  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.
