This paper describes a scheme for local computation in conditional Gaussian Bayesian networks that combines the approach of Lauritzen and Jensen (2001) with some elements of Shachter and Kenley (1989). Message passing takes place on an elimination tree structure rather than the more compact (and usual) junction tree of cliques. This yields a local computation scheme in which all calculations involving the continuous variables are performed by manipulating univariate regressions, and hence matrix operations are avoided.

Given a Bayesian network of discrete random variables with a hyper-Dirichlet prior, a method is proposed for assigning Dirichlet priors to the conditional probabilities of structurally different networks. It defines a distance measure between priors which is to be minimized for the assignment process. Intuitively one would expect that if two models priors are to qualify as being 'close' in some sense, then their posteriors should also be nearby after an observation. However one does not know in advance what will be observed next. Thus we are led to propose an expectation of Kullback-Leibler distances over all possible next observations to define a measure of distance between priors. In conjunction with the additional assumptions of global and local independence of the parameters, a number of theorems emerge which are usually taken as reasonable assumptions in the Bayesian network literature. A simple example is given to illustrate the technique.

Probabilistic expert systems based on Bayesian networks require initial specification of both qualitative graphical structure and quantitative conditional probability assessments. As (possibly incomplete) data accumulate on real cases, the parameters of the system may adapt, but it is also essential that the initial specifications be monitored with respect to their predictive performance. A range of monitors based on standardized scoring rules that are designed to detect both qualitative and quantitative departures from the specified model is presented. A simulation study demonstrates the efficacy of these monitors at uncovering such departures.>