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24
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.
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...
Markov properties for acyclic directed mixed graphs
 Scandinavian Journal of Statistics
, 2003
"... We consider acyclic directed mixed graphs, in which directed edges (x → y) and bidirected edges (x ↔ y) may occur. A simple extension of Pearl’s dseparation criterion, called mseparation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property r ..."
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Cited by 36 (5 self)
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We consider acyclic directed mixed graphs, in which directed edges (x → y) and bidirected edges (x ↔ y) may occur. A simple extension of Pearl’s dseparation criterion, called mseparation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the mseparation criterion.
A SINful approach to Gaussian graphical model selection
 Journal of Statistical Planning and Inference
"... Abstract. Multivariate Gaussian graphical models are defined in terms of Markov properties, i.e., conditional independences associated with the underlying graph. Thus, model selection can be performed by testing these conditional independences, which are equivalent to specified zeroes among certain ..."
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Cited by 25 (5 self)
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Abstract. Multivariate Gaussian graphical models are defined in terms of Markov properties, i.e., conditional independences associated with the underlying graph. Thus, model selection can be performed by testing these conditional independences, which are equivalent to specified zeroes among certain (partial) correlation coefficients. For concentration graphs, covariance graphs, acyclic directed graphs, and chain graphs (both LWF and AMP), we apply Fisher’s ztransformation, ˇ Sidák’s correlation inequality, and Holm’s stepdown procedure, to simultaneously test the multiple hypotheses obtained from the Markov properties. This leads to a simple method for model selection that controls the overall error rate for incorrect edge inclusion. In practice, we advocate partitioning the simultaneous pvalues into three disjoint sets, a significant set S, an indeterminate set I, and a nonsignificant set N. Then our SIN model selection method selects two graphs, a graph whose edges correspond to the union of S and I, and a more conservative graph whose edges correspond to S only. Prior information about the presence and/or absence of particular edges can be incorporated readily. 1.
Binary models for marginal independence
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B
, 2005
"... A number of authors have considered multivariate Gaussian models for marginal independence. In this paper we develop models for binary data with the same independence structure. The models can be parameterized based on Möbius inversion and maximum likelihood estimation can be performed using a versi ..."
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Cited by 16 (2 self)
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A number of authors have considered multivariate Gaussian models for marginal independence. In this paper we develop models for binary data with the same independence structure. The models can be parameterized based on Möbius inversion and maximum likelihood estimation can be performed using a version of the Iterated Conditional Fitting algorithm. The approach is illustrated on a simple example. Relations to multivariate logistic and dependence ratio models are discussed.
A new algorithm for maximum likelihood estimation in Gaussian graphical models for marginal independence
 In U. Kjærulff and C. Meek (Eds.), Proceedings of the 19th Conference on Uncertainty in Artificial Intelligence
, 2003
"... Graphical models with bidirected edges (↔) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood estimation in the case of continuous variables with a Gaussian ..."
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Cited by 15 (7 self)
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Graphical models with bidirected edges (↔) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood estimation in the case of continuous variables with a Gaussian joint distribution, sometimes termed a covariance graph model. We present a new fitting algorithm which exploits standard regression techniques and establish its convergence properties. Moreover, we contrast our procedure to existing estimation algorithms. 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 14 (11 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.
Covariance Chains
 Bernoulli
, 2006
"... Covariance matrices which can be arranged in tridiagonal form are called covariance chains. They are used to clarify some issues of parameter equivalence and of independence equivalence for linear models in which a set of latent variables influences a set of observed variables. For this purpose, ort ..."
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Cited by 12 (8 self)
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Covariance matrices which can be arranged in tridiagonal form are called covariance chains. They are used to clarify some issues of parameter equivalence and of independence equivalence for linear models in which a set of latent variables influences a set of observed variables. For this purpose, orthogonal decompositions for covariance chains are derived first in explicit form. Covariance chains are also contrasted to concentration chains, for which estimation is explicit and simple. For this purpose, maximumlikelihood equations are derived first for exponential families when some parameters satisfy zero value constraints. From these equations explicit estimates are obtained, which are asymptotically efficient, and they are applied to covariance chains. Simulation results confirm the satisfactory behaviour of the explicit covariance chain estimates also in moderatesize samples.
Cumulative distribution networks and the derivativesumproduct algorithm
"... We introduce a new type of graphical model called a ‘cumulative distribution network’ (CDN), which expresses a joint cumulative distribution as a product of local functions. Each local function can be viewed as providing evidence about possible orderings, or rankings, of variables. Interestingly, we ..."
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Cited by 12 (6 self)
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We introduce a new type of graphical model called a ‘cumulative distribution network’ (CDN), which expresses a joint cumulative distribution as a product of local functions. Each local function can be viewed as providing evidence about possible orderings, or rankings, of variables. Interestingly, we find that the conditional independence properties of CDNs are quite different from other graphical models. We also describe a messagepassing algorithm that efficiently computes conditional cumulative distributions. Due to the unique independence properties of the CDN, these messages do not in general have a onetoone correspondence with messages exchanged in standard algorithms, such as belief propagation. We demonstrate the application of CDNs for structured ranking learning using a previouslystudied multiplayer gaming dataset. 1
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 12 (2 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.