Maximum Likelihood Degree Research Articles

The maximum likelihood degree of a very affine variety

AbstractWe 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 conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern–Schwartz–MacPherson class. The strengthened version recovers the geometric deletion–restriction formula of Denhamet al. for arrangement complements, and generalizes Kouchnirenko’s theorem on the Newton polytope for nondegenerate hypersurfaces.

Maximum likelihood degree of variance component models

Most statistical software packages implement numerical strategies for computation of maximum likelihood estimates in random effects models. Little is known, however, about the algebraic complexity of this problem. For the one-way layout with random effects and unbalanced group sizes, we give formulas for the algebraic degree of the likelihood equations as well as the equations for restricted maximum likelihood estimation. In particular, the latter approach is shown to be algebraically less complex. The formulas are obtained by studying a univariate rational equation whose solutions correspond to the solutions of the likelihood equations. Applying techniques from computational algebra, we also show that balanced two-way layouts with or without interaction have likelihood equations of degree four. Our work suggests that algebraic methods allow one to reliably find global optima of likelihood functions of linear mixed models with a small number of variance components.

Linear precision for parametric patches

We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for Krasauskas’ toric patches, which we show is equivalent to a certain rational map on \({\mathbb C}{\mathbb P}^d\) being a birational isomorphism. Lastly, we establish the connection between linear precision for toric surface patches and maximum likelihood degree for discrete exponential families in algebraic statistics, and show how iterative proportional fitting may be used to compute toric patches.

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 problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses, the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms. Exact formulae in terms of degrees and Newton polytopes are given for polynomials with generic coefficients.

Solving the Likelihood Equations

Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include models specified by rank conditions on matrices and the Jukes–Cantor models of phylogenetics. The maximum likelihood degree of a generic complete intersection is also determined.