Propagation of Probabilities, Means, and Variances in Mixed Graphical Association Models

Abstract

A scheme is presented for modeling and local computation of exact probabilities, means, and variances for mixed qualitative and quantitative variables. The models assume that the conditional distribution of the quantitative variables, given the qualitative, is multivariate Gaussian. The computational architecture is set up by forming a tree of belief universes, and the calculations are then performed by local message passing between universes. The asymmetry between the quantitative and qualitative variables sets some additional limitations for the specification and propagation structure. Approximate methods when these are not appropriately fulfilled are sketched. It has earlier been shown how to exploit the local structure in the specification of a discrete probability model for fast and efficient computation, thereby paving the way for exploiting probability-based models as parts of realistic systems for planning and decision support. The purpose of this article is to extend this computational scheme to networks, where some vertices represent entities that are measured on a quantitative and some on a qualitative scale. An extension has the advantage of unifying several known techniques, but allows more flexible and faithful modeling and speeds computation as well. To handle this more general case, the properties of (CG) conditional Gaussian distributions are exploited. A fictitious but simple example is used for illustration throughout the paper, concerned with monitoring emissions from a waste incinerator. From optical measurements of the darkness of the smoke, the concentration of CO2—which are both on a continuous scale—and possible knowledge about qualitative characteristics such as the type of waste burned, one wants to infer about the state of the incinerator and the current emission of heavy metals.