Results 1  10
of
12
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 ..."
Abstract

Cited by 91 (7 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 74 (17 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 47 (4 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 25 (5 self)
 Add to MetaCart
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.
Multimodality of the likelihood in the bivariate seemingly unrelated regression model
, 2002
"... ..."
Iterative conditional fitting for Gaussian ancestral graph models
 In M. Chickering and J. Halpern (Eds.), Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence
, 2004
"... Ancestral graph models, introduced by Richardson and Spirtes (2002), generalize both Markov random fields and Bayesian networks to a class of graphs with a global Markov property that is closed under conditioning and marginalization. By design, ancestral graphs encode precisely the conditional indep ..."
Abstract

Cited by 18 (6 self)
 Add to MetaCart
Ancestral graph models, introduced by Richardson and Spirtes (2002), generalize both Markov random fields and Bayesian networks to a class of graphs with a global Markov property that is closed under conditioning and marginalization. By design, ancestral graphs encode precisely the conditional independence structures that can arise from Bayesian networks with selection and unobserved (hidden/latent) variables. Thus, ancestral graph models provide a potentially very useful framework for exploratory model selection when unobserved variables might be involved in the datagenerating process but no particular hidden structure can be specified. In this paper, we present the Iterative Conditional Fitting (ICF) algorithm for maximum likelihood estimation in Gaussian ancestral graph models. The name reflects that in each step of the procedure a conditional distribution is estimated, subject to constraints, while a marginal distribution is held fixed. This approach is in duality to the wellknown Iterative Proportional Fitting algorithm, in which marginal distributions are fitted while conditional distributions are held fixed. 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 ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
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
Separation An Completeness Properties For Amp Chain Graph Markov Models
 Ann. Statist
, 2000
"... This paper introduces ..."
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 ..."
Abstract
 Add to MetaCart
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.