: Belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the Kolmogorov axioms defining probability functions. We argue that there are (at least) two useful and quite different ways of understanding belief functions. The first is as a generalized probability function (which technically corresponds to the inner measure induced by a probability function). The second is as a way of representing evidence. Evidence, in turn, can be understood as a mapping from probability functions to probability functions. It makes sense to think of updating a belief if we think of it as a generalized probability. On the other hand, it makes sense to combine two beliefs (using, say, Dempster's rule of combination) only if we think of the belief functions as representing evidence. Many previous papers have pointed out problems with the belief function approach; the claim of this paper is that these problems can be explained as a consequence of confounding the...

: 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 P r(\DeltajB) is a probability function which is the result of conditioning on B being true, so too our conditional belief function Bel(\DeltajB) 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 closed-form expression for it. An alternate way of understanding our definition of conditional belief is provided by considering ideas from an earlier paper [FH91], where we connect belief functions with inner measures. In particular, we show here how to extend the definition of conditional pro...

: We introduce a new probabilistic approach to dealing with uncertainty, based on the observation that probability theory does not require that every event be assigned a probability. For a nonmeasurable event (one to which we do not assign a probability), we can talk about only the inner measure and outer measure of the event. In addition to removing the requirement that every event be assigned a probability, our approach circumvents other criticisms of probability-based approaches to uncertainty. For example, the measure of belief in an event turns out to be represented by an interval (defined by the inner and outer measure), rather than by a single number. Further, this approach allows us to assign a belief (inner measure) to an event E without committing to a belief about its negation :E (since the inner measure of an event plus the inner measure of its negation is not necessarily one). Interestingly enough, inner measures induced by probability measures turn out to correspond in a ...

This paper examines the applicability of belief functions methodology in three reasoning tasks: (1) representation of incomplete knowledge, (2) belief updating, and (3) evidence pooling. We find that belief functions have difficulties representing incomplete knowledge, primarily knowledge expressed in conditional sentences. In this context, we also show that the prevailing practices of encoding if-then rules as belief function expressions are inadequate, as they lead to counterintuitive con-clusions under chaining, contraposition, and reasoning by cases. Next, we examine the role of belief functions in updating states of belief and find that, if partial knowledge is encoded and updated by belief function methods, the updating pro-cess violates basic patterns of plausibility and the resulting beliefs cannot serve as a basis for rational decisions. Finally, assessing their role in evidence pooling, we find that belief functions offer a rich language for describing the evidence gath-ered, highly compatible with the way people summarize observations. However, the methods available for integrating evidence into a coherent state of belief ca-

This paper compares two formalisms for uncertain inference, Kyburg's Combinatorial Semantics and Dempster-Shafer belief function theory, on the basis of an example from the domain of medical diagnosis. I review Shafer's example about the imaginary disease ploxoma and show how it would be represented in Combinatorial Semantics. I conclude that belief function theory has a qualitative advantage because it offers greater flexibility of expression, and provides results about more specific classes of patients. Nevertheless, a quantitative comparison reveals that the inferences sanctioned by Combinatorial Semantics are more reliable than those of belief function theory. 5.1 Introduction This paper compares Kyburg's Combinatorial Semantics (CS) and Dempster-Shafer belief function theory (BFT), two formalisms for representing uncertain inference from statistical data. I reconsider a medical diagnosis example introduced by Shafer [Shafer82] concerning the imaginary disease ploxoma. In Section ...