  The multiple extension problem arises frequently in diagnostic and default
inference. That is, we can often use any of a number of sets of defaults or
possible hypotheses to explain observations or make Predictions. In default
inference, some extensions seem to be simply wrong and we use qualitative
techniques to weed out the unwanted ones. In the area of diagnosis, however,
the multiple explanations may all seem reasonable, however improbable. Choosing
among them is a matter of quantitative preference. Quantitative preference
works well in diagnosis when knowledge is modelled causally. Here we suggest a
framework that combines probabilities and defaults in a single unified
framework that retains the semantics of diagnosis as construction of
explanations from a fixed set of possible hypotheses. We can then compute
probabilities incrementally as we construct explanations. Here we describe a
branch and bound algorithm that maintains a set of all partial explanations
while exploring a most promising one first. A most probable explanation is
found first if explanations are partially ordered.
