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31
Stratified exponential families: Graphical models and model selection
 ANNALS OF STATISTICS
, 2001
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On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties
, 1999
"... 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 applicati ..."
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Cited by 29 (7 self)
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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 finitelypresented 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 finitelypresented groups are fundamental groups of smooth complex (not necessarily compact) algebraic varieties. The first examples of finitelypresented 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...
Graphical models and exponential families
 In Proceedings of the 14th Annual Conference on Uncertainty in Arti cial Intelligence (UAI98
, 1998
"... 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, includin ..."
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Cited by 19 (1 self)
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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 nonindependence 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
Universality of Nash equilibria
 Mathematics of Operations Research
, 2003
"... ABSTRACT. Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some threeperson game, and also to the set of totally mixed Nash equilibria of an Nperson game in which each player has two pure strategies. From the NashTognoli Theorem it follows that every compa ..."
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Cited by 10 (2 self)
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ABSTRACT. Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some threeperson game, and also to the set of totally mixed Nash equilibria of an Nperson game in which each player has two pure strategies. From the NashTognoli 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.
Algebraically constructible functions and signs of polynomials
 Manuscripta Math
, 1997
"... Abstract. Let W be a real algebraic set. We show that the following families of integervalued functions on W coincide: (i) the functions of the form w → χ(Xw), where Xw are the fibres of a regular morphism f: X → W of real algebraic sets, (ii) the functions of the form w → χ(Xw), where Xw are the fi ..."
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Cited by 10 (3 self)
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Abstract. Let W be a real algebraic set. We show that the following families of integervalued functions on W coincide: (i) the functions of the form w → χ(Xw), where Xw are the fibres of a regular morphism f: X → W of real algebraic sets, (ii) the functions of the form w → χ(Xw), where Xw are the fibres of a proper regular morphism f: X → W of real algebraic sets, (iii) the finite sums of signs of polynomials on W. Such functions are called algebraically constructible on W. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. 1.
Quantifier elimination for statistical problems
 In Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence (UAI99
, 1999
"... Recent improvements on Tarski's procedure for quantifier elimination in the first order theory of real numbers makes it feasible to solve small instances of the following problems completely automatically: 1. listing all equality and inequality constraints implied by a graphical model with hidden va ..."
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Cited by 8 (0 self)
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Recent improvements on Tarski's procedure for quantifier elimination in the first order theory of real numbers makes it feasible to solve small instances of the following problems completely automatically: 1. listing all equality and inequality constraints implied by a graphical model with hidden variables. 2. Comparing graphical models with hidden variables (i.e., model equivalence, inclusion, and overlap). 3. Answering questions about the identification of a model or portion of a model, and about bounds on quantities derived from a model. 4. Determining whether an independence assertion is implied from a given set of independence assertions. We discuss the foundations of quantifier elimination and demonstrate its application to these problems. 1
Algebraically constructible functions: real algebra and topology
, 2001
"... Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a description of local obstructions for a topological space ..."
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Cited by 7 (0 self)
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Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a description of local obstructions for a topological space to be homeomorphic to a real algebraic set.
Algebraically Constructible Functions
, 1997
"... 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 gen ..."
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Cited by 7 (3 self)
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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.
Moduli Spaces of Linkages and Arrangements
 Advances in Geometry, volume 172 of Progress in Mathematics
, 1997
"... We prove realizability theorems for vectorvalued polynomial mappings, realalgebraic 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 pap ..."
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Cited by 7 (1 self)
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We prove realizability theorems for vectorvalued polynomial mappings, realalgebraic 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 finitelypresented 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...