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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42 Citations
Highly Influential Citations
1
Background Citations
12
Methods Citations
6

42 Citations

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Citation Type

Defect of Euclidean distance degree
The Euclidean Distance Degree of an Algebraic Variety
TLDR
A theory of such nearest point maps of a real algebraic variety with respect to Euclidean distance from the perspective of computational algebraic geometry is developed.
The euclidean distance degree
TLDR
A theory of such nearest point maps of a real algebraic variety with respect to Euclidean distance from the perspective of computational algebraic geometry is developed.
The maximum likelihood degree of toric varieties
The maximum likelihood degree of Fermat hypersurfaces
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation.…
Rank one local systems and forms of degree one
Cohomology support loci of rank one local systems of a smooth quasiprojective complex algebraic variety are finite unions of torsion-translated complex subtori of the character variety of the…
Characteristic classes of affine varieties and Plucker formulas for affine morphisms
An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group…
Maximum likelihood degree of Fermat hypersurfaces via Euler characteristics
Maximum likelihood degree of a projective variety is the number of critical points of a general likelihood function. In this note, we compute the Maximum likelihood degree of Fermat hypersurfaces. We…
Critical points via monodromy and local methods
Solving the Likelihood Equations to Compute Euler Obstruction Functions
TLDR
This paper discusses a symbolic and a numerical implementation of algorithms to compute the Euler obstruction at a point using Macaulay2 and Bertini.
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References

SHOWING 1-10 OF 78 REFERENCES
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…
The entropic discriminant
A geometric deletion-restriction formula
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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