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45
Toric ideals of phylogenetic invariants
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 2005
"... 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 tra ..."
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Cited by 52 (13 self)
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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 minimal generators and Gröbner bases for these toric ideals. For the JukesCantor and Kimura models on a binary tree, our Gröbner basis consists of quadrics, cubics and quartics.
Phylogenetic ideals and varieties for the general Markov model
 Math. Biosciences
"... Abstract. The general Markov model of the evolution of biological sequences along a tree leads to a parameterization of an algebraic variety. Understanding this variety and the polynomials, called phylogenetic invariants, which vanish on it, is a problem within the broader area of Algebraic Statisti ..."
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Cited by 42 (5 self)
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Abstract. The general Markov model of the evolution of biological sequences along a tree leads to a parameterization of an algebraic variety. Understanding this variety and the polynomials, called phylogenetic invariants, which vanish on it, is a problem within the broader area of Algebraic Statistics. For an arbitrary trivalent tree, we determine the full ideal of invariants for the 2state model, establishing a conjecture of PachterSturmfels. For the κstate model, we reduce the problem of determining a defining set of polynomials to that of determining a defining set for a 3leaved tree. Along the way, we prove several new cases of a conjecture of GarciaStillmanSturmfels on certain statistical models on star trees, and reduce their conjecture to a family of subcases. 1.
ON THE IDEALS OF SECANT VARIETIES OF SEGRE VARIETIES
, 2003
"... We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture settheoretically for an arbitrary n ..."
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Cited by 40 (8 self)
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We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture settheoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.
On the toric algebra of graphical models
, 2006
"... We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a loglinear model, or other more general exponential models. For decomposable graphical models these conditions are equivalent to a set of con ..."
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Cited by 36 (6 self)
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We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a loglinear model, or other more general exponential models. For decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the Hammersley–Clifford theorem; however, we show that for nondecomposable graphical models they are not. We also show that nondecomposable models can have nonrational maximum likelihood estimates. These results are used to give several novel characterizations of decomposable graphical models.
Tropical geometry of statistical models
 Proceedings of the National Academy of Sciences, 101:16132–16137
, 2004
"... This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sumproduct algorit ..."
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Cited by 31 (4 self)
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This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sumproduct algorithm is an efficient tool for evaluating specific coordinates. The question addressed here is how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. A key role is played by the Newton polytope of a statistical model. Our results are applied to the hidden Markov model and to the general Markov model on a binary tree. 1 Algebraic Statistics, Tropical Geometry, and Inference This paper presents a unified mathematical framework for probabilistic inference with statistical models, such as graphical models. Our approach is summarized as follows: (a) Statistical models are algebraic varieties. (b) Every algebraic variety can be tropicalized. (c) Tropicalized statistical models are fundamental for parametric inference. By a statistical model we mean a family of joint probability distributions for a collection of discrete
RHODES,J.A.(2009). Identifiability of parameters in latent structure models with many observed variables
 Ann. Statist
"... While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones, we demonstr ..."
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Cited by 21 (4 self)
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While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones, we demonstrate a general approach for establishing identifiability utilizing algebraic arguments. A theorem of J. Kruskal for a simple latentclass model with finite state space lies at the core of our results, though we apply it to a diverse set of models. These include mixtures of both finite and nonparametric product distributions, hidden Markov models and random graph mixture models, and lead to a number of new results and improvements to old ones. In the parametric setting, this approach indicates that for such models, the classical definition of identifiability is typically too strong. Instead generic identifiability holds, which implies that the set of nonidentifiable parameters has measure zero, so that parameter inference is still meaningful. In particular, this sheds light on the properties of finite mixtures of Bernoulli products, which have been used for decades despite being known to have nonidentifiable parameters. In the nonparametric setting, we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed, but we explicitly describe the conditions. 1. Introduction. Statistical
Combinatorial secant varieties
 Quart. J. Pure Applied Math
"... Abstract. The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Per ..."
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Cited by 16 (1 self)
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Abstract. The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal of the secant ideal coincides with the secant ideal of the initial ideal. For toric varieties, this leads to the notion of delightful triangulations of convex polytopes. 1.
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 15 (1 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
On the ideals and singularities of secant varieties of segre varieties
 math.AG/0601452, Bull. London Math. Soc
"... Abstract. We find minimal generators for the ideals of secant varieties of Segre varieties in the cases of σk(P 1 × P n × P m) for all k, n, m, σ2(P n × P m × P p × P r) for all n, m, p, r (GSS conjecture for four factors), and σ3(P n ×P m ×P p) for all n, m, p and prove they are normal with rationa ..."
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Cited by 14 (4 self)
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Abstract. We find minimal generators for the ideals of secant varieties of Segre varieties in the cases of σk(P 1 × P n × P m) for all k, n, m, σ2(P n × P m × P p × P r) for all n, m, p, r (GSS conjecture for four factors), and σ3(P n ×P m ×P p) for all n, m, p and prove they are normal with rational singularities in the first case and arithmetically CohenMacaulay in the second two. 1.
Algebraic statistical models
 Statistica Sinica
"... Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semialgebraic subsets of the parameter space of a reference model with nice properti ..."
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Cited by 13 (4 self)
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Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semialgebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an ‘algebraic exponential family’. This new definition provides a unified framework for the study of statistical models with algebraic structure. In this paper we review the ingredients to this definition and illustrate in examples how computational algebraic geometry can be used to solve problems arising in statistical inference in algebraic models. Key words and phrases: Algebraic statistics, computational algebraic geometry, exponential family, maximum likelihood estimation, model invariants, singularities. 1.