Geometry of maximum likelihood estimation in Gaussian graphical models

TL;DR

This paper explores the geometric properties of maximum likelihood estimation in Gaussian graphical models, providing algebraic criteria for the existence of the MLE and analyzing its behavior across different graph structures.

Contribution

It introduces an algebraic elimination criterion to determine the minimum observations for MLE existence and examines the ML degree in various graph classes, including a novel example where MLE exists at the treewidth.

Findings

Lower bounds on observations for MLE existence established

Application to bipartite, grid, and colored graphs demonstrated

First example where MLE exists with observations equal to treewidth

Abstract

We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator (MLE) exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also study the ML degree, and we present the first instance of a graph for which the MLE exists with probability one, even when the number of observations equals the treewidth.