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25
An Alternative Markov Property for Chain Graphs
 Scand. J. Statist
, 1996
"... Graphical Markov models use graphs, either undirected, directed, or mixed, to represent possible dependences among statistical variables. Applications of undirected graphs (UDGs) include models for spatial dependence and image analysis, while acyclic directed graphs (ADGs), which are especially conv ..."
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Cited by 74 (5 self)
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Graphical Markov models use graphs, either undirected, directed, or mixed, to represent possible dependences among statistical variables. Applications of undirected graphs (UDGs) include models for spatial dependence and image analysis, while acyclic directed graphs (ADGs), which are especially convenient for statistical analysis, arise in such fields as genetics and psychometrics and as models for expert systems and Bayesian belief networks. Lauritzen, Wermuth, and Frydenberg (LWF) introduced a Markov property for chain graphs, which are mixed graphs that can be used to represent simultaneously both causal and associative dependencies and which include both UDGs and ADGs as special cases. In this paper an alternative Markov property (AMP) for chain graphs is introduced, which in some ways is a more direct extension of the ADG Markov property than is the LWF property for chain graph. 1 INTRODUCTION Graphical Markov models use graphs, either undirected, directed, or mixed, to represent...
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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Cited by 30 (4 self)
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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.
Partial inversion for linear systems and partial closure of independence graphs
 BIT, Numer. Math
"... We introduce and study a calculus for realvalued square matrices, called partial inversion, and an associated calculus for binary square matrices. The first, applied to systems of recursive linear equations, generates new sets of parameters for different types of statistical joint response models. ..."
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Cited by 21 (16 self)
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We introduce and study a calculus for realvalued square matrices, called partial inversion, and an associated calculus for binary square matrices. The first, applied to systems of recursive linear equations, generates new sets of parameters for different types of statistical joint response models. The corresponding generating graphs are directed and acyclic. The second calculus, applied to matrix representations of independence graphs, gives chain graphs induced by such a generating graph. Chain graphs are more complex independence graphs associated with recursive joint response models. Missing edges in independence graphs coincide with structurally zero parameters in linear systems. A wide range of consequences of an assumed independence structure can be derived by partial closure, but computationally efficient algorithms still need to be developed for applications to very large graphs.
On Recovery Algorithm for Chain Graphs
, 1997
"... The class of chain graphs (CGs) involving both undirected graphs (= Markov networks) and directed acyclic graphs (= Bayesian networks) was introduced in middle eighties for description of probabilistic conditional independence structures. Every class of Markov equivalent CGs (that is CGs describing ..."
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Cited by 18 (3 self)
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The class of chain graphs (CGs) involving both undirected graphs (= Markov networks) and directed acyclic graphs (= Bayesian networks) was introduced in middle eighties for description of probabilistic conditional independence structures. Every class of Markov equivalent CGs (that is CGs describing the same conditional independence structure) has a natural representative, which is called the largest CG. The paper presents socalled recovery algorithm, which on basis of the conditional independence structure given by a CG (in form of socalled dependency model) finds the largest CG, representing the corresponding class of Markov equivalent CGs. As a byproduct a graphical characterization of graphs, which are the largest CGs (for a class of Markov equivalent CGs) is obtained, and a simple algorithm changing every CG into the largest CG of the corresponding equivalence class is given. 1 INTRODUCTION Classic graphical approaches to description of probabilistic conditional independence stru...
Structural Learning of Chain Graphs via Decomposition
"... Chain graphs present a broad class of graphical models for description of conditional independence structures, including both Markov networks and Bayesian networks as special cases. In this paper, we propose a computationally feasible method for the structural learning of chain graphs based on the i ..."
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Cited by 12 (0 self)
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Chain graphs present a broad class of graphical models for description of conditional independence structures, including both Markov networks and Bayesian networks as special cases. In this paper, we propose a computationally feasible method for the structural learning of chain graphs based on the idea of decomposing the learning problem into a set of smaller scale problems on its decomposed subgraphs. The decomposition requires conditional independencies but does not require the separators to be complete subgraphs. Algorithms for both skeleton recovery and complex arrow orientation are presented. Simulations under a variety of settings demonstrate the competitive performance of our method, especially when the underlying graph is sparse.
An Inclusion Optimal Algorithm for Chain Graph Structure Learning
 In Proceedings of the 17th International Conference on Artificial Intelligence and Statistics
, 2014
"... This paper presents and proves an extension of Meek’s conjecture to chain graphs under the LauritzenWermuthFrydenberg interpretation. The proof of the conjecture leads to the development of a structure learning algorithm that finds an inclusion optimal chain graph for any given probability distr ..."
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Cited by 7 (7 self)
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This paper presents and proves an extension of Meek’s conjecture to chain graphs under the LauritzenWermuthFrydenberg interpretation. The proof of the conjecture leads to the development of a structure learning algorithm that finds an inclusion optimal chain graph for any given probability distribution satisfying the composition property. Finally, the new algorithm is experimentally evaluated. 1
Characterizing Markov equivalence classes for AMP chain graph models
 The Annals of Statistics
, 2005
"... 2 Chain graphs (CG) ( = adicyclic graphs) use undirected and directed edges to represent simultaneously both structural and associative dependences.. Like acyclic directed graphs (ADGs), the CG associated with a given statistical model may not be unique, so CGs fall into Markov equivalence classes, ..."
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Cited by 7 (0 self)
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2 Chain graphs (CG) ( = adicyclic graphs) use undirected and directed edges to represent simultaneously both structural and associative dependences.. Like acyclic directed graphs (ADGs), the CG associated with a given statistical model may not be unique, so CGs fall into Markov equivalence classes, which may be superexponentially large, leading to unidentifiability and computational inefficiency in model search and selection. It is shown here that under the AnderssonMadiganPerlman (AMP) Markov interpretation of a CG, each Markovequivalence class can be uniquely represented by a single distinguished CG, the AMP essential graph, that is itself simultaneously Markov equivalent to all CGs in the AMP Markov equivalence class. A complete characterization of AMP essential graphs is obtained. Like the essential graph previously introduced for ADGs, the AMP essential graph will play a fundamental role for inference and model search and selection for AMP CG models.
MINIMAL SUFFICIENT CAUSATION AND DIRECTED ACYCLIC GRAPHS
, 2009
"... Notions of minimal sufficient causation are incorporated within the directed acyclic graph causal framework. Doing so allows for the graphical representation of sufficient causes and minimal sufficient causes on causal directed acyclic graphs while maintaining all of the properties of causal directe ..."
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Cited by 6 (1 self)
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Notions of minimal sufficient causation are incorporated within the directed acyclic graph causal framework. Doing so allows for the graphical representation of sufficient causes and minimal sufficient causes on causal directed acyclic graphs while maintaining all of the properties of causal directed acyclic graphs. This in turn provides a clear theoretical link between two major conceptualizations of causality: one counterfactualbased and the other based on a more mechanistic understanding of causation. The theory developed can be used to draw conclusions about the sign of the conditional covariances among variables.