# Discrete Statistical Models with Rational Maximum Likelihood Estimator

## TL;DR

This paper characterizes discrete statistical models with rational maximum likelihood estimators using real algebraic geometry, providing an algorithm to construct such models, including Bayesian networks and graphical models.

## Contribution

It offers a complete characterization of models with rational MLEs and introduces an algorithm for constructing these models, connecting algebraic geometry with statistical modeling.

## Key findings

- Identified all models with rational MLEs using algebraic geometry.
- Developed an algorithm to construct models with rational MLEs.
- Demonstrated the approach on Bayesian networks and graphical models.

## Abstract

A discrete statistical model is a subset of a probability simplex. Its maximum likelihood estimator (MLE) is a retraction from that simplex onto the model. We characterize all models for which this retraction is a rational function. This is a contribution via real algebraic geometry which rests on results due to Huh and Kapranov on Horn uniformization. We present an algorithm for constructing models with rational MLE, and we demonstrate it on a range of instances. Our focus lies on models familiar to statisticians, like Bayesian networks, decomposable graphical models, and staged trees.