Propagation of variances in belief networks

The belief network is a graphical structure for representing independencies in a joint probability distribution. The methods which perform probabilistic inference in belief networks often treat the conditional probabilities which are stored in the network as certain values. If one takes either a subjectivistic or a limiting frequency approach to probability, one can never be certain of probability values. An algorithm should not only be capable of reporting the probabilities of the outcomes of remaining nodes when other nodes are instantiated, it should also be capable of reporting the uncertainty in these probabilities relative to the uncertainty in the probabilities which are stored in the network. In this paper a method for determining the variances in inferred probabilities is obtained under the assumption that a prosterior distribution on the uncertainty variables can be approximated by the prior distribution. This assumption is plausible if there is a reasonable amount of confidence in the probabilities stored in the network.