One-dimensional Discrete Models of Maximum Likelihood Degree One

TL;DR

This paper proves a conjecture that one-dimensional discrete models with rational maximum likelihood estimators have degrees bounded linearly by support size, linking algebraic statistics with Cauchy-Riemann geometry.

Contribution

It establishes a linear bound on the degree of such models and explores their combinatorial and geometric properties, including sharp models and their connections to monomial maps.

Findings

Degree of models is bounded linearly by support size

Finitely many fundamental models exist for each number of states

Sharp models have special geometric properties

Abstract

We settle a conjecture by Bik and Marigliano stating that the degree of a one-dimensional discrete model with rational maximum likelihood estimator is bounded above by a linear function in the size of its support, therefore showing that there are only finitely many fundamental such models for any given number of states. We study these models from a combinatorial perspective with regard to their existence and enumeration. In particular, sharp models, those whose degree attains the maximal bound, enjoy special properties and have been studied as monomial maps between unit spheres. In this way, we present a novel link between Cauchy-Riemann geometry and algebraic statistics.