2019/05/31 by Carlos Améndola, Dimitra Kosta, Kaie Kubjas
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Codimension #Combinatorics #Degree (music) #Fano plane #Geometry #Mathematics #Maximum likelihood #Polynomial and algebraic computation #Pure mathematics #Quadratic equation #Quintic function #Statistics #Surface (topology) #math.AG #math.ST #msc:13P25 #msc:14M25 #msc:14Q15 #msc:62F10 #stat.TH
paper · pdf · doi:10.2140/astat.2020.11.5
published as Alg. Stat. 11 (2020) 5-30 · 28 pages, 4 figures, 4 tables, this article supersedes arXiv:1602.08307
openalex created_date 2019/05/29 · arxiv created 2019/11/17 · openalex publication_date 2020/10/01 · arxiv updated 2020/10/07 · openalex updated_date 2026/08/05
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 singular points of type A1 and provide explicit expressions that allow 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.