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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 92 (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
ANCESTRAL GRAPH MARKOV MODELS
, 2002
"... This paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all DAG independence models. This class of graphs, called maximal ancestral graphs, has two attractive features: there is at most one edge between each pair of verti ..."
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Cited by 76 (18 self)
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This paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all DAG independence models. This class of graphs, called maximal ancestral graphs, has two attractive features: there is at most one edge between each pair of vertices; every missing edge corresponds to an independence relation. These features lead to a simple parameterization of the corresponding set of distributions in the Gaussian case.
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...
Normal Linear Regression Models with Recursive Graphical Markov Structure
 J. MULTIVARIATE ANAL
, 1998
"... A multivariate normal statistical model defined by the Markov pr er deter by an acyclic digric admits ar efactorof its likelihood function (LF) into the pr duct of conditional LFs, eachfactor having the for of a classical multivar  linear rear model (# MANOVA model).Her these modelsar extended ..."
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Cited by 13 (5 self)
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A multivariate normal statistical model defined by the Markov pr er deter by an acyclic digric admits ar efactorof its likelihood function (LF) into the pr duct of conditional LFs, eachfactor having the for of a classical multivar  linear rear model (# MANOVA model).Her these modelsar extended in anatur way tonor linear rear models whose LFs continue to admit suchr efactorr frr which maximum likelihoodestimator and likelihoodr (LR) test statistics can beder ed by classical linear methods. The centrdistr  of the LR test statisticfor testing one such multivariv norv linear rear model against another isder ed, and there of theseresesion models to blockr enor linear systems is established. It is shown how a collection of nonnested dependentnor linear rear models (# seemingly unringly ringly can be combined into a single multivariv norvlinear rn grear model by imposing apar set of graphical Markov (# conditional independence) restrictions.
A graphical characterization of lattice conditional independence models
 Ann. Math. and Artificial Intelligence
, 1997
"... Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of nonmonotone missing data patterns and of nonnested dependent linear regression models ( ≡ seemingly unrelated regressions). It is shown here that the class of LCI models coin ..."
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Cited by 9 (2 self)
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Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of nonmonotone missing data patterns and of nonnested dependent linear regression models ( ≡ seemingly unrelated regressions). It is shown here that the class of LCI models coincides with a subclass of the class of graphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graphtheoretic characterization of those ADGs that are Markov equivalent to some transitive ADG is obtained. This characterization allows one to determine whether a specific ADG D is Markov equivalent to some transitive ADG, hence to some LCI model, in polynomial time, without an exhaustive search of the (exponentially large) equivalence class [D]. These results do not require the existence or positivity of joint densities. 1. Introduction. The use of directed graphs to represent possible dependencies among statistical variables dates back to Wright (1921) and has generated considerable research activity in the social and natural sciences. Since 1980, particular attention has been directed at
The Wishart Distributions on Homogeneous Cones
, 2001
"... The classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distributions on a homogeneous cone using the theory of exponential families. The generalized Wishart distributions include all known fam ..."
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Cited by 4 (0 self)
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The classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distributions on a homogeneous cone using the theory of exponential families. The generalized Wishart distributions include all known families of Wishart distributions as special cases. The relations to graphical models and Bayesian statistics are indicated.
A Characterization of Moral Transitive Directed Acyclic Graph Markov models as trees and its properties
, 2000
"... It follows from the known relationships among the dierent classes of graphical Markov models for conditional independence that the intersection of the classes of moral directed acyclic graph models (or decomposable {DEC models), and transitive directed acyclic graph {TDAG models (or lattice cond ..."
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Cited by 3 (1 self)
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It follows from the known relationships among the dierent classes of graphical Markov models for conditional independence that the intersection of the classes of moral directed acyclic graph models (or decomposable {DEC models), and transitive directed acyclic graph {TDAG models (or lattice conditional independence {LCI models) is nonempty. This paper shows that the conditional independence models in the intersection can be characterized as labeled trees, where every vertex on the tree corresponds to a single random variable. This fact leads to the de nition of a speci c Markov property for trees and therefore to the introduction of trees as part of the family of graphical Markov Models.
CONDITIONAL INDEPENDENCE MODELS FOR SEEMINGLY UNRELATED REGRESSIONS WITH INCOMPLETE DATA
"... Abstract. We consider normal ≡ Gaussian seemingly unrelated regressions (SUR) models with incomplete data (ID). Imposing a natural minimal set of conditional independence constraints, we find restricted SUR/ID models for which the likelihood function and the parameter space factors into the product ..."
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Cited by 2 (1 self)
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Abstract. We consider normal ≡ Gaussian seemingly unrelated regressions (SUR) models with incomplete data (ID). Imposing a natural minimal set of conditional independence constraints, we find restricted SUR/ID models for which the likelihood function and the parameter space factors into the product of the likelihood functions and the parameter spaces of standard complete data multivariate analysis of variance models. Hence, the restricted model has a unimodal likelihood and permits explicit likelihood inference. The restricted model may be used to directly model the data actually observed. Alternatively, the maximum likelihood estimates in the restricted model can yield improved starting values for iterative methods to maximize the likelihood of the unrestricted SUR/ID model. In the development of our methodology, we review and extend existing results for complete data SUR models and the multivariate ID problem. The results are presented in the framework of both lattice conditional independence models and graphical Markov models based on acyclic directed graphs. Date: October 1, 2003. Key words and phrases. Acyclic directed graph, graphical model, incomplete data, lattice conditional independence model, MANOVA, maximum likelihood estimator, multivariate analysis, multivariate linear model, missing data, seemingly unrelated regressions.
In Admissibility of the Maximum Likelihood . . .
, 1998
"... Lattice conditional independence (LCI) models introduced by Andersson and Perlman [3] have pleasant feature of admitting explicit maximum likelihood estimators and likelihood ratio test statistics. This is because the likelihood function and parameter space for a LCI model can be factored into pro ..."
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Lattice conditional independence (LCI) models introduced by Andersson and Perlman [3] have pleasant feature of admitting explicit maximum likelihood estimators and likelihood ratio test statistics. This is because the likelihood function and parameter space for a LCI model can be factored into products of conditional likelihood functions and parameter spaces, where the standard multivariate techniques can be applied. In this paper we consider the problem of estimating the covariance matrices under LCI restriction in a decision theoretic setup. The Stein loss function is used in this study and, using the factorization mentioned above, minimax estimators are obtained. Since the maximum likelihood estimator has constant risk and is different from minimax estimator, this shows that the maximum likelihood estimator under LCI restriction is inadmissible. These results extend those obtained