Vol. 11, No. 1, 2020

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Maximum likelihood estimation of toric Fano varieties

Carlos Améndola, Dimitra Kosta and Kaie Kubjas

Vol. 11 (2020), No. 1, 5–30
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.

Keywords
algebraic statistics, maximum likelihood estimation, maximum likelihood degree, Fano varieties, toric varieties, toric fiber product
Mathematical Subject Classification
Primary: 62F10
Secondary: 13P25, 14M25, 14Q15
Milestones
Received: 20 May 2019
Revised: 17 January 2020
Accepted: 24 March 2020
Published: 1 October 2020
Authors
Carlos Améndola
Department of Mathematics
Technical University of Munich
Garching
Germany
Dimitra Kosta
School of Mathematics and Statistics
University of Glasgow
United Kingdom
Kaie Kubjas
Department of Mathematics and Systems Analysis
Aalto University
Finland