Seth Sullivant

Lectures on Algebraic Statistics

Markov Bases.- Likelihood Inference.- Conditional Independence.- Hidden Variables.- Bayesian Integrals.- Exercises.- Open Problems.

Toric Ideals of Phylogenetic Invariants

Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an Abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine generators and Gröbner bases for these toric ideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gröbner bases consist of certain explicitly constructed polynomials of degree at most four.

Algebraic factor analysis: tetrads, pentads and beyond

Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of positive definite matrices. This parameter space is studied from the perspective of computational algebraic geometry. Gröbner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space. These polynomials, known as model invariants, arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix. Besides revealing the geometry of the factor analysis model, the model invariants also furnish useful statistics for testing goodness-of-fit.

A Divide-and-Conquer Algorithm for Generating Markov Bases of Multi-way Tables

SummaryWe describe a divide-and-conquer technique for generating a Markov basis that connects all tables of counts having a fixed set of marginal totals. This procedure is based on decomposing the independence graph induced by these marginals. We discuss the practical imports of using this method in conjunction with other algorithms for determining Markov bases.

Gröbner bases and polyhedral geometry of reducible and cyclic models

This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Grobner bases of toric ideals associated to a subset of such models. We study the polytopes for Cyclic models, and we give a complete polyhedral description of these polytopes in the binary cyclic case. Further, we show how to build Grobner bases of a reducible model from the Grobner bases of its pieces. This result also gives a different proof that decomposable models have quadratic Grobner bases. Finally, we present the solution of a problem posed by Vlach (Discrete Appl. Math. 13 (1986) 61-78) concerning the dimension of fibers coming from models corresponding to the boundary of a simplex.

A finiteness theorem for Markov bases of hierarchical models

We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is greater than a computable bound, the Markov bases consist of elements from Markov bases of smaller tables. We give an explicit formula for this bound in terms of Graver bases. We also compute these Markov and Graver complexities for all Kx2x2x2 tables.

Emerging applications of algebraic geometry

Polynomial Optimization on Odd-Dimensional Spheres.- Engineering Systems and Free Semi-Algebraic Geometry.- Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation.- Using Invariants for Phylogenetic Tree Construction.- On the Algebraic Geometry of Polynomial Dynamical Systems.- A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals.- Sums of Squares, Moment Matrices and Optimization Over Polynomials.- Positivity and Sums of Squares: A Guide to Recent Results.- Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas.- Open Problems in Algebraic Statistics.

Trek separation for Gaussian graphical models

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar d-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials.

Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models

We provide a polyhedral description of the conditions for the existence of the maximum likelihood estimate (MLE) for a hierarchical log-linear model. The MLE exists if and only if the observed margins lie in the relative interior of the marginal cone. Using this description, we give an algorithm for determining if the MLE exists. If the tree width is bounded, the algorithm runs in polynomial time. We also perform a computational study of the case of three random variables under the no three-factor effect model.

Prolongations and Computational Algebra

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics. Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA 01075, U.S.A. e-mail: jsidman@mtholyoke.edu Department of Mathematics, North Carolina State University, Raleigh, NC, USA e-mail: smsulli2@ncsu.edu Received by the editors November 21, 2006; revised April 3, 2007. Sidman was partially supported by NSF grant DMS-0600471 and the Clare Boothe Luce Program. AMS subject classification: Primary: 13P10; secondary: 14M99. c ©Canadian Mathematical Society 2009. 930