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.

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,

Structural equation modeling (SEM) has dominated causal analysis in the social and behavioral sciences since the 1960s. Currently, many SEM practitioners are having difficulty articulating the causal content of SEM and are seeking foundational answers.

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...

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...

this paper, we will show how path diagrams can be used to solve a number of important problems in structural equation modelling. There are a number of problems associated with structural equation modeling. These problems include:

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We show that the d-separation criterion constitutes a valid test for conditional independence relationships that are induced by feedback systems involving discrete variables. 1 INTRODUCTION It is well known that the d-separation test is sound and complete relative to the independencies assumed in the construction of Bayesian networks [Verma and Pearl, 1988, Geiger et al., 1990]. In other words, any d-separation condition in the network corresponds to a genuine independence condition in the underlying probability distribution and, conversely, every d-connection corresponds to a dependency in at least one distribution compatible with the network. The situation with feedback systems is more complicated, primarily because the probability distributions associated with such systems do not lend themselves to a simple product decomposition. The joint distribution of feedback systems cannot be written as a product of the conditional distributions of each child variable, given its parents. Rath...

Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of non-monotone 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

In this article, we consider the computational aspects of deciding whether a conditional independence statement t is implied by a list of conditional independence statements L using the independence implication provided by the method of structural imsets. We present two algorithmic methods which have the interesting complementary properties that one method performs well to prove that t is implied by L, while the other performs well to prove that t is not implied by L. However, both methods do not well perform the opposite. This gives rise to a parallel algorithm in which both methods race against each other in order to determine effectively whether t is or is not implied. Some empirical evidence is provided that suggests this racing algorithms method performs considerably better than an existing method based on so-called skeletal characterization of the respective implication. Furthermore, unlike previous methods, the method is able to handle more than five variables.