We consider Bayesian inference on the spiked eigenstructures of high-dimensional covariance matrices; specifically, we focus on estimating the eigenvalues and corresponding eigenvectors of high-dimensional covariance matrices in which a few eigenvalues are significantly larger than the rest. We impose an inverse-Wishart prior distribution on the unknown covariance matrix and derive the posterior distributions of the eigenvalues and eigenvectors by transforming the posterior distribution of the covariance matrix. We prove that the posterior distribution of the spiked eigenvalues and corresponding eigenvectors converges to the true parameters under the spiked high-dimensional covariance assumption, and also that the posterior distribution of the spiked eigenvector attains the minimax optimality under the single spiked covariance model. Simulation studies and real data analysis demonstrate that our proposed method outperforms all existing methods in quantifying uncertainty.

翻译：本文研究高维协方差矩阵尖峰特征结构的贝叶斯推断问题，重点估计存在少量显著大于其余特征值的尖峰特征值及其对应特征向量。我们在未知协方差矩阵上施加逆Wishart先验分布，通过对协方差矩阵后验分布进行变换推导出特征值与特征向量的后验分布。我们证明：在尖峰高维协方差假设下，尖峰特征值及其对应特征向量的后验分布收敛于真实参数；在单尖峰协方差模型中，尖峰特征向量的后验分布达到极小极大最优性。模拟研究与实际数据分析表明，本文提出的方法在不确定性量化方面优于所有现有方法。