标题： 贝叶斯分析尖峰协方差模型：校正特征值偏差并确定尖峰数量 显示英文标题

标题： Bayesian Analysis of Spiked Covariance Models: Correcting Eigenvalue Bias and Determining the Number of Spikes

摘要： 我们研究在尖峰协方差模型中的贝叶斯推断，其中少量的尖峰特征值主导了谱。我们的目标是推断尖峰特征值、其对应的特征向量以及尖峰的数量，为具有不确定性量化的主要成分分析提供贝叶斯解决方案。我们在协方差矩阵上放置一个逆Wishart先验，以推导尖峰特征值和特征向量的后验分布。尽管由于共轭性后验采样计算效率高，但在高维设置下后验特征值估计可能存在偏差。为了解决这个问题，我们提出了两种偏差校正策略：(i) 超参数调整方法，和 (ii) 事后乘法校正。对于推断尖峰数量，我们开发了一个边际似然的BIC类型近似，并证明了在高维情况下后验一致性$p>n$。此外，我们建立了主要特征结构的集中不等式和后验收缩率，证明了在单尖峰情况下尖峰特征向量的最小最大最优性。模拟研究和一个实际数据应用显示，我们的方法在对特征结构估计和尖峰数量估计的不确定性量化方面优于现有方法。 显示英文摘要

摘要： We study Bayesian inference in the spiked covariance model, where a small number of spiked eigenvalues dominate the spectrum. Our goal is to infer the spiked eigenvalues, their corresponding eigenvectors, and the number of spikes, providing a Bayesian solution to principal component analysis with uncertainty quantification. We place an inverse-Wishart prior on the covariance matrix to derive posterior distributions for the spiked eigenvalues and eigenvectors. Although posterior sampling is computationally efficient due to conjugacy, a bias may exist in the posterior eigenvalue estimates under high-dimensional settings. To address this, we propose two bias correction strategies: (i) a hyperparameter adjustment method, and (ii) a post-hoc multiplicative correction. For inferring the number of spikes, we develop a BIC-type approximation to the marginal likelihood and prove posterior consistency in the high-dimensional regime $p>n$. Furthermore, we establish concentration inequalities and posterior contraction rates for the leading eigenstructure, demonstrating minimax optimality for the spiked eigenvector in the single-spike case. Simulation studies and a real data application show that our method performs better than existing approaches in providing accurate quantification of uncertainty for both eigenstructure estimation and estimation of the number of spikes.