Exact distribution theory for belief net responses

Bayesian belief networks provide estimates of conditional probabilities, called query responses. The precision of these estimates is assessed using posterior distributions. This paper discusses two claims and a conjecture by Kleiter (1996) concerning the exact posterior distribution of queries. The two claims provide conditions where a query has an exact beta distribution. The first claim is clarified by the following generalization. Assuming a BDe prior and complete data, a query has the same distribution under equivalent network structures. If the query can be represented as a network parameter under an equivalent structure, it must then have a beta distribution. Kleiter's second claim is contradicted by a counter-example. His conjecture, concerning finite mixtures of beta distributions, is also disproved.