Proof of a Conjecture of Drton, Sturmfels and Sullivant on the maximum likelihood degree of the Gaussian graphical model of a cycle

TL;DR

We address the problem of determining the maximum likelihood degree (ML-degree) of the Gaussian graphical model on the $n$-cycle $C_n$. The authors reduce the computation to the transverse intersection $L_{C_n}^{-1}\cap(Id+L_{C_n}^perp)$ and show, via fiber analysis over the identity and a detailed smoothness verification of all intersection points, that the ML-degree equals $$(n-3)\cdot 2^{n-2}+1$$. The method combines a graph of the projection with bihomogeneous equations, symmetry reductions, and Jacobian rank checks to enumerate and confirm the smooth, isolated intersection points. This result extends exact likelihood-geometry data beyond chordal graphs and provides a precise algebraic description for the cycle graph, with implications for algebraic statistics and likelihood-based inference on Gaussian graphical models.

Abstract

In this article, we compute the precise value of the maximum likelihood degree of the Gaussian graphical model of a cycle, confirming a conjecture due to Drton, Sturmfels and Sullivant.

Theorems & Definitions (23)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof