Results 1  10
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70
Optimal Structure Identification with Greedy Search
, 2002
"... In this paper we prove the socalled "Meek Conjecture". In particular, we show that if a is an independence map of another DAG then there exists a finite sequence of edge additions and covered edge reversals in such that (1) after each edge modification and (2) after all mod ..."
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Cited by 239 (1 self)
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In this paper we prove the socalled "Meek Conjecture". In particular, we show that if a is an independence map of another DAG then there exists a finite sequence of edge additions and covered edge reversals in such that (1) after each edge modification and (2) after all modifications H.
ANCESTRAL GRAPH MARKOV MODELS
, 2002
"... This paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all DAG independence models. This class of graphs, called maximal ancestral graphs, has two attractive features: there is at most one edge between each pair of verti ..."
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Cited by 115 (22 self)
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This paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all DAG independence models. This class of graphs, called maximal ancestral graphs, has two attractive features: there is at most one edge between each pair of vertices; every missing edge corresponds to an independence relation. These features lead to a simple parameterization of the corresponding set of distributions in the Gaussian case.
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 86 (7 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
Hierarchical Latent Class Models for Cluster Analysis
 Journal of Machine Learning Research
, 2002
"... Latent class models are used for cluster analysis of categorical data. Underlying such a model is the assumption that the observed variables are mutually independent given the class variable. A serious problem with the use of latent class models, known as local dependence, is that this assumption is ..."
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Cited by 61 (12 self)
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Latent class models are used for cluster analysis of categorical data. Underlying such a model is the assumption that the observed variables are mutually independent given the class variable. A serious problem with the use of latent class models, known as local dependence, is that this assumption is often untrue. In this paper we propose hierarchical latent class models as a framework where the local dependence problem can be addressed in a principled manner. We develop a searchbased algorithm for learning hierarchical latent class models from data. The algorithm is evaluated using both synthetic and realworld data.
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 54 (7 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 Hammersleyâ€“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.
Asymptotic Model Selection for Naive Bayesian Networks
 In Proc. of the 18th Conference on Uncertainty in Artificial Intelligence (UAI02
, 2002
"... We develop a closed form asymptotic formula to compute the marginal likelihood of data given a naive Bayesian network model with two hidden states and binary features. ..."
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Cited by 49 (3 self)
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We develop a closed form asymptotic formula to compute the marginal likelihood of data given a naive Bayesian network model with two hidden states and binary features.
Finding optimal bayesian networks
 In UAI
, 2002
"... In this paper, we derive optimality results for greedy Bayesiannetwork search algorithms that perform singleedge modications at each step and use asymptotically consistent scoring criteria. Our results extend those of Meek (1997) and Chickering (2002), who demonstrate that in the limit of large ..."
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Cited by 48 (3 self)
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In this paper, we derive optimality results for greedy Bayesiannetwork search algorithms that perform singleedge modications at each step and use asymptotically consistent scoring criteria. Our results extend those of Meek (1997) and Chickering (2002), who demonstrate that in the limit of large datasets, if the generative distribution is perfect with respect to a DAG dened over the observable variables, such search algorithms will identify this optimal (i.e. generatve) DAGmodel. We relax their assumption about the generative distribtion, and assume only that this distribution satises the composition property over the observable variables, which is a more realistic assumption for real domains. Under this assumption, we guarantee that the search algorithms identify an inclusionoptimal model; that is, a model that (1) contains the generative distribution and (2) has no submodel that contains this distribution. In addition, we show that the composition property is guaranteed to hold whenever the dependence relationships in the generative distribution can be characterized by paths between singleton elements in some generative graphical model (e.g. a DAG, a chain graph, or a Markov network) even when we allow the generative model to include unobserved variables, and the observed data to be subject to selection bias. 1
Algebraic factor analysis: tetrads, pentads and beyond
, 2008
"... Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of po ..."
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Cited by 39 (13 self)
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Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of positive definite matrices. This parameter space is studied from the perspective of computational algebraic geometry. GrÃ¶bner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space. These polynomials, known as model invariants, arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix. Besides revealing the geometry of the factor analysis model, the model invariants also furnish useful statistics for testing goodnessoffit.