Matroid Stratification of ML Degrees of Independence Models

TL;DR

This paper explores the ML degree of discrete exponential independence models and models from the second hypersimplex, revealing matroid invariants and connections to hyperplane arrangements, with computational evidence supporting conjectures.

Contribution

It introduces a matroid-based framework for understanding ML degrees in independence models and analyzes the principal A-determinant for models from the second hypersimplex.

Findings

ML degree is a matroid invariant for models with two variables

Connection established between A-determinant factors and ML degree

Computational evidence supports a lower bound conjecture for the second hypersimplex models

Abstract

We study the maximum likelihood (ML) degree of discrete exponential independence models and models defined by the second hypersimplex. For models with two independent variables, we show that the ML degree is an invariant of a matroid associated to the model. We use this description to explore ML degrees via hyperplane arrangements. For independence models with more variables, we investigate the connection between the vanishing of factors of its principal $A$-determinant and its ML degree. Similarly, for models defined by the second hypersimplex, we determine its principal $A$-determinant and give computational evidence towards a conjectured lower bound of its ML degree.