Marginal maximum likelihood estimation and asymptotic theory for latent variable models in high dimensions

2026-09-15

AI Summary

本文解决了高维潜变量模型中边际最大似然估计的统计基础问题，通过新方法建立了高维参数估计的一致性和渐近正态性。

Abstract

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.

Problem

• marginal maximum likelihood estimation
• latent variable models
• high dimensions
• asymptotic theory

Innovation

• marginal maximum likelihood estimation
• asymptotic theory
• high-dimensional latent variable models
• asymptotic normality
• uncertainty quantification