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10
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 49 (4 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...
A reconstruction algorithm for the essential graph
, 2008
"... A standard graphical representative of a Bayesian network structure is a special chain graph, known as an essential graph. An alternative algebraic approach to the mathematical description of this statistical model uses instead a certain integervalued vector, known as a standard imset. We give a di ..."
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Cited by 5 (2 self)
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A standard graphical representative of a Bayesian network structure is a special chain graph, known as an essential graph. An alternative algebraic approach to the mathematical description of this statistical model uses instead a certain integervalued vector, known as a standard imset. We give a direct formula for the translation of any chain graph describing a Bayesian network structure into the standard imset. Moreover, we present a twostage algorithm which makes it possible to reconstruct the essential graph on the basis of the standard imset. The core of this paper is the proof of the correctness of the algorithm.
A Graphical Representation of Equivalence Classes of AMP Chain Graphs
, 2005
"... This paper deals with chain graph models under alternative AMP interpretation. A new representative of an AMP Markov equivalence class, called the largest deflagged graph, is proposed. The representative is based on revealed internal structure of the AMP Markov equivalence class. More specifically, ..."
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Cited by 4 (2 self)
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This paper deals with chain graph models under alternative AMP interpretation. A new representative of an AMP Markov equivalence class, called the largest deflagged graph, is proposed. The representative is based on revealed internal structure of the AMP Markov equivalence class. More specifically, the AMP Markov equivalence class decomposes into finer strong equivalence classes and there exists a distinguished strong equivalence class among those forming the AMP Markov equivalence class. The largest deflagged graph is the largest chain graph in that distinguished strong equivalence class. A composed graphical procedure to get the largest deflagged graph on the basis of any AMP Markov equivalent chain graph is presented. In general, the largest deflagged graph differs from the AMP essential graph, which is another representative of the AMP Markov equivalence class.
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 4 (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.
New Algorithm for Learning Decomposable Models
, 2000
"... There exist a lot of algorithms for the construction of Bayesian Networks (BN). But almost all computations in BN are carried out by transforming them to another special type of probabilistic models  decomposable models (DM). This task of transformation is known to be a NP complex problem and ..."
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Cited by 1 (1 self)
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There exist a lot of algorithms for the construction of Bayesian Networks (BN). But almost all computations in BN are carried out by transforming them to another special type of probabilistic models  decomposable models (DM). This task of transformation is known to be a NP complex problem and todays algorithms for the construction of BN cannot guarantee the existence of reasonably small DM (it is necessary to fit this DM to the memory of computer).
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 1 (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.
Academy of Sciences of the Czech Republic
"... involved with basic research in systems, control, and information sciences. This report gives an overview of our research activities in 1997. It is of course impossible to give a full account of the research results here. The results selected are divided into sections representing the seven research ..."
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involved with basic research in systems, control, and information sciences. This report gives an overview of our research activities in 1997. It is of course impossible to give a full account of the research results here. The results selected are divided into sections representing the seven research departments of the Institute. Each department is briefly introduced and its overall activity is described. The report is completed by a list of works published and/or accepted for publication. 1
Chain Graph, Multivariate Regression
"... This paper deals with multivariate regression chain graphs, which were introduced by Cox and Wermuth (1993, 1996) to represent linear causal models with correlated errors. Specifically, we present a constraint based algorithm for learning a chain graph a given probability distribution is faithful to ..."
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This paper deals with multivariate regression chain graphs, which were introduced by Cox and Wermuth (1993, 1996) to represent linear causal models with correlated errors. Specifically, we present a constraint based algorithm for learning a chain graph a given probability distribution is faithful to. We also show that for each Markov equivalence class of multivariate regression chain graphs there exists a set of chain graphs with a unique minimal set of lines. Finally, we show that this set of lines can be identified from any member of the class by repeatedly splitting its connectivity components according to certain conditions. Keywords:
Academy of Sciences of the Czech Republic
"... In this paper we study two operations of merging components in a chain graph, which appear to be elementary operations yielding an equivalent graph in the respective sense. At first, we recall basic results on the operation of feasible merging components, which is related to classic LWF Markov equiv ..."
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In this paper we study two operations of merging components in a chain graph, which appear to be elementary operations yielding an equivalent graph in the respective sense. At first, we recall basic results on the operation of feasible merging components, which is related to classic LWF Markov equivalence of chain graphs. These results are used to get a graphical characterisation of factorisation equivalence of classic chain graphs. As another example of the use of this operation, we derive some important invariants of LWF Markov equivalence of chain graphs. Last, we recall analogous basic results on the operation of legal merging components. This operation is related to the socalled strong equivalence of chain graphs, which includes both classic LWF Markov equivalence and alternative AMP Markov equivalence.
2 Characterization of inclusion neighbourhood
, 2004
"... xxx (2004) xxx–xxx www.elsevier.com/locate/ijar ..."