A causal network is used in a number of areas as a depiction of patterns of 'influence' among sets of variables. In expert systems it is common to perform 'inference' by means of local computations on such large but sparse networks. In general, non-probabilistic methods are used to handle uncertainty when propagating the effects of evidence, and it has appeared that exact probabilistic methods are not computationally feasible. Motivated by an application in electromyography, we counter this claim by exploiting a range oflocal representations for the joint probability distribution, combined with topological changes to the original network termed 'marrying' and 'filling-in'. The resulting structure allows efficient algorithms for transfer between representations, providing rapid absorption and propagation of evidence. The scheme is first illustrated on a small, fictitious but challenging example, and the underlying theory and computational aspects are then discussed.

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We investigate directed Markov fields over finite graphs without positivity assumptions on the densities involved. A criterion for conditional independence of two groups of variables given a third is given and named as the directed, global Markov property. We give a simple proof of the fact that the directed, local Markov property and directed, global Markov property are equivalent and -in the case of absolute continuity w.r.t. a product measure -equivalent to the recursive factorization of densities. It is argued that our criterion is easy to use, it is sharper than that given by Kiiveri, Speed, and Carlin and equivalent to that of Pearl. It follows that our criterion cannot be sharpened.

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A directed acyclic graph or influence diagram is frequently used as a representation for qualitative knowledge in some domains in which expert system techniques have been applied, and conditional probability tables on appropriate sets of variables form the quantitative part of the accumulated experience. It is shown how one can introduce imprecision into such probabilities as a data base of cases accumulates. By exploiting the graphical structure, the updating can be performed locally, either approximately or exactly, and the setup makes it possible to take advantage of a range of well-established statistical techniques. As examples we discuss discrete models, models based on Dirichlet distributions and models of the logistic regression type.