[论文] Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Dis...

论文概要

研究领域: ML 作者: Shayan Hundrieser, Tudor Manole 发布时间: 2026-10-21 arXiv: 2610.30109

中文摘要

我们分析了基于 n 个独立同分布样本、使用非参数最大似然估计量（NPMLE）估计单变量高斯混合密度的问题。我们证明，当真实混合测度是次高斯的，无约束 NPMLE 在平方 Hellinger 距离下达到极小化最优速率 log(n)/n。此外，当真实混合测度具有紧支撑时，约束在紧区间上的 NPMLE 也达到极小化最优速率，其阶略快，为 log(n)/(n log log n)。我们的证明包括精确刻画有限高斯混合类的局部 Hellinger 括号熵。我们的关键技术创新之一是精确界定有界次数多项式球的 L² 括号熵，这可能具有独立意义。作为应用，我们建立了基于 NPMLE 的 g-建模程序在有界先验的高斯经验贝叶斯问题中的遗憾最优性。

原文摘要

We analyze the problem of estimating univariate Gaussian mixture densities using the nonparametric maximum likelihood estimator (NPMLE) based on n independent and identically distributed samples. We show that the unconstrained NPMLE achieves the minimax optimal rate log(n)/n in squared Hellinger distance, when the true mixing measure is sub-Gaussian. In addition, we show that when the true mixing measure has compact support, the NPMLE constrained to a compact interval also achieves the minimax optimal rate, which is slightly faster and of order log(n)/(n log log n). Our proof consists of sharply characterizing the local Hellinger bracketing entropy of the class of finite Gaussian mixtures. One of our key technical innovations is to sharply bound the L^2 bracketing entropy of balls of bound...

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