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...

We introduce and study a calculus for real-valued 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.

The class of chain graphs (CGs) involving both undirected graphs (= Markov networks) and directed acyclic graphs (= Bayesian networks) was introduced in middle eighties for description of probabilistic conditional independence structures. Every class of Markov equivalent CGs (that is CGs describing the same conditional independence structure) has a natural representative, which is called the largest CG. The paper presents socalled recovery algorithm, which on basis of the conditional independence structure given by a CG (in form of so-called dependency model) finds the largest CG, representing the corresponding class of Markov equivalent CGs. As a byproduct a graphical characterization of graphs, which are the largest CGs (for a class of Markov equivalent CGs) is obtained, and a simple algorithm changing every CG into the largest CG of the corresponding equivalence class is given. 1 INTRODUCTION Classic graphical approaches to description of probabilistic conditional independence stru...

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.

Independence models induced by some uncertainty measures (e.g. conditional probability, possibility) do not obey the usual graphoid properties, since they do not satisfy the symmetry property. They are efficiently representable through directed acyclic l-graphs by using L-separation criterion.

2 Chain graphs (CG) ( = adicyclic graphs) use undirected and directed edges to represent simultaneously both structural and associative dependences.. Like acyclic directed graphs (ADGs), the CG associated with a given statistical model may not be unique, so CGs fall into Markov equivalence classes, which may be superexponentially large, leading to unidentifiability and computational inefficiency in model search and selection. It is shown here that under the Andersson-Madigan-Perlman (AMP) Markov interpretation of a CG, each Markov-equivalence class can be uniquely represented by a single distinguished CG, the AMP essential graph, that is itself simultaneously Markov equivalent to all CGs in the AMP Markov equivalence class. A complete characterization of AMP essential graphs is obtained. Like the essential graph previously introduced for ADGs, the AMP essential graph will play a fundamental role for inference and model search and selection for AMP CG models.

This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that the strictly positive discrete probability distributions with the prescribed sample space that factorize according to a chain graph G with dimension d have positive Lebesgue measure wrt R d, whereas those that factorize according to G but are not faithful to it have zero Lebesgue measure wrt R d. This means that, in the measuretheoretic sense described, almost all the strictly positive discrete probability distributions with the prescribed sample space that factorize according to G are faithful to it.

. Irrelevance relations are sets of statements of the form: given that the `value' of Z is known, the `values' of Y can add no further information about the `values' of X. Undirected Graphs (UGs), Directed Acyclic Graphs (DAGs) and Chain Graphs (CGs) were used and investigated as schemes for the purpose of representing irrelevance relations. It is known that, although all three schemes can approximate irrelevance, they are inadequate in the sense that there are relations which cannot be fully represented by anyone of them. In this paper annotated graphs are defined and suggested as a new model for graphical representation. It is shown that this new model is a proper generalization of the former Several of the results presented in this paper are included in the MSc Thesis of the second author, done under the supervision of the first author, submitted to the Graduate School of the Technion, Israel Institute of Technology. The contribution of the first author was supported by the ...

Within orthopædics, clinicians routinely take multiple measurements on patients during the course of their treatment, often repeating the same measurements before and after operations, and subsequently at periodic follow-up consultations. This data combined with additional factors gives a wealth of information, resulting in a high-dimensional data set with a mixture of data types and a longitudinal aspect; all of which can be problematic in statistical analysis. Therefore, general statistical methods for the investigation and analysis of a generic medical data set are presented and developed. Methods are proposed for supporting exploratory analysis of the data via novel visualisations of the patient’s status over time across multiple variables, thus giving an easily interpretable overview of this evolution. To address the problem of high dimensionality of the data, a new approach to variable selection is proposed and developed using principal variables. The method is further extended by the use of temporal smoothing to tackle data with this repeated measures aspect allowing for the simultaneous reduction of the patient status variables over time. The ultimate goal of these analyses is to determine an appropriate model for the orthopædic data, with a focus on the modelling of the time series of patient progress. The techniques of graphical modelling and, in particular, those of chain graphs lend themselves to this problem. Additionally, they have the added benefit of a simple and intuitive visualisation which is of benefit to clinicians. All of these methods are illustrated via their application to two large-scale case study data sets concerning total joint replacement.

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 state-space 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