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45
On the ideals of Secant Varieties to certain rational varieties
, 2006
"... If X ⊂ Pn is a reduced and irreducible projective variety, it is interesting to find the equations describing the (higher) secant varieties of X. In this paper we find those equations in the following cases: • X = Pn1 ×... × Pnt × Pn is the Segre embedding of the product and n is “large ” with res ..."
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Cited by 15 (1 self)
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If X ⊂ Pn is a reduced and irreducible projective variety, it is interesting to find the equations describing the (higher) secant varieties of X. In this paper we find those equations in the following cases: • X = Pn1 ×... × Pnt × Pn is the Segre embedding of the product and n is “large ” with respect to the ni (Theorem 2.4); • X is a SegreVeronese embedding of some products with 2 or three factors; • X is a Del Pezzo surface.
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.
Discrete chain graph models
 Bernoulli
, 2009
"... The statistical literature discusses different types of Markov properties for chain graphs that lead to four possible classes of chain graph Markov models. The different models are rather well understood when the observations are continuous and multivariate normal, and it is also known that one mode ..."
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Cited by 11 (1 self)
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The statistical literature discusses different types of Markov properties for chain graphs that lead to four possible classes of chain graph Markov models. The different models are rather well understood when the observations are continuous and multivariate normal, and it is also known that one model class, referred to as models of LWF (Lauritzen–Wermuth–Frydenberg) or block concentration type, yields discrete models for categorical data that are smooth. This paper considers the structural properties of the discrete models based on the three alternative Markov properties. It is shown by example that two of the alternative Markov properties can lead to nonsmooth models. The remaining model class, which can be viewed as a discrete version of multivariate regressions, is proven to comprise only smooth models. The proof employs a simple change of coordinates that also reveals that the model’s likelihood function is unimodal if the chain components of the graph are complete sets.
Likelihood ratio tests and singularities
 Ann. Statist
, 2008
"... Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semialgebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisf ..."
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Cited by 9 (3 self)
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Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semialgebraic 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 semialgebraic 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 chisquare distributions may arise. While boundary points often lead to mixtures of chisquare distributions, singularities give rise to nonstandard limits. We demonstrate that minima of chisquare 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.
Generalized measurement models
, 2004
"... Given a set of random variables, it is often the case that their associations can be explained by hidden common causes. We present a set of welldefined assumptions and a provably correct algorithm that allow us to identify some of such hidden common causes. The assumptions are fairly general and so ..."
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Cited by 7 (4 self)
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Given a set of random variables, it is often the case that their associations can be explained by hidden common causes. We present a set of welldefined assumptions and a provably correct algorithm that allow us to identify some of such hidden common causes. The assumptions are fairly general and sometimes weaker than those used in practice by, for instance, econometricians, psychometricians, social scientists and in many other fields where latent variable models are important and tools such as factor analysis are applicable. The goal is automated knowledge discovery: identifying latent variables that can be used across diferent applications and causal models and throw new insights over a data generating process. Our approach is evaluated throught simulations and three realworld cases.
Toric statistical models: Parametric and binomial representations, Preprint 498, Dipartimento di Matematica, Università di
, 2004
"... Abstract Toric models have been recently introduced in the analysis of statistical models for categorical data. The main improvement with respect to classical loglinear models is shown to be a simple representation of structural zeros. In this paper we analyze the geometry of toric models, showing ..."
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Cited by 6 (1 self)
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Abstract Toric models have been recently introduced in the analysis of statistical models for categorical data. The main improvement with respect to classical loglinear models is shown to be a simple representation of structural zeros. In this paper we analyze the geometry of toric models, showing that a toric model is the disjoint union of a number of loglinear models. Moreover, we discuss the connections between the parametric and algebraic representations. The notion of Hilbert basis of a lattice is proved to allow a special representation among all possible parametrizations.
On the Number of Samples Needed to Learn the Correct Structure of a Bayesian Network
"... Bayesian Networks (BNs) are useful tools giving a natural and compact representation of joint probability distributions. In many applications one needs to learn a Bayesian Network (BN) from data. In this context, it is important to understand the number of samples needed in order to guarantee a succ ..."
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Bayesian Networks (BNs) are useful tools giving a natural and compact representation of joint probability distributions. In many applications one needs to learn a Bayesian Network (BN) from data. In this context, it is important to understand the number of samples needed in order to guarantee a successful learning. Previous works have studied BNs sample complexity, yet they mainly focused on the requirement that the learned distribution will be close to the original distribution which generated the data. In this work, we study a different aspect of the learning task, namely the number of samples needed in order to learn the correct structure of the network. We give both asymptotic results (lower and upperbounds) on the probability of learning a wrong structure, valid in the large sample limit, and experimental results, demonstrating the learning behavior for feasible sample sizes. 1
Factorized Normalized Maximum Likelihood Criterion for Learning Bayesian Network Structures
, 2008
"... This paper introduces a new scoring criterion, factorized normalized maximum likelihood, for learning Bayesian network structures. The proposed scoring criterion requires no parameter tuning, and it is decomposable and asymptotically consistent. We compare the new scoring criterion to other scoring ..."
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Cited by 6 (4 self)
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This paper introduces a new scoring criterion, factorized normalized maximum likelihood, for learning Bayesian network structures. The proposed scoring criterion requires no parameter tuning, and it is decomposable and asymptotically consistent. We compare the new scoring criterion to other scoring criteria and describe its practical implementation. Empirical tests confirm its good performance.
Maximum likelihood estimation in latent class models for contingency table data
 In Algebraic and Geometric Methods in Statistics
, 2008
"... 1 page 1 ..."
On the Validity of the Likelihood Ratio and Maximum Likelihood Methods
 J. Statist. Plann. Inference
, 2000
"... When the null or alternative hypothesis of a statistical testing problem is a composite of regions of varying dimensionality, the likelihood ratio test is statistically inappropriate. Its inappropriateness is revealed not by its performance under the NeymanPearson criterion but by the fact that it ..."
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Cited by 5 (0 self)
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When the null or alternative hypothesis of a statistical testing problem is a composite of regions of varying dimensionality, the likelihood ratio test is statistically inappropriate. Its inappropriateness is revealed not by its performance under the NeymanPearson criterion but by the fact that it yields incorrect inferences in certain regions of the sample space due to its inability to adapt to the diering dimensions in the composite hypothesis. Maximum likelihood estimators and associated model selection procedures also are inappropriate for such composite models. Tests and estimators based on the pvalues associated with the various regions that determine the composite model are more appropriate for this geometry. Similar issues arise when the boundary of the null or alternative hypothesis is a composite of regions of varying dimensionality. Corresponding author: Michael D. Perlman, Department of Statistics, University of Washington, Seattle, WA 98195. email: michael@ms.washin...