Invariant Theory and Scaling Algorithms for Maximum Likelihood Estimation

We show that maximum likelihood estimation in statistics is equivalent to finding the capacity in invariant theory, in two statistical settings: log-linear models and Gaussian transformation families.The former includes the classical independence model while the latter includes 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.