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10
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.
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 8 (3 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.
THE GEOMETRY OF INDEPENDENCE TREE MODELS WITH HIDDEN VARIABLES
"... Abstract. In this paper we investigate the geometry of undirected discrete graphical models of trees when all the variables in the system are binary, where leaves represent the observable variables and where the inner nodes are unobserved. We obtain a full geometric description of these models which ..."
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Abstract. In this paper we investigate the geometry of undirected discrete graphical models of trees when all the variables in the system are binary, where leaves represent the observable variables and where the inner nodes are unobserved. We obtain a full geometric description of these models which is given by polynomial equations and inequalities. We also give exact formulas for their parameters in terms of the marginal probability over the observed variables. Our analysis is based on combinatorial results generalizing the notion of cumulants and introduce a novel use of Möbius functions on partially ordered sets. The geometric structure we obtain links to the notion of a tree metric considered in phylogenetic analysis and to some interesting determinantal formulas involving hyperdeterminants of 2 × 2 × 2 tables as defined in [19]. 1.
DOI: 10.3150/07BEJ6144 Symmetric measures via moments
"... 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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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.
LIKELIHOOD RATIO TESTS AND SINGULARITIES 1
, 2009
"... 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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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. 1. Introduction. Let PΘ
CRiSM Paper No. 0713, www.warwick.ac.uk/go/crism ALGEBRAIC CAUSALITY: BAYES NETS AND BEYOND
"... Abstract. The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of Causal Bayesian Networks has also been investigated in ..."
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Abstract. The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of Causal Bayesian Networks has also been investigated in this context. After reviewing these newer relationships, we proceed to demonstrate that many of the ideas embodied in the concept of a “causal model ” can be more generally expressed directly in terms of a partial order and a family of polynomial maps. The more conventional graphical constructions, when available, remain a powerful tool. Key words. Bayesian networks, causality, computational commutative algebra. AMS(MOS) subject classifications. 1. Introduction. There
Maximum Entropy in the framework of Algebraic Statistics: A First Step
, 2007
"... Abstract. Algebraic statistics is a recently evolving field, where one would treat statistical models as algebraic objects and thereby use tools from computational commutative algebra and algebraic geometry in the analysis and computation of statistical models. In this approach, calculation of param ..."
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Abstract. Algebraic statistics is a recently evolving field, where one would treat statistical models as algebraic objects and thereby use tools from computational commutative algebra and algebraic geometry in the analysis and computation of statistical models. In this approach, calculation of parameters of statistical models amounts to solving set of polynomial equations in several variables, for which one can use celebrated Gröbner basis theory. Owing to the important role of information theory in statistics, this paper as a first step, explores the possibility of describing maximum and minimum entropy (ME) models in the framework of algebraic statistics. We show that MEmodels are toric models (a class of algebraic statistical models) when the constraint functions (that provide the information about the underlying random variable) are integer valued functions, and maximum entropy distributions can be calculated by solving set of (Laurent) polynomial equations when expected values of constraint functions are supplied as sample means. 2 1.