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

Arthur Bik and Orlando Marigliano: 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.

Section6.ipynb

Section7.ipynb

Section8.ipynb

Project page created: 20/05/2022

Project contributors: Arthur Bik, Orlando Marigliano

Code written by: Arthur Bik

Software used: SageMath (Version 9.2)