The maximum likelihood degree of toric varieties

被引:11
作者
Amendola, Carlos [1 ]
Bliss, Nathan [2 ]
Burke, Isaac [3 ]
Gibbons, Courtney R. [4 ]
Helmer, Martin [5 ]
Hosten, Serkan [6 ]
Nash, Evan D. [7 ]
Rodriguez, Jose Israel [8 ]
Smolkin, Daniel [9 ]
机构
[1] Tech Univ Berlin, Berlin, Germany
[2] Univ Illinois, Chicago, IL 60680 USA
[3] Natl Univ Ireland, Galway, Ireland
[4] Hamilton Coll, Clinton, NY 13323 USA
[5] Univ Calif Berkeley, Berkeley, CA 94720 USA
[6] San Francisco State Univ, San Francisco, CA 94132 USA
[7] Ohio State Univ, Columbus, OH 43210 USA
[8] Univ Chicago, Chicago, IL 60637 USA
[9] Univ Utah, Salt Lake City, UT 84112 USA
基金
美国国家科学基金会;
关键词
Maximum likelihood degree; Toric variety; A-discriminant; HOMOTOPY; SYSTEMS;
D O I
10.1016/j.jsc.2018.04.016
中图分类号
TP301 [理论、方法];
学科分类号
081202 ;
摘要
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. (C) 2018 Elsevier Ltd. All rights reserved.
引用
收藏
页码:222 / 242
页数:21
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