Investigation of Variances in Belief Networks

The belief network is a well-known graphi­ cal structure for representing independences in a joint probability distribution. The meth­ ods, which perform probabilistic inference in belief networks, often treat the conditional probabilities which are stored in the network as certain values. However, if one takes ei­ ther 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 alternatives of remain­ ing nodes when other nodes are instanti­ ated; 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 in­ ferred probabilities is obtained under the as­ sumption that a posterior distribution on the uncertainty variables can be approximated by the prior distribution. It is shown that this assumption is plausible if their is a reason­ able amount of confidence in the probabili­ ties which are stored in the network. Fur­ thermore in this paper, a surprising upper bound for the prior variances in the proba­ bilities of the alternatives of all nodes is ob­ tained in the case where the probability dis­ tributions of the probabilities of the alterna­ tives are beta distributions. It is shown that the prior variance in the probability at an al­ ternative of a node is bounded above by the largest variance in an element of the condi­ tional probability distribution for that node.