Abstract. 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 well-quasi-ordering 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 finite-dimensional 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. 1.

Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic 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 semi-algebraic 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 chi-square distributions may arise. While boundary points often lead to mixtures of chi-square distributions, singularities give rise to non-standard limits. We demonstrate that minima of chi-square 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.

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 hill-climbing methods. 1.

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.

Abstract. 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: off-diagonal 3 × 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model. Our proof is computational: inspired by work of Aschenbrenner and Hillar we introduce the concept of G-Gröbner basis, where G is a monoid acting on an infinite set of variables, and we report on a computation that yielded a finite G-Gröbner basis of I relative to the monoid G of strictly increasing functions N → N. 1. Introduction and

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,

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

Abstract. A symmetric ideal I ⊆ R = K[x1, x2,...] is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gröbner bases for symmetric ideals in the infinite dimensional polynomial ring R. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of R. 1.

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.

Abstract. Let R = K[X] be the polynomial ring in infinitely many indeterminates X over a field K, and let SX be the symmetric group of X. The group SX acts naturally on R, and this in turn gives R the structure of a module over the group ring R[SX]. A recent theorem of Aschenbrenner and Hillar states that the module R is Noetherian. We address whether submodules of R can have any number of minimal generators, answering this question positively. Let R = K[X] be the polynomial ring in infinitely many indeterminates X over a field K. Write SX (resp. SN) for the symmetric group of X (resp. {1,..., N}) and R[SX] for its (left) group ring, which acts naturally on R. A symmetric ideal I ⊆ R is an R[SX]-submodule of R. Aschenbrenner and Hillar recently proved [1] that all symmetric ideals are finitely generated over R[SX]. They were motivated by finiteness questions in chemistry [2] and algebraic statistics [4]. In proving the Noetherianity of R, it was shown that a symmetric ideal I has a special, finite set of generators called a minimal Gröbner