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11
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 75 (17 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.
Chain Graph Models and their Causal Interpretations
 B
, 2001
"... Chain graphs are a natural generalization of directed acyclic graphs (DAGs) and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are a number of simple and apparently plausible, but ultim ..."
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Cited by 46 (4 self)
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Chain graphs are a natural generalization of directed acyclic graphs (DAGs) and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are a number of simple and apparently plausible, but ultimately fallacious interpretations of chain graphs that are often invoked, implicitly or explicitly. These interpretations also lead to awed methods for applying background knowledge to model selection. We present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated as the equilibrium distribution of dynamic models with feedback. These dynamic interpretations lead to a simple theory of intervention, extending the theory developed for DAGs. Finally, we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feedback in econometrics. Keywords: Causal model; cha...
Markov equivalence for ancestral graphs
, 2004
"... Ancestral graph models can encode conditional independence relations that arise in directed acyclic graph (DAG) models with latent and selection variables. However, for any 3JJ.cestral graph, there may be several other graphs to which it is Markov equivalent. We state and prove conditions under whic ..."
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Cited by 16 (5 self)
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Ancestral graph models can encode conditional independence relations that arise in directed acyclic graph (DAG) models with latent and selection variables. However, for any 3JJ.cestral graph, there may be several other graphs to which it is Markov equivalent. We state and prove conditions under which two maximal ancestral graphs are Markov equivalent to each other, thereby extending analogous results for DAGs given by other authors. 'University of W2k'lhi.ng1;on Technical No. 466. Contents
Probability distributions with summary graph structure
, 2008
"... A joint density of many variables may satisfy a possibly large set of independence statements, called its independence structure. Often the structure of interest is representable by a graph that consists of nodes representing variables and of edges that couple node pairs. We consider joint densities ..."
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Cited by 4 (2 self)
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A joint density of many variables may satisfy a possibly large set of independence statements, called its independence structure. Often the structure of interest is representable by a graph that consists of nodes representing variables and of edges that couple node pairs. We consider joint densities of this type, generated by a stepwise process in which all variables and dependences of interest are included. Otherwise, there are no constraints on the type of variables or on the form of the generating conditional densities. For the joint density that then results after marginalising and conditioning, we derive what we name the summary graph. It is seen to capture precisely the independence structure implied by the generating process, it identifies dependences which remain undistorted due to direct or indirect confounding and it alerts to such, possibly severe distortions in other parametrizations. Summary graphs preserve their form after marginalising and conditioning and they include multivariate regression chain graphs as special cases. We use operators for matrix representations of graphs to derive matrix results and translate these into special types of path. 1. Introduction. Graphical Markov
Sequences of regressions and their independences
, 2012
"... Ordered sequences of univariate or multivariate regressions provide statistical modelsfor analysingdata fromrandomized, possiblysequential interventions, from cohort or multiwave panel studies, but also from crosssectional or retrospective studies. Conditional independences are captured by what we ..."
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Cited by 4 (1 self)
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Ordered sequences of univariate or multivariate regressions provide statistical modelsfor analysingdata fromrandomized, possiblysequential interventions, from cohort or multiwave panel studies, but also from crosssectional or retrospective studies. Conditional independences are captured by what we name regression graphs, provided the generated distribution shares some properties with a joint Gaussian distribution. Regression graphs extend purely directed, acyclic graphs by two types of undirected graph, one type for components of joint responses and the other for components of the context vector variable. We review the special features and the history of regression graphs, prove criteria for Markov equivalence anddiscussthenotion of simpler statistical covering models. Knowledgeof Markov equivalence provides alternative interpretations of a given sequence of regressions, is essential for machine learning strategies and permits to use the simple graphical criteria of regression graphs on graphs for which the corresponding criteria are in general more complex. Under the known conditions that a Markov equivalent directed acyclic graph exists for any given regression graph, we give a polynomial time algorithm to find one such graph.
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 2 (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.
Submitted to the Annals of Statistics PROBABILITY DISTRIBUTIONS WITH SUMMARY GRAPH
"... A joint density of several variables may satisfy a possibly large set of independence statements, called its independence structure. Often this structure is fully representable by a graph that consists of nodes representing variables and of edges that couple node pairs. We consider joint densities o ..."
Abstract
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A joint density of several variables may satisfy a possibly large set of independence statements, called its independence structure. Often this structure is fully representable by a graph that consists of nodes representing variables and of edges that couple node pairs. We consider joint densities of this type, generated by a stepwise process in which all variables and dependences of interest are included. Otherwise, there are no constraints on the type of variables or on the form of the distribution generated. For densities that then result after marginalising and conditioning, we derive what we name the summary graph. It is seen to capture precisely the independence structure implied by the generating process, it identifies dependences which remain undistorted due to direct or indirect confounding and it alerts to possibly severe distortions of these two types in other parametrizations. Summary graphs preserve their form after marginalising and conditioning and they include multivariate regression chain graphs as special cases. We use operators for matrix representations of graphs to derive matrix results and translate these into special types of path. 1. Introduction. Graphical Markov
Random Effects Graphical Models for Multiple Site Sampling ∗†
, 2003
"... We propose a two component graphical chain model, the discrete regression distribution, in which a set of categorical (or discrete) random variables is modeled as a response to a set of categorical and continuous covariates. We examine necessary and sufficient conditions for a discrete regression di ..."
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We propose a two component graphical chain model, the discrete regression distribution, in which a set of categorical (or discrete) random variables is modeled as a response to a set of categorical and continuous covariates. We examine necessary and sufficient conditions for a discrete regression distribution to be described by a given graph. The discrete regression formulation is extended to a statespace representation for the analysis of data collected at many random sites. In addition, some new results concerning marginalization in chain graph models are explored. Using the new results, we examine the Markov properties of the extended model as well as the marginal model of covariates and responses. Key words and phrases: chain graph, contingency table, discrete regression model, graphical models, marginalization, random effects 1
GRAPHICAL MARKOV MODELS
"... Graphical Markov models are multivariate statistical models which are currently under vigorous development and which combine two simple but most powerful notions, generating processes in single and joint response variables and conditional independences captured by graphs. The development of graphica ..."
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Graphical Markov models are multivariate statistical models which are currently under vigorous development and which combine two simple but most powerful notions, generating processes in single and joint response variables and conditional independences captured by graphs. The development of graphical Markov started with work by Wermuth (1976, 1980) and Darroch, Lauritzen and Speed (1980) which built on early results in 1920 to 1930 by geneticist Sewall Wright and probabilist Andrej Markov as well as on results for loglinear models by Birch (1963), Goodman (1970), Bishop, Fienberg and Holland (1973) and for covariance selection by Dempster (1972). Wright used graphs, in which nodes represent variables and arrows indicate linear dependence, to describe hypotheses about stepwise processes in single responses that could have generated his data. He developed a method, called path analysis, to estimate linear dependences and to judge whether the hypotheses are well compatible with his data which he summarized in terms of simple and partial correlations. With this approach he was far ahead of his time, since corresponding