Maximum Likelihood Degree Research Articles

Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

We establish connections between the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.

The Maximum Likelihood Degree of Sparse Polynomial Systems

We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood (ML) degree counts the number of critical points of the likelihood function restricted to the model. We prove that the ML degree of a generic sparse polynomial system is determined by its Newton polytopes and equals the mixed volume of a related Lagrange system of equations.

Likelihood degenerations

Computing all critical points of a monomial on a very affine variety is a fundamental task in algebraic statistics, particle physics and other fields. The number of critical points is known as the maximum likelihood (ML) degree. When the variety is smooth, it coincides with the Euler characteristic. We introduce degeneration techniques that are inspired by the soft limits in CEGM theory, and we answer several questions raised in the physics literature. These pertain to bounded regions in discriminantal arrangements and to moduli spaces of point configurations. We present theory and practise, connecting complex geometry, tropical combinatorics, and numerical nonlinear algebra.

Computing maximum likelihood estimates for Gaussian graphical models with Macaulay2

We introduce the package "GraphicalModelsMLE" for computing the maximum likelihood estimates (MLEs) of a Gaussian graphical model in the computer algebra system Macaulay2. This package allows the computation of MLEs for the class of loopless mixed graphs. Additional functionality allows the user to explore the underlying algebraic structure of the model, such as its maximum likelihood degree and the ideal of score equations.

The leading coefficient of Lascoux polynomials

Lascoux polynomials have been recently introduced to prove polynomiality of the maximum-likelihood degree of linear concentration models. We find the leading coefficient of the Lascoux polynomials (type C) and their generalizations to the case of general matrices (type A) and skew symmetric matrices (type D). In particular, we determine the degrees of such polynomials. As an application, we find the degree of the polynomial δ(m,n,n−s) of the algebraic degree of semidefinite programming, and when s=1 we find its leading coefficient for types C, A and D.

Maximum Likelihood Degree of Surjective Rational Maps

With any surjective rational map \(f: \mathbb {P}^n \dashrightarrow \mathbb {P}^n\) of the projective space, we associate a numerical invariant (ML degree) and compute it in terms of a naturally defined vector bundle \(E_f \longrightarrow \mathbb {P}^n\).

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 the variety of complete quadrics. This allows us to pr...

Maximum likelihood degree of the two-dimensional linear Gaussian covariance model

In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of $n\times n$ Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is $2n-3$.

Computing Euler Obstruction Functions Using Maximum Likelihood Degrees

Abstract We give a numerical algorithm computing Euler obstruction functions using maximum likelihood degrees. The maximum likelihood degree is a well-studied property of a variety in algebraic statistics and computational algebraic geometry. In this article we use this degree to give a new way to compute Euler obstruction functions. We define the maximum likelihood obstruction function and show how it coincides with the Euler obstruction function. With this insight, we are able to bring new tools of computational algebraic geometry to study Euler obstruction functions.

Estimating linear covariance models with numerical nonlinear algebra

Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.

Maximum likelihood estimation of toric Fano varieties

We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all $2$-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two ordinary double points and provide explicit expressions that allow one to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using $A$-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees.

Autocovariance varieties of moving average random fields

We study the autocovariance functions of moving average random fields over the integer lattice Zd from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry.

Moment maps, strict linear precision, and maximum likelihood degree one

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.

The maximum likelihood degree of toric varieties

We study the maximum likelihood (ML) degree of toric varieties, known as discrete exponential models in statistics. By introducing scaling coefficients to the monomial parameterization of the toric variety, one can change the ML degree. We show that the ML degree is equal to the degree of the toric variety for generic scalings, while it drops if and only if the scaling vector is in the locus of the principal A-determinant. We also illustrate how to compute the ML estimate of a toric variety numerically via homotopy continuation from a scaled toric variety with low ML degree. Throughout, we include examples motivated by algebraic geometry and statistics. We compute the ML degree of rational normal scrolls and a large class of Veronese-type varieties. In addition, we investigate the ML degree of scaled Segre varieties, hierarchical log-linear models, and graphical models.

The Maximum Likelihood Degree of Mixtures of Independence Models

The maximum likelihood degree (ML degree) measures the algebraic complexity of a fundamental optimization problem in statistics: maximum likelihood estimation. In this problem, one maximizes the likelihood function over a statistical model. The ML degree of a model is an upper bound to the number of local extrema of the likelihood function and can be expressed as a weighted sum of Euler characteristics. The independence model (i.e. rank one matrices over the probability simplex) is well known to have an ML degree of one, meaning their is a unique local maxima of the likelihood function. However, for mixtures of independence models (i.e. rank two matrices over the probability simplex), it was an open question as to how the ML degree behaved. In this paper, we use Euler characteristics to prove an outstanding conjecture by Hauenstein, the first author, and Sturmfels; we give recursions and closed form expressions for the ML degree of mixtures of independence models.

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 give a formula of the maximum likelihood degree in terms of the constants β μ , ν \beta _{\mu , \nu } , which is defined to be the number of complex solutions to the system of equations z 1 ν = z 2 ν = ⋯ = z μ ν = 1 z_1^\nu =z_2^\nu =\cdots =z_\mu ^\nu =1 and z 1 + ⋯ + z μ + 1 = 0 z_1+\cdots +z_\mu +1=0 .

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. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.

Bounding the maximum likelihood degree

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.

Varieties with maximum likelihood degree one

We show that algebraic varieties with maximum likelihood degree one are exactlythe images of reduced A-discriminantal varieties under monomial maps with nite bers. Themaximum likelihood estimator corresponding to such a variety is Kapranov's Horn uniformization.This extends Kapranov's characterization of A-discriminantal hypersurfaces to varieties of arbitrarycodimension.

Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental optimization problem in statistics. Westudy this problem on manifolds of matrices with bounded rank. These represent mixtures of distributionsof two independent discrete random variables. We determine the maximum likelihood degree for a rangeof determinantal varieties, and we apply numerical algebraic geometry to compute all critical points oftheir likelihood functions. This led to the discovery of maximum likelihood duality between matrices ofcomplementary ranks, a result proved subsequently by Draisma and Rodriguez.