In high-dimensional Bayesian statistics, several methods have been developed, including many prior distributions that lead to the sparsity of estimated parameters. However, such priors have limitations in handling the spectral eigenvector structure of data, and as a result, they are ill-suited for analyzing over-parameterized models (high-dimensional linear models that do not assume sparsity) that have been developed in recent years. This paper introduces a Bayesian approach that uses a prior dependent on the eigenvectors of data covariance matrices, but does not induce the sparsity of parameters. We also provide contraction rates of derived posterior distributions and develop a truncated Gaussian approximation of the posterior distribution. The former demonstrates the efficiency of posterior estimation, while the latter enables quantification of parameter uncertainty using a Bernstein-von Mises-type approach. These results indicate that any Bayesian method that can handle the spectrum of data and estimate non-sparse high dimensions would be possible.

翻译：在高维贝叶斯统计中，已发展出多种方法，其中包括许多能导致估计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，因此难以适用于近年来发展起来的过参数化模型（即不假设稀疏性的高维线性模型）。本文提出一种贝叶斯方法，该方法采用依赖于数据协方差矩阵特征向量的先验分布，但不会诱导参数稀疏性。我们进一步给出了推导所得后验分布的收缩率，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效率，后者则通过Bernstein-von Mises型方法实现了参数不确定性的量化。这些结果表明，任何能够处理数据谱结构并估计非稀疏高维参数的贝叶斯方法均具备可行性。