A new approach to updating beliefs

We define a new notion of conditional belief, which plays the same role for Dempster-Shafer belief functions as conditional probability does for probability functions. Our definition is different from the standard definition given by Dempster, and avoids many of the well-known problems of that definition. Just as the conditional probability Pr(.IB) is a probability function which is the result of conditioning on B being true, so too our conditional belief function BeZ(-IB) is a belief function which is the result of conditioning on B being true. We define the conditional belief as the lower envelope (that is, the inf) of a family of conditional probability functions, and provide a closedform expression for it. An alternate way of understanding our definition of conditional belief is provided by considering ideas from an earlier paper [Fagin and Halpern, 19891, where we connect belief functions with inner measures. In particular, we show here how to extend the definition of conditional probability to nonmeasurable sets, in order to get notions of inner and outer conditional probabilities, which can be viewed as best approximations to the true conditional probability, given our lack of information. Our definition of conditional belief turns out to be an exact analogue of our definition of inner conditional probability.