# Maximum Likelihood Estimation of Toric Fano Varieties

## TL;DR

This paper investigates the maximum likelihood estimation problem for toric Fano varieties, providing explicit formulas, analyzing ML degrees, and establishing connections to phylogenetic trees and intersection theory.

## Contribution

It offers explicit ML estimates for certain toric Fano varieties, analyzes ML degree variations, and links ML degrees to phylogenetic models and intersection theory.

## Key findings

- ML degree equals the surface degree for most 2D Gorenstein toric Fano varieties
- Explicit formulas for ML estimates when ML degree < 5
- Toric Fano varieties from 3-valent phylogenetic trees have ML degree one

## Abstract

We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all $2$-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two ordinary double points and provide explicit expressions that allow one to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using $A$-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees.