  The belief network is a well-known graphical structure for representing
independences 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. However, 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 alternatives
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 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 reasonable amount of confidence in the probabilities
which are stored in the network. Furthermore in this paper, a surprising upper
bound for the prior variances in the probabilities of the alternatives of all
nodes is obtained in the case where the probability distributions of the
probabilities of the alternatives are beta distributions. It is shown that the
prior variance in the probability at an alternative of a node is bounded above
by the largest variance in an element of the conditional probability
distribution for that node.
