Bayesian updating and belief functions

In a wide class of situations of uncertainty, the available information concerning the event space can be described as follows. There exists a true probability that is only known to belong to a certain set P of probabilities: moreover, the lower envelope f of P is a belief function, i.e., a nonadditive measure of a particular type, and characterizes P, i.e., P is the set of all probabilities that dominate f. The effect of conditioning on such situations is examined. The natural conditioning rule in this case is the Bayesian rule. An explicit expression for the Mobius transform phi /sup E/ of f/sup E/ in terms of phi , the transform of f, is found, and an earlier finding that the lower envelope f/sup E/ of P/sup E/ is itself a belief function is derived from it. However, f/sup E/ no longer characterizes P/sup E/ unless f satisfies further stringent conditions that are both necessary and sufficient. The difficulties resulting from this fact are discussed, and suggestions to cope with them are made.>