  As belief networks are used to model increasingly complex situations, the
need to automatically construct them from large databases will become
paramount. This paper concentrates on solving a part of the belief network
induction problem: that of learning the quantitative structure (the conditional
probabilities), given the qualitative structure. In particular, a theory is
presented that shows how to propagate inference distributions in a belief
network, with the only assumption being that the given qualitative structure is
correct. Most inference algorithms must make at least this assumption. The
theory is based on four network transformations that are sufficient for any
inference in a belief network. Furthermore, the claim is made that contrary to
popular belief, error will not necessarily grow as the inference chain grows.
Instead, for QBN belief nets induced from large enough samples, the error is
more likely to decrease as the size of the inference chain increases.
