Jeffrey's rule of conditioning generalized to belief functions

Jeffrey's rule of conditioning has been proposed in order to revise a probability measure by another probability function. We generalize it within the framework of the models based on belief functions. We show that several forms of Jeffrey's conditionings can be defined that correspond to the geometrical rule of conditioning and to Dempster's rule of conditioning, respectively. 1. Jeffrey's rule in probability theory. In probability theory conditioning on an event . is classically obtained by the application of Bayes' rule. Let (Q, � , P) be a probability space where P(A) is the probability of the event Ae � where� is a Boolean algebra defined on a finite2 set n. P(A) quantified the degree of belief or the objective probability, depending on the interpretation given to the probability measure, that a particular arbitrary element m of n which is not a priori located in any of the sets of� belongs to a particular set Ae�. Suppose it is known that m belongs to Be� and P(B)>O. The probability measure P must be updated into PB that quantifies the same event as previously but after taking in due consideration the know ledge that me B. PB is obtained by Bayes' rule of conditioning: This rule can be obtained by requiring that: 81: VBE�. PB(B) = 1 82: VBe�, VX,Ye� such that X.Y�B. and PJ3(X) _ P(X) PB(Y)P(Y) PB(Y) = 0 ifP(Y)>O