@article{AMENDOLA2019222,
title = {The maximum likelihood degree of toric varieties},
journal = {Journal of Symbolic Computation},
volume = {92},
pages = {222-242},
year = {2019},
issn = {0747-7171},
doi = {https://doi.org/10.1016/j.jsc.2018.04.016},
url = {https://www.sciencedirect.com/science/article/pii/S0747717118300476},
author = {Carlos Améndola and Nathan Bliss and Isaac Burke and Courtney R. Gibbons and Martin Helmer and Serkan Hoşten and Evan D. Nash and Jose Israel Rodriguez and Daniel Smolkin},
keywords = {Maximum likelihood degree, Toric variety, -discriminant},
abstract = {We study the maximum likelihood (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 log-linear models, and graphical models.}
}