Likelihood Geometry

@inproceedings{Huh2013LikelihoodG,
  title={Likelihood Geometry},
  author={June Huh and Bernd Sturmfels},
  year={2013}
}
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, Adiscriminants, hyperplane arrangements, Grassmannians, and… 
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References

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The maximum likelihood degree of a very affine variety
  • June Huh
  • Mathematics
    Compositio Mathematica
  • 2013
Abstract We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s
The maximum likelihood degree
Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization
Maximum Likelihood Duality for Determinantal Varieties
In a recent paper, Hauenstein, Sturmfels, and the second author discovered a conjectural bijection between critical points of the likelihood func- tion on the complex variety of matrices of rank r
Varieties with maximum likelihood degree one
We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum likelihood
Discriminants, Horn uniformization, and varieties with maximum likelihood degree one
We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum likelihood
Euler characteristics of general linear sections and polynomial Chern classes
We obtain a precise relation between the Chern–Schwartz–MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a
The number of critical points of a product of powers of linear functions
be the complement of .A. Given a complex n-vector 2 = (21,...,2n) E Cn, consider the multivalued holomorphic function defined on M by Studying Bethe vectors in statistical mechanics, A. Varchenko
Critical Points of the Product of Powers of Linear Functions and Families of Bases of Singular Vectors
The quasiclassical asymptotics of the Knizhnik-Zamolodchikov equation with values in the tensor product of sl(2)- representations are considered. The first term of asymptotics is an eigenvector of a
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