A new look at conditional probability with belief functions

We discuss repeatable experiments about which various agents may have different information. This information can vary from a full probabilistic description of the experiment in the sense that the probabilities of all outcomes are known to the agent, to having no information whatsoever, except the collection of possible outcomes. We argue that belief functions are very suitable for modeling the type of information we have in mind. We redevelop and rederive various notions of conditional belief functions, using a viewpoint of relative frequencies. We call the two main forms of conditioning contingent and necessary conditioning, respectively. The former is used when the conditioning event may also have not occurred, whereas the latter is used when it turns out that the event on which we condition occurs necessarily. Our approach unifies various notions in the literature into one conceptual framework, namely, the updated belief functions of Fagin and Halpern, the unnormalized conditional belief function of Smets, and the notions of updating and focusing as used by Dubois and Prade. We show that the original Dempster–Shafer definition of conditional belief functions cannot be interpreted directly in our framework. We give a number of examples illustrating our interpretation, as well as the differences between the various notions of conditioning.