On the maximum likelihood degree for Gaussian graphical models

TL;DR

This paper investigates the likelihood geometry of Gaussian graphical models, proving properties of ML-degree behavior, characterizing models with rational MLE, and confirming a conjectured formula for cycles.

Contribution

It provides a detailed proof of ML-degree monotonicity, completes the characterization of models with rational MLE, and verifies a lower bound formula for cycles.

Findings

ML-degree is monotonic on induced subgraphs

Only Gaussian models with rational MLE are from chordal graphs

The conjectured ML-degree formula for cycles is correct

Abstract

In this paper we revisit the likelihood geometry of Gaussian graphical models. We give a detailed proof that the ML-degree behaves monotonically on induced subgraphs. Furthermore, we complete a missing argument that the ML-degree of the $n$-th cycle is larger than one for any $n\geq 4$, therefore completing the characterization that the only Gaussian graphical models with rational maximum likelihood estimator are the ones corresponding to chordal (decomposable) graphs. Finally, we prove that the formula for the ML-degree of a cycle conjectured by Drton, Sturmfels and Sullivant provides a correct lower bound.