The Maximum Likelihood Degree of Toric Varieties

Type of Work

Article

Date

4-11-2018

Journal Title

Journal of Symbolic Computation

Journal ISSN

0747-7171

DOI

10.1016/j.jsc.2018.04.016

Abstract

We study the maximum likelihood degree (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 loglinear models, and graphical models.

Notes

This page links to a version of the paper posted at arXiv.org on November 8, 2017.

The article was published in: Journal of Symbolic Computation, In Press, Corrected Proof (April 11, 2017). doi: 10.1016/j.jsc.2018.04.016

Hamilton Areas of Study

Mathematics

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