Maximum likelihood estimation for matrix normal models via quiver representations

TL;DR

This paper analyzes the properties of maximum likelihood estimation in matrix normal models using quiver representation theory, establishing sample size conditions for boundedness, existence, and uniqueness of MLEs, and proving a related conjecture.

Contribution

It introduces a novel approach using quiver representations to study MLE properties in matrix normal models and proves a conjecture about the conditions for MLE existence.

Findings

Sample size conditions for bounded log-likelihood

Almost sure existence of MLEs under certain conditions

Proof of a conjecture relating boundedness and MLE existence

Abstract

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.