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24
Finite generation of symmetric ideals
 TRANS. AMER. MATH. SOC
, 2005
"... Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R ..."
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Cited by 13 (8 self)
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Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R[SX]. We prove that all ideals of R invariant under the action of SX are finitely generated as R[SX]modules. The proof involves introducing a certain wellquasiordering on monomials and developing a theory of Gröbner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finitedimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.
Likelihood ratio tests and singularities
 Ann. Statist
, 2008
"... Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semialgebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisf ..."
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Cited by 9 (3 self)
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Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semialgebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff’s theorem hold for semialgebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point. At boundary points or singularities, the tangent cone need not be a linear space and limiting distributions other than chisquare distributions may arise. While boundary points often lead to mixtures of chisquare distributions, singularities give rise to nonstandard limits. We demonstrate that minima of chisquare random variables are important for locally identifiable models, and in a study of the factor analysis model with one factor, we reveal connections to eigenvalues of Wishart matrices.
Algebraic geometry of Gaussian Bayesian networks. Adv
 Department of Mathematics North Carolina State University Raleigh
"... Abstract. Conditional independence models in the Gaussian case are algebraic varieties in the cone of positive definite covariance matrices. We study these varieties in the case of Bayesian networks, with a view towards generalizing the recursive factorization theorem to situations with hidden varia ..."
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Cited by 7 (1 self)
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Abstract. Conditional independence models in the Gaussian case are algebraic varieties in the cone of positive definite covariance matrices. We study these varieties in the case of Bayesian networks, with a view towards generalizing the recursive factorization theorem to situations with hidden variables. In the case when the underlying graph is a tree, we show that the vanishing ideal of the model is generated by the conditional independence statements implied by graph. We also show that the ideal of any Bayesian network is homogeneous with respect to a multigrading induced by a collection of upstream random variables. This has a number of important consequences for hidden variable models. Finally, we relate the ideals of Bayesian networks to a number of classical constructions in algebraic geometry including toric degenerations of the Grassmannian, matrix Schubert varieties, and secant varieties. 1.
Bayesian learning of measurement and structural models
 23rd International Conference on Machine Learning
, 2006
"... We present a Bayesian search algorithm for learning the structure of latent variable models of continuous variables. We stress the importance of applying search operators designed especially for the parametric family used in our models. This is performed by searching for subsets of the observed vari ..."
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Cited by 5 (3 self)
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We present a Bayesian search algorithm for learning the structure of latent variable models of continuous variables. We stress the importance of applying search operators designed especially for the parametric family used in our models. This is performed by searching for subsets of the observed variables whose covariance matrix can be represented as a sum of a matrix of low rank and a diagonal matrix of residuals. The resulting search procedure is relatively efficient, since the main search operator has a branch factor that grows linearly with the number of variables. The resulting models are often simpler and give a better fit than models based on generalizations of factor analysis or those derived from standard hillclimbing methods. 1.
Algebraic Techniques for Gaussian Models
, 2006
"... Abstract: Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be employed to solve equation systems arising in maxim ..."
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Cited by 5 (2 self)
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Abstract: Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be employed to solve equation systems arising in maximum likelihood estimation and parameter identification, but they also permit to study model singularities at which standard asymptotic approximations to the distribution of estimators and test statistics may no longer be valid. This paper demonstrates such applications of algebraic geometry in selected examples of Gaussian models, thereby complementing the existing literature on models for discrete variables. MSC 2000: 62H05, 62H12 Key words: Algebraic statistics, multivariate normal distribution, parameter identification, singularities 1
Equivariant Gröbner bases and the Gaussian twofactor model
, 2009
"... We show that the kernel I of the ring homomorphism R[yij  i, j ∈ N, i> j] → R[si, ti  i ∈ N] determined by yij ↦ → sisj +titj is generated by two types of polynomials: offdiagonal 3 × 3minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian twofactor m ..."
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Cited by 5 (0 self)
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We show that the kernel I of the ring homomorphism R[yij  i, j ∈ N, i> j] → R[si, ti  i ∈ N] determined by yij ↦ → sisj +titj is generated by two types of polynomials: offdiagonal 3 × 3minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian twofactor model. Our proof is computational: inspired by work of Aschenbrenner and Hillar we introduce the concept of GGröbner basis, where G is a monoid acting on an infinite set of variables, and we report on a computation that yielded a finite GGröbner basis of I relative to the monoid G of strictly increasing functions N → N.
FINITENESS FOR THE kFACTOR MODEL AND CHIRALITY VARIETIES
, 2008
"... This paper deals with two families of algebraic varieties arising from applications. First, the kfactor model in statistics, consisting of n×n covariance matrices of n observed Gaussian variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired ..."
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Cited by 5 (1 self)
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This paper deals with two families of algebraic varieties arising from applications. First, the kfactor model in statistics, consisting of n×n covariance matrices of n observed Gaussian variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired by applications in chemistry. A point in such a chirality variety records chirality measurements of all ksubsets among an nset of ligands. Both classes of varieties are given by a parameterisation, while for applications having polynomial equations would be desirable. For instance, such equations could be used to test whether a given point lies in the variety. We prove that in a precise sense, which is different for the two classes of varieties, these equations are finitely characterisable when k is fixed and n grows.
Open problems in algebraic statistics
"... Abstract. 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 mathematical problems in this emerging field, with main emphasis on graphical models with hidden v ..."
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Cited by 4 (1 self)
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Abstract. 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 mathematical problems in this emerging field, with main emphasis on graphical models with hidden variables, maximum likelihood estimation, and multivariate Gaussian distributions. These are notes from a lecture presented at the IMA in Minneapolis during the 2006/07 program on Applications of Algebraic Geometry. Key words. Algebraic statistics, contingency tables, hidden variables, Schur modules,
FINITE GRÖBNER BASES IN INFINITE DIMENSIONAL POLYNOMIAL RINGS AND APPLICATIONS
"... Abstract. We introduce the theory of monoidal Gröbner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Gröbner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finitenes ..."
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Cited by 3 (1 self)
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Abstract. We introduce the theory of monoidal Gröbner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Gröbner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finiteness results in commutative algebra and applications. A basic theorem of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry. Using this machinery, we give new streamlined proofs of some classical finiteness theorems in algebraic statistics as well as a proof of the independent set conjecture of Ho¸sten and the second author. 1.
Three notions of tropical rank for symmetric matrices
 In preparation
, 2009
"... Abstract. We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certa ..."
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Cited by 2 (1 self)
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Abstract. We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull. Résumé. Nous introduisons et étudions trois notions différentes de rang tropical pour des matrices symétriques et des matrices de dissimilarité, en utilisant des décompositions minimales en matrices symétriques de rang 1, en matrices d’arbres étoiles, et en matrices d’arbres. Nos résultats donnent lieu à une étude détaillée des ensembles des sécantes tropicales de certaines jolies variétés tropicales, y compris la grassmannienne tropicale. En particulier, nous déterminons la dimension de chaque ensemble des sécantes, l’enveloppe convexe de la variété, ainsi que, dans la plupart des cas, le plus petit ensemble des sécantes qui est égal à l’enveloppe convexe. Resumen. Introducimos y estudiamos tres nociones diferentes de rango tropical para matrices simétricas y matrices de disimilaridad, utilizando las decomposiciones minimales en matrices simétricas de rango 1 en matrices de árboles estrella y en matrices de árboles. Nuestros resultados brindan un estudio detallado de conjuntos de secantes de ciertas variedades tropicales clásicas, incluyendo la grassmanniana tropical. En particular, determinamos la dimensión de cada conjunto de dichas secantes, la cápsula convexa de la variedad, y también, en la mayoridad de los casos, el conjunto más pequeño de secantes que coincide con la cápsula convexa.