Results 1  10
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36
Algebraic Geometry of Bayesian Networks
 Journal of Symbolic Computation
, 2005
"... We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. 1 ..."
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Cited by 61 (8 self)
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We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. 1
Solving the likelihood equations
, 2004
"... Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include ..."
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Cited by 31 (7 self)
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Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include models specified by rank conditions on matrices and the JukesCantor models of phylogenetics. The maximum likelihood degree of a generic complete intersection is also determined.
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 14 (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.
Three counterexamples on semigraphoids
"... Semigraphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group Sn. In this paper we resolve two problems on semigraphoids posed in Studen´y’s book [ ..."
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Cited by 9 (5 self)
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Semigraphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group Sn. In this paper we resolve two problems on semigraphoids posed in Studen´y’s book [19], and we answer a related question of Postnikov, Reiner, and Williams on generalized permutohedra [18]. We also study the semigroup and the toric ideal associated with semigraphoids.
Conjunctive bayesian networks
 Bernoulli
, 2007
"... Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. (1999) by allowing th ..."
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Cited by 8 (2 self)
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Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. (1999) by allowing the occurrence of an event to depend on more than one predecessor event. The present paper studies the statistical and algebraic properties of CBNs. We determine the maximum likelihood parameters and present a combinatorial solution to the model selection problem. Our method performs well on two datasets where the events are HIV mutations associated with drug resistance. Concluding with a study of the algebraic properties of CBNs, we show that CBNs are toric varieties after a coordinate transformation and that their ideals possess a quadratic Gröbner basis.
Markov Bases of Binary Graph Models
, 2008
"... This paper is concerned with the topological invariant of a graph given by the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We describe a degree four Markov basis for the model when the underlying graph is a cycle and generalize this resul ..."
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Cited by 7 (5 self)
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This paper is concerned with the topological invariant of a graph given by the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We describe a degree four Markov basis for the model when the underlying graph is a cycle and generalize this result to the complete bipartite graph K2,n. We also give a combinatorial classification of degree two and three Markov basis moves as well as a Buchbergerfree algorithm to compute moves of arbitrary given degree. Finally, we compute the algebraic degree of the model when the underlying graph is a forest.
Graphical methods for efficient likelihood inference in gaussian covariance models
 Journal of Machine Learning
, 2008
"... Abstract. In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bidirected graph into a maximal ancestral graph that (i) represents the same independence structure as the origi ..."
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Cited by 7 (2 self)
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Abstract. In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bidirected graph into a maximal ancestral graph that (i) represents the same independence structure as the original bidirected graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bidirected edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts. 1.
Symmetric measures via moments
, 2006
"... Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a fini ..."
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Cited by 5 (1 self)
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Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a finite group of (nonsingular) linear transformations and polynomials invariant under the same group. Two specific aspects of the connection are discussed: generalization of the (uniqueness part of the multivariate) problem of moments and loglinear, or toric, modeling by expansion of invariant terms. A distribution of minuscule subimages extracted from a large database of natural images is analyzed to illustrate the above concepts.
Markov chains, quotient ideals and connectivity with positive margins
 In Algebraic and Geometric Methods in Statistics
, 2010
"... ..."
CUMULANT VARIETIES
, 2005
"... For discrete distributions in R d on a finite support D probabilities and moments are algebraically related. Intuitively, if there are n = D  support points then there are n probabilities p(x), x ∈ D and n basic moments. By suitable interpolation of the probabilities using a Gröbner method, high o ..."
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Cited by 3 (0 self)
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For discrete distributions in R d on a finite support D probabilities and moments are algebraically related. Intuitively, if there are n = D  support points then there are n probabilities p(x), x ∈ D and n basic moments. By suitable interpolation of the probabilities using a Gröbner method, high order moments can be express linearly in terms of n basic moments, relative to the Gröbner basis. A main result is that high order cumulates can also be expressed as polynomial functions of n low order moments and cumulates. This means that statistical models which can be expressed via an algebraically variety for the basic probabilities and moments, such as graphical models, therefore induce a variety for the basic cumulants, which we shall call the cumulant variety. In is important to stress that the cumulant variety depends on the monomial ordering defining the original Gröbner basis.