Maximum Likelihood Estimation for Matrix Normal Models via Quiver Representations

Abstract

We study the log-likelihood function and maximum likelihood estimate (MLE) for the matrix normal model for both real and complex models. We describe the exact number of samples needed to achieve (almost surely) three conditions, namely a bounded log-likelihood function, existence of MLEs, and uniqueness of MLEs. As a consequence, we observe that almost sure boundedness of the log-likelihood function guarantees almost sure existence of an MLE, thereby proving a conjecture of Drton, Kuriki, and Hoff [Existence and Uniqueness of the Kronecker Covariance MLE, preprint, arXiv:2003.06024, 2020]. The main tools we use are from the theory of quiver representations, in particular, results of Kac, Kin..