We provide a classification of graphical models according to their representation as exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical (DAG) models and chain graphs with no hidden variables, including DAG models with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). A SEF is a finite union of CEFs of various dimensions satisfying some regularity conditions. The main results of this paper are that graphical models are SEFs and that many graphical models are not CEFs. That is, roughly speaking, graphical models when viewed as exponential families correspond to a set of smooth manifolds of various dimensions and usually not to a single smooth manifold. These results are discussed in the context of model selection. Keywords : Bayesian networks, graphical models, hidden variables, cur...

We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G;PO(3)) = Hom(G;PO(3))==PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties. 1 Introduction The goal of this paper is to understand representation varieties of Artin and Shephard groups and thereby obtain information on Serre's problem of determining which finitely-presented groups are fundamental groups of smooth complex (not necessarily compact) algebraic varieties. The first examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties were given by J. Morgan [Mo1], [Mo2]. We find a new class of such examples which consists of certain Artin and Shephard groups. Since all Artin and Shephard...

We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements. 1. Introduction This paper deals with moduli spaces of planar linkages. An abstract linkage (L; `) is a graph L with a positive real number `(e) assigned to each edge e. We assume that we have chosen a distinguished oriented edge e = [v 1 v 2 ] in L. The moduli space M(L) of planar realizations of L := (L; `; e ) is the set 1 of maps OE from the vertex set of L into the Euclidean plane R 2 (which will be identified with the complex plane C ) such that ffl jOE(v) \Gamma OE(w)j 2 = (`[vw]) 2 for each edge [vw] of L. ffl OE(v 1 ) = (0; 0). ffl OE(v 2 ) = (`(e ); 0). Clearly these conditions give M(L) a natural structure of a real-algebraic set in R 2r where r is the number of vertices...

We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). An SEF is a finite union of CEFs satisfying a frontier condition. In addition, we illustrate how one can automatically generate independence and non-independence constraints on the distributions over the observable variables implied by a Bayesian network with hidden variables. The relevance of these results for model selection is examined. 1 Introduction A graphical model is a family of probability distributions. The set of distributions associated with a graphical model are usually define...

ABSTRACT. Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some three-person game, and also to the set of totally mixed Nash equilibria of an N-person game in which each player has two pure strategies. From the Nash-Tognoli Theorem it follows that every compact differentiable manifold can be encoded as the set of totally mixed Nash equilibria of some game. Moreover, there exist isolated Nash equilibria of arbitrary topological degree. 1.

In this paper we describe the trivial summand for monodromy around a fibre of a polynomial map C n → C, generalising and clarifying work of Artal Bartolo, Cassou-Noguès and Dimca [2], who proved similar results under strong restrictions on the homology of the general fibre and singularities of the other fibres. They also

We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages and arrangements of lines in the projective plane. 1 Introduction In this paper we describe the results of our papers [KM6] and [KM8]. Both papers deal with moduli spaces of elementary geometric objects, the first with arrangements of lines in the projective plane, the second with linkages in the Euclidean plane. We conclude the paper with a brief sketch from [KM6] of how the study of arrangements of lines leads to examples of Artin and Shephard groups which are not fundamental groups of smooth (not necessarily compact) complex algebraic varieties (Theorem 14.1). The problem of deciding which finitely-presented groups are the fundamental groups of smooth complex algebraic varieties is called "Serre's problem" in [Mo]. Our contribution to this problem is based on our discovery of the connection between configuration spaces of elemen...

An algebraic version of Kashiwara and Schapira's calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and King's numerical conditions for a stratified set of dimension three to be algebraic. These properties, which include generalizations of the invariants modulo 4, 8, and 16 of Coste and Kurdyka, are defined using the link operator on the ring of constructible functions.

Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield 2

Abstract. We revisit Akbulut and King’s first example of a compact semialgebraic set which satisfies Sullivan’s local Euler characteristic condition, but which is not homeomorphic to an algebraic set. A nontrivial obstruction is computed using the link operator on the ring of constructible functions. There are many local topological conditions satisfied by real algebraic sets. The simplest of these was discovered by Sullivan [5] more than thirty years ago: The link of every point has even Euler characteristic. In low dimensions this is the only obstruction for a set to be algebraic. More precisely, if a compact semialgebraic set of dimension less than three satisfies Sullivan’s condition, then it is homeomorphic to an algebraic set. Akbulut and King [1] found an example of a compact 3-dimensional semialgebraic set satisfying Sullivan’s condition which is not homeomorphic to an algebraic set, and this is the example we will discuss. The method we use to compute the obstruction is due to Parusiński and the author [2]. We have also found an enormous list of independent obstructions in dimension four [3]. The yoga of algebraically constructible functions used to prove our obstructions vanish for algebraic sets is presented in the paper of Isabelle Bonnard in this volume.