The Maximum Likelihood Degree of Toric Models is Monotonic

TL;DR

This paper proves that the ML degree of a facial submodel of a toric model does not exceed that of the original model, with implications for data zeros and applications to graphical models.

Contribution

It establishes the monotonicity of ML degree for facial submodels of toric models, resolving a conjecture and linking to tropical likelihood degenerations.

Findings

ML degree of submodels is at most that of the original model

Zeros in data affect ML degree analysis

Applications to graphical and quasi-independence models

Abstract

We settle a conjecture by Coons and Sullivant stating that the maximum likelihood (ML) degree of a facial submodel of a toric model is at most the ML degree of the model itself. We discuss the impact on the ML degree from observing zeros in the data. Moreover, we connect this problem to tropical likelihood degenerations, and show how the results can be applied to discrete graphical and quasi-independence models.