Updating probabilistic knowledge using imprecise and uncertain evidence

This paper examines how to update a priori knowledge which is representable by a multi-dimensional probability distribution, when one learns that the observation is representable by a cluster of random sets or bodies of evidence defined on different one-dimensional space. In order to resolve this problem, firstly, a set of marginal probability distributions is derived from the set of random sets, where each marginal probability distribution is compatible with the corresponding random set, and is 'close' to a priori probability distribution's marginalization with respect to the corresponding universe in the sense of cross-entropy. Then an additively constrained set is derived from all random sets. Lastly, the iterative proportional fitting procedure (IPFP) is used to search the desired probability distribution in the additively constrained set with respect to a priori probability distribution.