Bayesian updating and belief functions

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Abstract
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 belieffunction, i.e., a nonadditive measure of a particular type, and characterizes P, i.e., P is the set of all probabilities that dominate f. This is in particular the case when data result from large-scale sampling with incomplete observations. This study is concerned with the effect of conditioning on such situations. The natural conditioning rule is here the Bayesian rule; there exists a posterior probability after the observation of event E, and it is known to be located in P(E), the set of conditionals of the members of P. An explicit expression for the Mobius transform phi(E) of f(E) in terms of phi, the transform of f, is found and Fagin and Halpern's earlier finding that the lower envelop f(E) of P(E) is itself a belief function is derived from it. However, f(E) no longer characterizes P(E) (not all probabilities dominating f(E) belong to it), unless f satisfy 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.

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