Varieties with maximum likelihood degree one

@article{Huh2013VarietiesWM,
  title={Varieties with maximum likelihood degree one},
  author={June Huh},
  journal={arXiv: Algebraic Geometry},
  year={2013}
}
  • June Huh
  • Published 12 January 2013
  • Mathematics
  • arXiv: Algebraic Geometry
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 estimator corresponding to such a variety is Kapranov's Horn uniformization. This extends Kapranov's characterization of A-discriminantal hypersurfaces to varieties of arbitrary codimension. 
Moment maps, strict linear precision, and maximum likelihood degree one
Likelihood Geometry
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
Maximum Likelihood Degree, Complete Quadrics, and ℂ*-Action
TLDR
An explicit, basic, albeit of high computational complexity, formula is provided for the maximum likelihood degree of linear concentration models in algebraic statistics by relating an intersection problem on the variety of complete quadrics.
Maximum likelihood degree, complete quadrics and ${\mathbb C}^*$-action
We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on a smooth compact moduli space of orbits of a ${\mathbb
Maximum likelihood degree and space of orbits of a ${\mathbb C}^*$ action
We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on a smooth compact moduli space of orbits of a ${\mathbb
Maximum Likelihood Estimation from a Tropical and a Bernstein--Sato Perspective
In this article, we investigate Maximum Likelihood Estimation with tools from Tropical Geometry and Bernstein--Sato theory. We investigate the critical points of very affine varieties and study their
Geometry of the Gaussian graphical model of the cycle
We prove a conjecture due to Sturmfels and Uhler concerning the degree of the projective variety associated to the Gaussian graphical model of the cycle. We involve new methods based on the
Discrete statistical models with rational maximum likelihood estimator
TLDR
This work presents an algorithm for constructing models with rational MLE, and demonstrates it on a range of instances of models familiar to statisticians, like Bayesian networks, decomposable graphical models, and staged trees.
A note on Maximum Likelihood Estimation for cubic and quartic canonical toric del Pezzo Surfaces
This article focuses on the study of toric algebraic statistical models which correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form for the Maximum Likelihood Estimate of
...
...

References

SHOWING 1-10 OF 31 REFERENCES
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
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
Some results on inhomogeneous discriminants
We study generalized Horn-Kapranov rational parametrizations of inhomogeneous sparse discriminants from both a theoretical and an algorithmic perspective. We show that all these parametrizations are
Tropical Discriminants
Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the
Solving the Likelihood Equations
TLDR
Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex, and the maximum likelihood degree of a generic complete intersection is determined.
The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties
For an irreducible subvariety Z in an algebraic group G we define a nonnegative integer gdeg(Z) as the degree, in a certain sense, of the Gauss map of Z. It can be regarded as a substitution for the
On a conjecture of V archenko
In this note we generalize and prove a recent conjecture of Varchenko concerning the number of critical points of a (multivalued) meromorphic function $\phi$ on an algebraic manifold. Under certain
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
Open Problems in Algebraic Statistics
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open
...
...