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Model selection and accounting for model uncertainty in graphical models using Occam's window
, 1993
"... We consider the problem of model selection and accounting for model uncertainty in highdimensional contingency tables, motivated by expert system applications. The approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic Pvalues leading to the selection o ..."
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Cited by 364 (48 self)
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We consider the problem of model selection and accounting for model uncertainty in highdimensional contingency tables, motivated by expert system applications. The approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic Pvalues leading to the selection of a single model; inference is then conditional on the selected model. The sampling properties of such a strategy are complex, and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest. In principle, a panacea is provided by the standard Bayesian formalism which averages the posterior distributions of the quantity of interest under each of the models, weighted by their posterior model probabilities. Furthermore, this approach is optimal in the sense of maximising predictive ability. However, this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large (often greater than 1011). We argue that the standard Bayesian formalism is unsatisfactory and we propose an alternative Bayesian approach that, we contend, takes full account of the true model uncertainty byaveraging overamuch smaller set of models. An efficient search algorithm is developed for nding these models. We consider two classes of graphical models that arise in expert systems: the recursive causal models and the decomposable
A characterization of Markov equivalence classes for acyclic digraphs
, 1995
"... Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow e ..."
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Cited by 122 (7 self)
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Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may, however, be many ADGs that determine the same dependence ( = Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markovequivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Here it is shown that each Markovequivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADGs in the equivalence class. Essential graphs are characterized, a polynomialtime algorithm for their construction is given, and their applications to model selection and other statistical
Bayesian model selection in structural equation models
, 1993
"... A Bayesian approach to model selection for structural equation models is outlined. This enables us to compare individual models, nested or nonnested, and also to search through the (perhaps vast) set of possible models for the best ones. The approach selects several models rather than just one, whe ..."
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Cited by 62 (10 self)
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A Bayesian approach to model selection for structural equation models is outlined. This enables us to compare individual models, nested or nonnested, and also to search through the (perhaps vast) set of possible models for the best ones. The approach selects several models rather than just one, when appropriate, and so enables us to take account, both informally and formally, of uncertainty about model structure when making inferences about quantities of interest. The approach tends to select simpler models than strategies based on multiple Pvaluebased tests. It may thus help to overcome the criticism of structural
On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs
 Scandinavian Journal of Statistics
, 1994
"... Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or fo ..."
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Cited by 39 (5 self)
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Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or for general chain graphs (which include both UDGs and ADGs as special cases). This paper addresses three questions regarding the equivalence of graphical Markov models: when is a given chain graph Markov equivalent (1) to some UDG? (2) to some (at least one) ADG? (3) to some decomposable UDG? The answers are obtained by means of an extension of Frydenberg's (1990) elegant graphtheoretic characterization of the Markov equivalence of chain graphs. 1 Introduction The use of graphs to represent dependence relations among random variables, first introduced by Wright (1921), has generated considerable research activity, especially since the early 1980s. Particular attention has been devoted to gra...
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 37 (2 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.
Triangular systems for symmetric binary variables
 Electr. J. Statist
, 2009
"... Abstract We introduce and study distributions of sets of binary variables that are symmetric, that is each has equally probable levels. The joint distribution of these special types of binary variables, if generated by a recursive process of linear main effects is essentially parametrized in terms ..."
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Cited by 12 (8 self)
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Abstract We introduce and study distributions of sets of binary variables that are symmetric, that is each has equally probable levels. The joint distribution of these special types of binary variables, if generated by a recursive process of linear main effects is essentially parametrized in terms of marginal correlations. This contrasts with the loglinear formulation of joint probabilities in which parameters measure conditional associations given all remaining variables. The new formulation permits useful comparisons of different types of graphical Markov models and leads to a close approximation of Gaussian orthant probabilities.
Graphical Model Structure Learning with `1Regularization
, 2010
"... This work looks at fitting probabilistic graphical models to data when the structure is not known. The main tool to do this is `1regularization and the more general group `1regularization. We describe limitedmemory quasiNewton methods to solve optimization problems with these types of regularize ..."
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Cited by 4 (1 self)
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This work looks at fitting probabilistic graphical models to data when the structure is not known. The main tool to do this is `1regularization and the more general group `1regularization. We describe limitedmemory quasiNewton methods to solve optimization problems with these types of regularizers, and we examine learning directed acyclic graphical models with `1regularization, learning undirected graphical models with group `1regularization, and learning hierarchical loglinear models with overlapping group `1regularization. ii
BIFROST  Block recursive models Induced From Relevant knowledge, Observations, and Statistical Techniques
 Computational Statistics and Data Analysis
, 1993
"... The theoretical background for a program for establishing expert systems on the basis of observations and expert knowledge is presented. Block recursive models form the basis of the statistical modelling. These models, together with various model selection methods for automatic model selection, a ..."
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Cited by 4 (0 self)
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The theoretical background for a program for establishing expert systems on the basis of observations and expert knowledge is presented. Block recursive models form the basis of the statistical modelling. These models, together with various model selection methods for automatic model selection, are presented. Additionally, the connection between a block recursive model and expert systems based on causal probabilistic networks is treated. A medical example concerning diagnosis of coronary artery disease forms the basis for an evaluation of the expert systems established. Keywords: causal probabilistic networks, graphical association models, machine learning, model selection, selection criteria, selection strategies. 1 Introduction BIFROST is a program for semiautomatic knowledge acquisition and is a continuation developments made in (Greve, Hjsgaard, Skjth and Thiesson 1990). The objective is to obtain preliminary causal models for use in the HUGIN expert system shell (Ander...
Highdimensional Graphical Model Search with gRapHD R Package
, 909
"... This paper presents the R package gRapHD for efficient selection of highdimensional undirected graphical models. The package provides tools for selecting trees, forests and decomposable models minimizing information criteria such as AIC or BIC, and for displaying the independence graphs of the model ..."
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Cited by 3 (0 self)
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This paper presents the R package gRapHD for efficient selection of highdimensional undirected graphical models. The package provides tools for selecting trees, forests and decomposable models minimizing information criteria such as AIC or BIC, and for displaying the independence graphs of the models. It has also some useful tools for analysing graphical structures. It supports the use of discrete, continuous, or both types of variables. 1