We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We focus on Gaussian transformation families, which include matrix normal models and Gaussian graphical models given by transitive directed acyclic graphs. We use stability under group actions to characterize boundedness of the likelihood, and existence and uniqueness of the maximum likelihood estimate. Our approach reveals promising consequences of the interplay between invariant theory and statistics. In particular, existing scaling algorithms from statistics can be used in invariant theory, and vice versa.

翻译：我们发现统计中的最大可能性估算与在不定理论中将一个组轨道上的规范最小化之间存在联系。我们侧重于高斯变种,其中包括矩阵正常模型和由中转定向环绕图提供的高斯图形模型。我们使用组合行动下的稳定性来描述最大可能性估算的界限、存在和独特性。我们的方法揭示了不变化理论与统计相互作用的有希望的后果。特别是,现有来自统计的缩放算法可以在不变理论中使用,反之亦然。