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25
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.
NeighborhoodBased Models for Social Networks
 Sociological Methodology
, 2002
"... Harrison White and several anonymous reviewers for valuable comments on the work. We argue that social networks can be modeled as the outcome of processes that occur in overlapping local regions of the network, termed local social neighborhoods. Each neighborhood is conceived as a possible site of i ..."
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Cited by 55 (4 self)
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Harrison White and several anonymous reviewers for valuable comments on the work. We argue that social networks can be modeled as the outcome of processes that occur in overlapping local regions of the network, termed local social neighborhoods. Each neighborhood is conceived as a possible site of interaction and corresponds to a subset of possible network ties. In this paper, we discuss hypotheses about the form of these neighborhoods, and we present two new and theoretically plausible ways in which neighborhoodbased models for networks can be constructed. In the first, we introduce the notion of a setting structure, a directly hypothesized (or observed) set of exogenous constraints on possible neighborhood forms. In the second, we propose higherorder neighborhoods that are generated, in part, by the outcome of interactive network processes themselves. Applications of both approaches to model construction are presented, and the developments are considered within a general conceptual framework of locale for social networks. We show how assumptions about neighborhoods can be cast within a hierarchy of increasingly complex models; these models represent a progressively greater capacity for network processes to “reach ” across a network through long cycles or semipaths. We argue that this class of models holds new promise for the development of empirically plausible models for networks and networkbased processes. 2 1.
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 46 (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...
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...
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
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.
Algebraic statistical models
 Statistica Sinica
"... Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semialgebraic subsets of the parameter space of a reference model with nice properti ..."
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Cited by 12 (4 self)
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Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semialgebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an ‘algebraic exponential family’. This new definition provides a unified framework for the study of statistical models with algebraic structure. In this paper we review the ingredients to this definition and illustrate in examples how computational algebraic geometry can be used to solve problems arising in statistical inference in algebraic models. Key words and phrases: Algebraic statistics, computational algebraic geometry, exponential family, maximum likelihood estimation, model invariants, singularities. 1.
Conditioning as disintegration
 Statistica Neerlandica
, 1997
"... Conditional probability distributions seem to have a bad reputation when it comes to rigorous treatment of conditioning. Technical arguments are published as manipulations of Radon±Nikodym derivatives, although we all secretly perform heuristic calculations using elementary de®nitions of conditional ..."
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Cited by 9 (0 self)
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Conditional probability distributions seem to have a bad reputation when it comes to rigorous treatment of conditioning. Technical arguments are published as manipulations of Radon±Nikodym derivatives, although we all secretly perform heuristic calculations using elementary de®nitions of conditional probabilities. In print, measurability and averaging properties substitute for intuitive ideas about random variables behaving like constants given particular conditioning information. One way to engage in rigorous, guiltfree manipulation of conditional distributions is to treat them as disintegrating measuresÐfamilies of probability measures concentrating on the level sets of a conditioning statistic. In this paper we present a little theory and a range of examplesÐfrom EM algorithms and the Neyman factorization, through Bayes theory and marginalization paradoxesÐto suggest that disintegrations have both intuitive appeal and the rigor needed for many problems in mathematical statistics.
Beware of the DAG!
 NIPS 2008 WORKSHOP ON CAUSALITY
, 2008
"... Directed acyclic graph (DAG) models are popular tools for describing causal relationships and for guiding attempts to learn them from data. In particular, they appear to supply a means of extracting causal conclusions from probabilistic conditional independence properties inferred from purely observ ..."
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Cited by 7 (0 self)
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Directed acyclic graph (DAG) models are popular tools for describing causal relationships and for guiding attempts to learn them from data. In particular, they appear to supply a means of extracting causal conclusions from probabilistic conditional independence properties inferred from purely observational data. I take a critical look at this enterprise, and suggest that it is in need of more, and more explicit, methodological and philosophical justification than it typically receives. In particular, I argue for the value of a clean separation between formal causal language and intuitive causal assumptions.
Algebraic Techniques for Gaussian Models
, 2006
"... Abstract: Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be employed to solve equation systems arising in maxim ..."
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Cited by 5 (2 self)
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Abstract: Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be employed to solve equation systems arising in maximum likelihood estimation and parameter identification, but they also permit to study model singularities at which standard asymptotic approximations to the distribution of estimators and test statistics may no longer be valid. This paper demonstrates such applications of algebraic geometry in selected examples of Gaussian models, thereby complementing the existing literature on models for discrete variables. MSC 2000: 62H05, 62H12 Key words: Algebraic statistics, multivariate normal distribution, parameter identification, singularities 1