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.

In a previous paper [8] we presented an algorithm for extracting causal influences from independence information, where a causal influence was defined as the existence of a directed arc in all minimal causal models consistent with the data. In this paper we address the question of deciding whether there exists a causal model that explains ALL the observed dependencies and independencies. Formally, given a list M of conditional independence statements, it is required to decide whether there exists a directed acyclic graph D that is perfectly consistent with M, namely, every statement in M, and no other, is reflected via d-separation in D. We present and analyze an effective algorithm that tests for the existence of such a dag, and produces one, if it exists. Key words: Causal modeling, graphoids, conditional independence. 1 1 Introduction Directed acyclic graphs (dags) have been widely used for modeling statistical data. Starting with the pioneering work of Sewal Wright [...

this paper, Rumelhart presented compelling evidence that text comprehension must be a distributed process that combines both top-down and bottom-up inferences. Strangely, this dual mode of inference, so characteristic of Bayesian analysis, did not match the capabilities of either the "certainty factors" calculus or the inference networks of PROSPECTOR -- the two major contenders for uncertainty management in the 1970s. I thus began to explore the possibility of achieving distributed computation in a "pure" Bayesian framework, so as not to compromise its basic capacity to combine bi-directional inferences (i.e., predictive and abductive) . Not caring much about generality at that point, I picked the simplest structure I could think of (i.e., a tree) and tried to see if anything useful can be computed by assigning each variable a simple processor, forced to communicate only with its neighbors. This gave rise to the tree-propagation algorithm reported in [15] and, a year later, the Kim-Pearl algorithm [12], which supported not only bi-directional inferences but also intercausal interactions, such as "explaining-away." These two algorithms were described in Section 2 of Fusion.

INTRODUCTION This chapter surveys the development of graphical models known as Bayesian networks, summarizes their semantical basis and assesses their properties and applications to reasoning and planning. Bayesian networks are directed acyclic graphs (DAGs) in which the nodes represent variables of interest (e.g., the temperature of a device, the gender of a patient, a feature of an object, the occurrence of an event) and the links represent causal influences among the variables. The strength of an influence is represented by conditional probabilities that are attached to each cluster of parents-child nodes in the network. Figure 1 illustrates a simple yet typical Bayesian network. It describes the causal relationships among the season of the year (X 1 ), whether rain falls (X 2 ) during the season, whether the sprinkler is on (X 3 ) during that season, whether the pavement would get wet (X<F28.21

For nearly a century, investigators in the social sciences have used regression models to deduce cause-and-effect relationships from patterns of association. Path models and automated search procedures are more recent developments. In my view, this enterprise has not been successful. The models tend to neglect the difficulties in establishing causal relations, and the mathematical complexities tend to obscure rather than clarify the assumptions on which the analysis is based. Formal statistical inference is, by its nature, conditional. If maintained hypotheses A, B, C,... hold, then H can be tested against the data. However, if A, B, C,... remain in doubt, so must inferences about H. Careful scrutiny of maintained hypotheses should therefore be a critical part of empirical work-- a principle honored more often in the breach than the observance.

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This paper demonstrates the use of graphs as a mathematical tool for expressing independenices, and as a formal language for communicating and processing causal information in statistical analysis. We show how complex information about external interventions can be organized and represented graphically and, conversely, how the graphical representation can be used to facilitate quantitative predictions of the effects of interventions. We first review the Markovian account of causation and show that directed acyclic graphs (DAGs) offer an economical scheme for representing conditional independence assumptions and for deducing and displaying all the logical consequences of such assumptions. We then introduce the manipulative account of causation and show that any DAG defines a simple transformation which tells us how the probability distribution will change as a result of external interventions in the system. Using this transformation it is possible to quantify, from non-experimental data...

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ABSTRACT. By applying the minimality principle for model selection, one should seek the model that describes the data by a code of minimal length. Learning is viewed as data compression that exploits the regularities or qualitative properties found in the data, in order to build a model containing the meaningful information. The theory of causal modeling can be interpreted by this approach. The regularities are the conditional independencies reducing a factorization and the v-structure regularities. In the absence of other regularities, a causal model is faithful and offers a minimal description of a probability distribution. The causal interpretation of a faithful Bayesian network is motivated by the canonical representation it offers and faithfulness. A causal model decomposes the distribution into independent atomic blocks and is able to explain all qualitative properties found in the data. The existence of faithful models depends on the additional regularities in the data. Local structure of the conditional probability distributions allow further compression of the model. Interfering regularities, however, generate conditional independencies that do not follow from the Markov condition. These regularities has to be incorporated into an augmented model for which the inference algorithms are adapted to take into account their influences. But for other regularities, like patterns in a string, causality does not offer a modeling framework that leads to a minimal description. 1