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Multiple testing and error control in Gaussian graphical model selection
 Statistical Science
"... Abstract. Graphical models provide a framework for exploration of multivariate dependence patterns. The connection between graph and statistical model is made by identifying the vertices of the graph with the observed variables and translating the pattern of edges in the graph into a pattern of cond ..."
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Abstract. Graphical models provide a framework for exploration of multivariate dependence patterns. The connection between graph and statistical model is made by identifying the vertices of the graph with the observed variables and translating the pattern of edges in the graph into a pattern of conditional independences that is imposed on the variables ’ joint distribution. Focusing on Gaussian models, we review classical graphical models. For these models the defining conditional independences are equivalent to vanishing of certain (partial) correlation coefficients associated with individual edges that are absent from the graph. Hence, Gaussian graphical model selection can be performed by multiple testing of hypotheses about vanishing (partial) correlation coefficients. We show and exemplify how this approach allows one to perform model selection while controlling error rates for incorrect edge inclusion. Key words and phrases: Acyclic directed graph, Bayesian network, bidirected graph, chain graph, concentration graph, covariance graph, DAG, graphical model, multiple testing, undirected graph. 1.
Simulation of hyperinverse Wishart distributions in graphical models
, 2007
"... We introduce and exemplify an efficient method for direct sampling from hyperinverse Wishart distributions. The method relies very naturally on the use of standard junctiontree representation of graphs, and couples these with matrix results for inverse Wishart distributions. We describe the theory ..."
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Cited by 11 (3 self)
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We introduce and exemplify an efficient method for direct sampling from hyperinverse Wishart distributions. The method relies very naturally on the use of standard junctiontree representation of graphs, and couples these with matrix results for inverse Wishart distributions. We describe the theory and resulting computational algorithms for both decomposable and nondecomposable graphical models. An example drawn from financial time series demonstrates application in a context where inferences on a structured covariance model are required. We discuss and investigate questions of scalability of the simulation methods to higherdimensional distributions. The paper concludes with general comments about the approach, including its use in connection with existing Markov chain Monte Carlo methods that deal with uncertainty about the graphical model structure.
Efficient Model Determination for Discrete Graphical Models
, 2000
"... We present a novel methodology for bayesian model determination in discrete decomposable graphical models. We assign, for each given graph, a Hyper Dirichlet distribution on the matrix of cell probabilities. To ensure compatibility across models such prior distributions are obtained by marginalis ..."
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Cited by 10 (1 self)
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We present a novel methodology for bayesian model determination in discrete decomposable graphical models. We assign, for each given graph, a Hyper Dirichlet distribution on the matrix of cell probabilities. To ensure compatibility across models such prior distributions are obtained by marginalisation from the prior conditional on the complete graph. This leads to a prior distribution automatically satisfying the hyperconsistency criterion. Our contribution is twofold. On one hand we improve an existing methodology, the MC 3 algorithm by Madigan and York (1995). On the other hand we introduce an original methodology based on the use of the Reversible jump sampler by Green (1995) and Giudici and Green (1999). Legal movement, that is leading to a decomposable graph, are identied making use of the junction tree representation of the considered graph. Keywords: Bayesian model selection; Contingency table; Dirichlet distribution; Hyper Markov distribution; Junction tree; Marko...
Bayesian structural learning and estimation in Gaussian graphical models
"... We propose a new stochastic search algorithm for Gaussian graphical models called the mode oriented stochastic search. Our algorithm relies on the existence of a method to accurately and efficiently approximate the marginal likelihood associated with a graphical model when it cannot be computed in c ..."
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Cited by 7 (2 self)
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We propose a new stochastic search algorithm for Gaussian graphical models called the mode oriented stochastic search. Our algorithm relies on the existence of a method to accurately and efficiently approximate the marginal likelihood associated with a graphical model when it cannot be computed in closed form. To this end, we develop a new Laplace approximation method to the normalizing constant of a GWishart distribution. We show that combining the mode oriented stochastic search with our marginal likelihood estimation method leads to excellent results with respect to other techniques discussed in the literature. We also describe how to perform inference through Bayesian model averaging based on the reduced set of graphical models identified. Finally, we give a novel stochastic search technique for multivariate regression models.
Bayesian analysis of matrix normal graphical models
 Biometrika
, 2009
"... We develop Bayesian analysis of matrixvariate normal data with conditional independence graphical structuring of the characterising variance matrix parameters. This leads to fully Bayesian analysis of matrix normal graphical models, including discussion of novel prior specifications, the resulting ..."
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Cited by 6 (3 self)
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We develop Bayesian analysis of matrixvariate normal data with conditional independence graphical structuring of the characterising variance matrix parameters. This leads to fully Bayesian analysis of matrix normal graphical models, including discussion of novel prior specifications, the resulting problems of posterior computation addressed using Markov chain Monte Carlo methods, and graphical model assessment that involves approximate evaluation of marginal likelihood functions under specified graphical models. Modelling and inference for spatial/image data via a novel class of Markov random fields that arise as natural examples of matrix normal graphical models is discussed. This is complemented by the development of a broad class of dynamic models for matrixvariate time series within which stochastic elements defining time series errors and structural changes over time are subject to graphical model structuring. Three examples illustrate these developments and highlight questions of graphical model uncertainty and comparison in matrix data contexts.
Association Models For Web Mining
, 2001
"... We describe how statistical association models and, specifically, graphical models, can be usefully employed to model web mining data. We describe some methodological problems related to the implementation of discrete graphical models for web mining data. In particular, we discuss model selection pr ..."
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Cited by 5 (2 self)
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We describe how statistical association models and, specifically, graphical models, can be usefully employed to model web mining data. We describe some methodological problems related to the implementation of discrete graphical models for web mining data. In particular, we discuss model selection procedures.
Supervised classification with conditional gaussian networks: Increasing the structure complexity from naive bayes
 International Journal of Approximate Reasoning
"... Most of the Bayesian networkbased classifiers are usually only able to handle discrete variables. However, most realworld domains involve continuous variables. A common practice to deal with continuous variables is to discretize them, with a subsequent loss of information. This work shows how disc ..."
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Most of the Bayesian networkbased classifiers are usually only able to handle discrete variables. However, most realworld domains involve continuous variables. A common practice to deal with continuous variables is to discretize them, with a subsequent loss of information. This work shows how discrete classifier induction algorithms can be adapted to the conditional Gaussian network paradigm to deal with continuous variables without discretizing them. In addition, three novel classifier induction algorithms and two new propositions about mutual information are introduced. The classifier induction algorithms presented are ordered and grouped according to their structural complexity: naive Bayes, tree augmented naive Bayes, kdependence Bayesian classifiers and semi naive Bayes. All the classifier induction algorithms are empirically evaluated using predictive accuracy, and they are compared to linear discriminant analysis, as a continuous classic statistical benchmark classifier. Besides, the accuracies for a set of stateoftheart classifiers are included in order to justify the use of linear discriminant analysis as the benchmark algorithm. In order to understand the behavior of the conditional Gaussian networkbased classifiers better, the results include biasvariance decomposition of the expected misclassification rate. The study suggests that semi naive Bayes structure based classifiers and, especially, the novel wrapper condensed semi naive Bayes backward, outperform the behavior of the rest of the presented classifiers. They also obtain quite competitive results compared to the stateoftheart algorithms included. Key words: conditional Gaussian network, Bayesian network, naive Bayes, tree augmented naive Bayes, kdependence Bayesian classifiers, semi naive Bayes, filter, wrapper.
Bayesian covariance matrix estimation using a mixture of decomposable graphical models. Unpublished manuscript
, 2005
"... Summary. Estimating a covariance matrix efficiently and discovering its structure are important statistical problems with applications in many fields. This article takes a Bayesian approach to estimate the covariance matrix of Gaussian data. We use ideas from Gaussian graphical models and model sele ..."
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Cited by 5 (2 self)
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Summary. Estimating a covariance matrix efficiently and discovering its structure are important statistical problems with applications in many fields. This article takes a Bayesian approach to estimate the covariance matrix of Gaussian data. We use ideas from Gaussian graphical models and model selection to construct a prior for the covariance matrix that is a mixture over all decomposable graphs, where a graph means the configuration of nonzero offdiagonal elements in the inverse of the covariance matrix. Our prior for the covariance matrix is such that the probability of each graph size is specified by the user and graphs of equal size are assigned equal probability. Most previous approaches assume that all graphs are equally probable. We give empirical results that show the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs, both in identifying the correct decomposable graph and in more efficiently estimating the covariance matrix. The advantage is greatest when the number of observations is small relative to the dimension of the covariance matrix. Our method requires the number of decomposable graphs for each graph size. We show how to estimate these numbers using simulation and that the simulation results agree with analytic results when such results are known. We also show how
A SINful Approach to Model Selection for Gaussian Concentration Graphs
, 2003
"... A multivariate Gaussian graphical Markov model for an undirected graph G, also called a covariance selection model or concentration graph model, is defined in terms of the Markov properties, i.e., conditional independences associated with G, which in turn are equivalent to specified zeroes among t ..."
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Cited by 5 (1 self)
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A multivariate Gaussian graphical Markov model for an undirected graph G, also called a covariance selection model or concentration graph model, is defined in terms of the Markov properties, i.e., conditional independences associated with G, which in turn are equivalent to specified zeroes among the set of pairwise partial correlation coe#cients. By means of Fisher's ztransformation and Sidak's correlation inequality, conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained, leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion. The simultaneous pvalues corresponding to the partial correlations are partitioned into three disjoint sets, a significant set S, an indeterminate set I, and a nonsignificant set N. Our SIN model selection method selects two graphs, a graph GSI whose edges correspond to the set I, and a more conservative graph GS whose edges correspond to S only. Prior information about the presence and/or absence of particular edges can be incorporated readily. Similar considerations apply to covariance graph models, which are defined in terms of marginal independence rather than conditional independence. 1.