Introduction The introduction of Bayesian networks (Pearl 1986b) and associated local computation algorithms (Lauritzen and Spiegelhalter 1988, Shenoy and Shafer 1990, Jensen, Lauritzen and Olesen 1990) has initiated a renewed interest for understanding causal concepts in connection with modelling complex stochastic systems. It has become clear that graphical models, in particular those based upon directed acyclic graphs, have natural causal interpretations and thus form a base for a language in which causal concepts can be discussed and analysed in precise terms. As a consequence there has been an explosion of writings, not primarily within mainstream statistical literature, concerned with the exploitation of this language to clarify and extend causal concepts. Among these we mention in particular books by Spirtes, Glymour and Scheines (1993), Shafer (1996), and Pearl (2000) as well as the collection of papers in Glymour and Cooper (1999). Very briefly, but fundamentally,

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...

The present set of lecture notes are prepared for the course “Statistik 2” at the University of Copenhagen. It is a revised version of notes prepared in connection with a series of lectures at the Swedish summerschool in Särö, June 11–17, 1979. The notes do by no means give a complete account of the theory of contingency tables. They are based on the idea that the graph theoretic methods in Darroch, Lauritzen and Speed (1978) can be used directly to develop this theory and, hopefully, with some pedagogical advantages. My thanks are due to the audience at the Swedish summerschool for patiently listening to the first version of these lectures, to Joseph Verducci, Stanford, who read the manuscript and suggested many improvements and corrections, and to Ursula Hansen, who typed the manuscript.

Recently, a class of multiscale tree-structured models was introduced in terms of scale-recursive dynamics defined on trees. The main advantage of these models is their association with a fast, recursive, Kalmanfilter prediction algorithm. In this article, we propose a more general class of multiscale graphical models over acyclic directed graphs, for use in command and control problems. Moreover, we derive the generalized-Kalmanfilter algorithm for graphical Markov models, which can be used to obtain the optimal predictors and prediction variances for multiscale graphical models. 1 Introduction Almost every aspect of command and control (C2) involves dealing with information in the presence of uncertainty. Since information in a battlefield is never precise, its status is rarely known exactly. In the face of this uncertainty, commanders must make decisions, issue orders, and monitor the consequences. The uncertainty may come from noisy data or, indeed, regions of the battle space whe...

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small exluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed. MARKOV RANDOM FIELDS, MARKOV PROPERTIES, CONDITIONAL INDEPENDENCE 1. Introduction. Conditional independence restrictions of the Markov type together with factorizations of probability distributions of Markov fields have been investigated over two decades. It is well known that if the distribution factorizes with respect to an undirected graph then it has the corresponding global Markov property which implies the local Markov property yet stronger than the pairwise one [see Lauritzen (1989),Whittaker (1990)]. Alternatively, under a positivity assumption on the density of the distribution a famous result, ascribed to Hammersley and Clifford, asserts the equi...

this paper is a study of well-known classes of CI-relations from the viewpoint of minors with a special emphasis on graphical visualization. After some preliminaries in Section 2 we characterize the classes of semigraphoids, pseudographoids and graphoids by means of their forbidden or compulsory 3-minors in Section 3. Then we provide in the next section a succint minor characterization of separation graphoids and an axiomatic characterization of boundary semigraphoids which originate from simple graphs according to the global and local Markov assumptions, respectively. Further we mention d-separation graphoids and their minors. In the second part of the paper we concentrate on semimatroids and especially on a new class of semimatroids called simple semimatroids. This class is of special interest for us because it is relatively easy to handle and, at the same time, the presented results have strong consequences for the structure of the classes of all semimatroids and probabilistically (p-) representable semimatroids. Let us recall that the p-representable semimatroids originate from systems of random variables ¸ = (¸ i ) i2N : a relation M is p-representable if there exists ¸ such that a triple (I; J; K)