Mathematical Research Letters

Volume 22 (2015)

Number 6

Bounding the maximum likelihood degree

Pages: 1613 – 1620



Nero Budur (Department of Mathematics, KU Leuven, Belgium; and Department of Mathematics, University of Notre Dame, Indiana, U.S.A.)

Botong Wang (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.)


Maximum likelihood estimation is a fundamental computational problem in statistics. In this note, we give a bound for the maximum likelihood degree of algebraic statistical models for discrete data. As usual, such models are identified with special very affine varieties. Using earlier work of Franecki and Kapranov, we prove that the maximum likelihood degree is always less or equal to the signed intersection-cohomology Euler characteristic. We construct counterexamples to a bound in terms of the usual Euler characteristic conjectured by Huh and Sturmfels.


very affine variety, intersection cohomology, algebraic statistics, maximum likelihood estimation

2010 Mathematics Subject Classification

14F45, 32S60, 55N33, 62E10, 62H12

Accepted 14 April 2015

Published 23 May 2016