# The Maximum Likelihood Degree of Toric Varieties

## TL;DR

This paper investigates the maximum likelihood degree of toric varieties, revealing how scaling affects it and providing methods for numerical computation and examples from algebraic geometry and statistics.

## Contribution

It establishes the relationship between ML degree and the degree of toric varieties under generic scalings, and explores the impact of the principal A-determinant locus.

## Key findings

- ML degree equals the degree of the toric variety for generic scalings.
- ML degree drops when the scaling vector lies in the principal A-determinant locus.
- Numerical methods for ML estimation via homotopy continuation are demonstrated.

## Abstract

We study the maximum likelihood degree (ML degree) of toric varieties, known as discrete exponential models in statistics. By introducing scaling coefficients to the monomial parameterization of the toric variety, one can change the ML degree. We show that the ML degree is equal to the degree of the toric variety for generic scalings, while it drops if and only if the scaling vector is in the locus of the principal $A$-determinant. We also illustrate how to compute the ML estimate of a toric variety numerically via homotopy continuation from a scaled toric variety with low ML degree. Throughout, we include examples motivated by algebraic geometry and statistics. We compute the ML degree of rational normal scrolls and a large class of Veronese-type varieties. In addition, we investigate the ML degree of scaled Segre varieties, hierarchical loglinear models, and graphical models.