Self-Conditional Probabilities and Probabilistic Interpretations of Belief Functions

We present an interpretation of belief functions within a pure probabilistic framework, namely as normalized self-conditional expected probabilities, and study their mathematical properties. Interpretations of belief functions appeal to partial knowledge. The self-conditional interpretation does this within the traditional probabilistic framework by considering surplus belief in an event emerging from a future observation, conditional on the event occurring. Dempster's original interpretation, in contrast, involves partial knowledge of a belief state. The modal interpretation, currently gaining popularity, models the probability of a proposition being believed (or proved, or known). The versatility of the belief function formalism is demonstrated by the fact that it accommodates very different intuitions.