AI 中文总结
本研究利用奇异学习理论工具，推导了概率主成分分析（PPCA）的边缘似然渐近性质，提出结合奇异贝叶斯信息准则（sBIC）的有效模型选择方法，可选择主成分数量，还可在PPCA与因子分析间选择，经模拟和真实数据验证有效。
AI 中文摘要
主成分分析的概率公式为选择主成分数量的问题提供了具有统计依据的解决方案。然而，概率主成分分析（PPCA）模型在模型奇点处表现出非标准的大样本渐近性质，此时费希尔信息矩阵不具有满秩，这使得开发易于处理的模型选择方法变得复杂。在本研究中，我们使用奇异学习理论的工具来完整描述PPCA的边缘似然渐近性质。奇异贝叶斯信息准则（sBIC）结合我们的渐近结果，提供了一种选择主成分数量的有效方法。特别地，sBIC纠正了基于标准维度的BIC的过度惩罚问题，该问题会导致选择的主成分数量过少，同时还能以概率收敛到1的方式选择最小的真实模型。我们的框架不仅适用于PPCA，还提供了可用于在PPCA和因子分析模型之间进行选择的sBIC表达式。在模拟和真实数据中，所提出的sBIC方法的有效性得到了验证。
英文摘要
The probabilistic formulation of principal component analysis promises statistically grounded solutions to the problem of selecting the number of principal components. However, developing tractable model selection methods is complicated by the fact that the probabilistic principal component analysis (PPCA) model exhibits non-standard large sample asymptotics at model singularities, where the Fisher information matrix does not have full rank. In this work, tools from singular learning theory are used to provide a complete description of the marginal likelihood asymptotics for PPCA. The singular Bayesian information criterion (sBIC) along with our asymptotic results provides an effective procedure for selecting the number of principal components. In particular, the sBIC corrects for the overpenalization of the standard dimension-based BIC, which leads to too few principal components being selected, while also selecting the smallest true model with probability converging to one. Our framework extends beyond PPCA, where we also provide expressions for the sBIC that can be used to select between PPCA and factor analysis models. In both simulations and on real data the effectiveness of the proposed sBIC methodology is demonstrated.
Comments37 pages, 5 figures
