SIAM Review, Volume 66, Issue 4, Page 721-747, November 2024.
We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We present a dictionary that relates notions of stability from geometric invariant theory to the existence and uniqueness of a maximum likelihood estimate. Our dictionary holds for both discrete and continuous statistical models: we discuss log-linear models and Gaussian models, including matrix normal models and directed Gaussian graphical models. Our approach reveals promising consequences of the interplay between invariant theory and statistics. For instance, algorithms from statistics can be used in invariant theory, and vice versa.

中文翻译：

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不变理论和最大似然估计之间的桥梁


SIAM 评论，第 66 卷，第 4 期，第 721-747 页，2024 年 11 月。

我们揭示了统计学中的最大似然估计与不变理论中群轨道上的范数最小化之间的联系。我们提出了一个字典，它将几何不变理论中的稳定性概念与最大似然估计的存在性和唯一性联系起来。我们的字典适用于离散和连续统计模型：我们讨论了对数线性模型和高斯模型，包括矩阵正态模型和有向高斯图形模型。我们的方法揭示了不变理论和统计学之间相互作用的有希望的后果。例如，统计算法可用于不变论，反之亦然。