A note on Maximum Likelihood Estimation for cubic and quartic canonical toric del Pezzo Surfaces

TL;DR

This paper derives closed-form maximum likelihood estimates for certain toric Del Pezzo surfaces with singularities and computes their ML degrees, advancing algebraic statistical modeling for these geometric structures.

Contribution

It provides explicit formulas for MLE and ML degrees for cubic and quartic toric Del Pezzo surfaces with Du Val singularities, a novel contribution in algebraic statistics.

Findings

Closed-form MLE for cubic and quartic toric Del Pezzo surfaces with singularities.

ML degrees calculated for surfaces of degree ≤ 6, mostly matching the degree.

Special case identified where ML degree differs from the surface degree.

Abstract

This article focuses on the study of toric algebraic statistical models which correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form for the Maximum Likelihood Estimate of algebraic statistical models which correspond to cubic and quartic toric Del Pezzo surfaces with Du Val singular points is given. Also, we calculate the ML degrees of some toric Del Pezzo surfaces of degree less than or equal to six, which equals the degree of the surface in all the case but one, namely the quintic with two points of type $\mathbb{A}_1$.