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Likelihood Geometry

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Combinatorial Algebraic Geometry

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2108))

Abstract

We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its maximum likelihood degree. We present an introduction to this theory and its statistical motivations. Many favorite objects from combinatorial algebraic geometry are featured: toric varieties, A-discriminants, hyperplane arrangements, Grassmannians, and determinantal varieties. Several new results are included, especially on the likelihood correspondence and its bidegree. This article represents the lectures given by the second author at the CIME-CIRM course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.

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Acknowledgements

We thank Paolo Aluffi and Sam Payne for helpful communications, and the Mathematics Department at KAIST, Daejeon, for hosting both authors in May 2013. Bernd Sturmfels was supported by NSF (DMS-0968882) and DARPA (HR0011-12-1-0011).

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    June Huh

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Bernd Sturmfels

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  1. June Huh
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  2. Bernd Sturmfels
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Correspondence to June Huh .

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Huh, J., Sturmfels, B. (2014). Likelihood Geometry. In: Combinatorial Algebraic Geometry. Lecture Notes in Mathematics(), vol 2108. Springer, Cham. https://doi.org/10.1007/978-3-319-04870-3_3

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