  The analysis of practical probabilistic models on the computer demands a
convenient representation for the available knowledge and an efficient
algorithm to perform inference. An appealing representation is the influence
diagram, a network that makes explicit the random variables in a model and
their probabilistic dependencies. Recent advances have developed solution
procedures based on the influence diagram. In this paper, we examine the
fundamental properties that underlie those techniques, and the information
about the probabilistic structure that is available in the influence diagram
representation. The influence diagram is a convenient representation for
computer processing while also being clear and non-mathematical. It displays
probabilistic dependence precisely, in a way that is intuitive for decision
makers and experts to understand and communicate. As a result, the same
influence diagram can be used to build, assess and analyze a model,
facilitating changes in the formulation and feedback from sensitivity analysis.
The goal in this paper is to determine arbitrary conditional probability
distributions from a given probabilistic model. Given qualitative information
about the dependence of the random variables in the model we can, for a
specific conditional expression, specify precisely what quantitative
information we need to be able to determine the desired conditional probability
distribution. It is also shown how we can find that probability distribution by
performing operations locally, that is, over subspaces of the joint
distribution. In this way, we can exploit the conditional independence present
in the model to avoid having to construct or manipulate the full joint
distribution. These results are extended to include maximal processing when the
information available is incomplete, and optimal decision making in an
uncertain environment. Influence diagrams as a computer-aided modeling tool
were developed by Miller, Merkofer, and Howard [5] and extended by Howard and
Matheson [2]. Good descriptions of how to use them in modeling are in Owen [7]
and Howard and Matheson [2]. The notion of solving a decision problem through
influence diagrams was examined by Olmsted [6] and such an algorithm was
developed by Shachter [8]. The latter paper also shows how influence diagrams
can be used to perform a variety of sensitivity analyses. This paper extends
those results by developing a theory of the properties of the diagram that are
used by the algorithm, and the information needed to solve arbitrary
probability inference problems. Section 2 develops the notation and the
framework for the paper and the relationship between influence diagrams and
joint probability distributions. The general probabilistic inference problem is
posed in Section 3. In Section 4 the transformations on the diagram are
developed and then put together into a solution procedure in Section 5. In
Section 6, this procedure is used to calculate the information requirement to
solve an inference problem and the maximal processing that can be performed
with incomplete information. Section 7 contains a summary of results.
