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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 30 (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...
Graphical Explanation in Belief Networks
 In Journal of Computational and Graphical Statistics
, 1997
"... Belief networks provide an important bridge between statistical modeling and expert systems. In this paper we present methods for visualizing probabilistic "evidence flows" in belief networks, thereby enabling belief networks to explain their behavior. Building on earlier research on explanation in ..."
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Cited by 16 (4 self)
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Belief networks provide an important bridge between statistical modeling and expert systems. In this paper we present methods for visualizing probabilistic "evidence flows" in belief networks, thereby enabling belief networks to explain their behavior. Building on earlier research on explanation in expert systems, we present a hierarchy of explanations, ranging from simple colorings to detailed displays. Our approach complements parallel work on textual explanations in belief networks. GRAPHICALBELIEF, Mathsoft Inc.'s belief network software, implements the methods. 1 Introduction A fundamental reason for building a mathematical or statistical model is to foster deeper understanding of complex, realworld systems. Consequently, explanationsdescriptions of the mechanisms which comprise such modelsform an important part of model validation, exploration, and use. Early tests of rulebased expert system models indicated the critical need for detailed explanations in that setting (...
Graphical methods for efficient likelihood inference in gaussian covariance models
 Journal of Machine Learning
, 2008
"... Abstract. In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bidirected graph into a maximal ancestral graph that (i) represents the same independence structure as the origi ..."
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Cited by 8 (3 self)
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Abstract. In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bidirected graph into a maximal ancestral graph that (i) represents the same independence structure as the original bidirected graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bidirected edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts. 1.
Graphical Answers to . . .
, 2004
"... In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph (alternatively graphs with dashed edges have been used for this purpose). Bidirected graphs are special instances of ancestral graphs, which are mixed ..."
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In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph (alternatively graphs with dashed edges have been used for this purpose). Bidirected graphs are special instances of ancestral graphs, which are mixed graphs with undirected, directed, and bidirected edges. In this paper, we show how simplicial sets and the newly defined orientable edges can be used to construct a maximal ancestral graph that is Markov equivalent to a given bidirected graph, i.e. the independence models associated with the two graphs coincide, and such that the number of arrowheads is minimal. Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bidirected edges. This construction yields an immediate check whether the original bidirected graph is Markov equivalent to a directed acyclic graph (Bayesian network) or an undirected graph (Markov random field). Moreover, the ancestral graph construction allows for computationally more efficient maximum likelihood fitting of covariance graph models, i.e. Gaussian bidirected graph models. In particular, we give a necessary and sufficient graphical criterion for determining when an entry of the maximum likelihood estimate of the covariance matrix must equal its empirical counterpart.