Maximum likelihood estimation for matrix normal models via quiver representations

Understand

In this paper, 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 log-likelihood function guarantees almost sure existence of an MLE, thereby proving a conjecture of Drton, Kuriki and Hoff.
• The main tools we use are from the theory of quiver representations, in particular, results of Kac, King and Schofield on canonical decomposition and stability.