# Classifying one-dimensional discrete models with maximum likelihood degree one

In the paper, we prove for \(n=2,3,4\) that a one-dimensional model in \(\Delta_n\) with a rational maximum likelihood estimator is parametrized by scalar multiples of monomials in \(t,1-t\) of degree \(\leq 2(n+1)-3\). We prove the cases \(n=2,3,4\) in Sections \(5,6,7\) respectively. In Section 8, we generate a list of all fundamental models in \(\Delta_n\) parametrized by monomials of degree \(\leq9\) for \(n\leq 5\).

In Section 5, we introduce the Invertibility Criterion and use it prove that a one-dimensional model in \(\Delta_n\) with a rational MLE are parametrized by scalar multiples of monomials of degree \(\leq 2(n+1)-3\) for \(n=2\). As the number of possible configurations of \(3\) points in a grid is small, we can do this by hand. In Sections 6 and 7, we introduce the Hyperfield and Hexagon Criteria and prove the result for \(n=3,4\). As the number of possible configurations is now to big to handle by hand, we let a computer go through them all. The same holds for the computations in Section 8. The code can be viewed below.

The source code can be downloaded here:

Project page created: 20/05/2022

Project contributors: Arthur Bik, Orlando Marigliano

Code written by: Arthur Bik

Software used: SageMath (Version 9.2)

Corresponding author of this page: Arthur Bik, arthur.bik@mis.mpg.de