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14
Algebraic Algorithms for Sampling from Conditional Distributions
 Annals of Statistics
, 1995
"... We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so a ..."
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Cited by 192 (16 self)
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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.
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
Bayesian Model Averaging And Model Selection For Markov Equivalence Classes Of Acyclic Digraphs
 Communications in Statistics: Theory and Methods
, 1996
"... Acyclic digraphs (ADGs) are widely used to describe dependences 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 B ..."
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Cited by 38 (5 self)
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Acyclic digraphs (ADGs) are widely used to describe dependences 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. 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. Recent results have shown that each Markovequivalence class is uniquely determined by a single chain graph, the essential graph, that is itself Markovequivalent simultaneously to all ADGs in the equivalence clas...
Remarks concerning graphical models for time series and point processes
 Revista de Econometria
, 1996
"... Uma rede estatística é uma cole,cão de nós representando variáveis aleatórias e um conjunto de arestas que ligam os nós. Um modelo estocástico por isso e chamado um modelo gráfico. Estes modelos, de gráficos e redes, sáo particularmente úteis para examinar as dependéncias estatísticas baseadas em co ..."
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Cited by 21 (3 self)
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Uma rede estatística é uma cole,cão de nós representando variáveis aleatórias e um conjunto de arestas que ligam os nós. Um modelo estocástico por isso e chamado um modelo gráfico. Estes modelos, de gráficos e redes, sáo particularmente úteis para examinar as dependéncias estatísticas baseadas em condi,coes do tipo das que ocorrem frequentemente em economia e estatística. Neste artigo as variáveis aleatórias dos nós serão séries temporais ou processos pontuais. Os casos de gráfos direcionados e nãodirecionados são apresentados. A statistical network is a collection of nodes representing random variables and a set of edges that connect the nodes. A probabilistic model for such is called a graphical model. These models, graphs and networks are particularly useful for examining statistical dependencies based on conditioning as often occurs in economics and statistics. In this paper the nodal random variables will be time series or point proceses. The cases of undirected and directed graphs are focussed on.
On Chain Graph Models For Description Of Conditional Independence Structures
 Ann. Statist
, 1998
"... This paper deals with chain graphs (CGs) which allow both directed and undirected edges. This class of graphs, introduced by Lauritzen and Wermuth [15], generalizes both UGs and DAGs. To establish the semantics of CGs one should associate an independency model to every CG. Some steps were already ma ..."
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Cited by 19 (3 self)
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This paper deals with chain graphs (CGs) which allow both directed and undirected edges. This class of graphs, introduced by Lauritzen and Wermuth [15], generalizes both UGs and DAGs. To establish the semantics of CGs one should associate an independency model to every CG. Some steps were already made. Lauritzen and Wermuth [16] intended to use CGs to describe independency models for strictly positive probability distributions and introduced the concept of the chain Markov property which is analogous to the concept of causal input list for DAGs. Lauritzen and Frydenberg [17, 9] generalized the concept of moral graph and introduced a moralization criterion for reading independency statements from a CG. Frydenberg [9] characterized CGs with the same Markov ON CHAIN GRAPH MODELS 3 property (that is producing the same CGmodel) and Andersson, Madigan and Perlman [3] used special CGs to represent uniquely classes of Markov equivalent DAGs. Whittaker [31] in his book gave several examples of the use of CGs, and other recent works also deal with them [6, 20, 23, 30], the most comprehensive account is provided by the book [19]. Several results proved here were already presented (without proof) in our previous conference contribution [5]. An alternative approach to the generalization of UGs and DAGs was started by Cox and Wermuth [7] who introduced a wider class of jointresponse chain graphs which allow also 'dashed' directed and undirected edges in addition to the classic 'solid' directed and undirected edges treated in this paper. Andersson, Madigan and Perlman [1] introduced an alternative Markov property to give an interpretation to those jointresponse CGs which combine dashed directed edges with solid undirected edges (of course, another independency model is associated...
Statistical notions of data disclosure avoidance and their relationship to traditional statistical methodology: Data swapping and loglinear models
 Proc. Bureau of the Census
, 1996
"... For most data releases especially those from censuses, the U. S. Bureau of the Census has either released data at high levels of aggregation or applied a data disclosure avoidance procedure such as data swapping or cell suppression before preparing microdata or tables for release. In this paper, we ..."
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Cited by 12 (3 self)
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For most data releases especially those from censuses, the U. S. Bureau of the Census has either released data at high levels of aggregation or applied a data disclosure avoidance procedure such as data swapping or cell suppression before preparing microdata or tables for release. In this paper, we present a general statistical characterization of the goal of a statistical agency in releasing confidential data subject to the application of disclosure avoidance procedures. We use this characterization to provide a framework for the study of data disclosure avoidance procedures for categorical variables. Consider a sample of n observations on p variables, which may be discrete or continuous. Our general characterization is in terms of the smoothing of a multidimensional empirical distribution function (an ordered version of the data), and sampling from it using bootstraplike selection. Both the smoothing and the sampling introduce alterations to the data and thus a bootstrap sample will not necessarily be the same as the original sample this works to preserve the confidentiality of individuals providing the original data. Two obvious questions are: How well confidentiality is preserved by such a process? Have the smoothing and sampling disguised fundamental relationships among the p variables of interest to others who will work only with the altered data? Rubin (1993) has provided a closely related characterization and approach based on multiple imputation. We explain some of these ideas in greater detail in the context of categorical random variables and compare them to methods in current use for data disclosure avoidance such as data swapping and cell suppression. We also relate this approach the data disclosure avoidance methods to statistical analysis associated with the use of loglinear models for crossclassified categorical data.
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
Chain Graphs: Semantics and Expressiveness
, 1995
"... . A chain graph (CG) is a graph admitting both directed and undirected edges with forbidden directed cycles. It generalizes both the concept of undirected graph (UG) and the concept of directed acyclic graph (DAG). CGs can be used efficiently to store graphoids, that is, independency knowledge of th ..."
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Cited by 8 (2 self)
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. A chain graph (CG) is a graph admitting both directed and undirected edges with forbidden directed cycles. It generalizes both the concept of undirected graph (UG) and the concept of directed acyclic graph (DAG). CGs can be used efficiently to store graphoids, that is, independency knowledge of the form "X is independent of Y given Z" obeying a set of five properties (axioms). Two equivalent criteria for reading independencies from a CG are formulated, namely the moralization criterion and the separation criterion. These criteria give exactly the graphoid closure of the input list for the CG. Moreover, a construction of a CG from a graphoid (through an input list), which produces a minimal Imap of that graphoid, is given. 1 Introduction Using graphs to describe independency structure arising among variables has a long and rich tradition. One can distinguish two classic approaches (for details see the book [11]): using undirected graphs (UGs), called also Markov networks, or using...
Algorithms for Learning Decomposable Models and Chordal Graphs
, 1997
"... Decomposable dependency models and their graphical counterparts, i.e., chordal graphs, possess a number of interesting and useful properties. On the basis of two characterizations of decomposable models in terms of independence relationships, we develop an exact algorithm for recovering the chordal ..."
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Cited by 5 (0 self)
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Decomposable dependency models and their graphical counterparts, i.e., chordal graphs, possess a number of interesting and useful properties. On the basis of two characterizations of decomposable models in terms of independence relationships, we develop an exact algorithm for recovering the chordal graphical representation of any given decomposable model. We also propose an algorithm for learning chordal approximations of dependency models isomorphic to general undirected graphs. 1 INTRODUCTION Graphical models are knowledge representation tools used by an increasing number of researchers, particularly from the Uncertainty in Artificial Intelligence community. The reason for the success of graphical models is their capacity to represent and handle independence relationships (which have proved crucial for the efficient management and storage of information), as well as uncertain information. Among the different kinds of graphical models, we are particularly interested in undirected and...