Invariant theory and scaling algorithms for maximum likelihood estimation

The task of fitting data to a model is fundamental in statistics. For this, a widespread approach is finding a maximum likelihood estimate (MLE), where one maximizes the likelihood of observing the data as we range over the model. For two common statistical settings (log-linear models and Gaussian transformation families), we show that this approach is equivalent to a capacity problem in invariant theory: finding a point of minimal norm in an orbit under a corresponding group action. The existence of the MLE can then be characterized by stability notions under the action. This dictionary between invariant theory and statistics has already led to the solution of long-standing questions concerning the MLE of matrix normal models. Moreover, algorithms from statistics can be used in invariant theory, and vice versa.

This talk is based on joint work with Carlos Améndola, Philipp Reichenbach and Anna Seigal.