AI 中文总结
本研究为高维潜变量模型中的边际最大似然估计建立了严格的渐近理论，包括一致性和渐近正态性，并为潜变量提供了频率学派和贝叶斯不确定性量化。
AI 中文摘要
本研究解决了高维潜变量模型中边际最大似然估计统计基础的一个长期存在的空白。边际最大似然估计被广泛用于社会科学、生态学和机器学习中的潜变量模型拟合。尽管应用广泛，但当样本量和观测变量数量同时发散时，非线性模型的严格渐近理论仍然有限。这一空白主要源于对潜变量进行积分会产生一个非线性目标函数，该函数紧密耦合了高维模型参数。为解决此问题，我们首先证明边际似然即使在真实参数处也表现出多个近乎平坦的方向，这与固定维情况下的行为形成对比。基于这一几何特征，我们开发了新技巧来建立高维参数边际估计量的一致性和渐近正态性。对于潜变量，我们提供了频率学派和贝叶斯不确定性量化，证明了最大后验估计量的渐近正态性以及插件后验的Bernstein-von Mises型结果。这些结果共同为高维模型中的边际估计和潜变量推断提供了严格的基础。
英文摘要
This work addresses a longstanding gap in the statistical foundations of marginal maximum likelihood estimation for high-dimensional latent variable models. Marginal maximum likelihood estimation is widely used to fit latent variable models across the social sciences, ecology, and machine learning. Despite its broad use, rigorous asymptotic theory for nonlinear models remains limited when both the sample size and the number of observed variables diverge. The gap arises largely from the fact that integration over the latent variables creates a nonlinear objective that tightly couples the high-dimensional model parameters. To address this issue, we first show that the marginal likelihood exhibits multiple nearly flat directions even at the true parameter, in contrast to the behavior in the fixed-dimensional case. Building on this geometric characterization, we develop new techniques to establish consistency and asymptotic normality for the marginal estimator of the high-dimensional parameters. For the latent variables, we provide frequentist and Bayesian uncertainty quantification, proving asymptotic normality of the maximum a posteriori estimator and a Bernstein-von Mises-type result for the plug-in posterior. Together, these results provide rigorous foundations for marginal estimation and latent-variable inference in high-dimensional models.
