty quantification of parameters via a Bernstein--von Mises-type approach. These findings suggest that Bayesian methods capable of handling data spectra and estimating non-sparse high-dimensional parameters are feasible.

翻译：在高维贝叶斯统计领域，已发展出大量方法论，包括各类导致参数稀疏性的先验分布。然而，此类先验在处理数据的谱特征向量结构方面存在局限性，使得对近年发展的过参数化模型（不假设稀疏性的高维线性模型）进行分析时估计效果欠佳。本研究提出一种贝叶斯方法，采用基于数据协方差矩阵特征向量但不会诱发参数稀疏性的先验分布。我们同时推导了后验估计的收缩率，并建立了后验分布的截断高斯近似。前者展示了后验估计的有效性，而后者则通过Bernstein--von Mises类方法促进了参数的不确定性量化。这些发现表明，能够处理数据谱并估计非稀疏高维参数的贝叶斯方法是可行的。