Uncertainty, belief, and probability 1

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 measures), 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 precise sense to Dempster‐Shafer belief functions. Hence, in addition to providing promising new conceptual tools for dealing with uncertainty, our approach shows that a key part of the important Dempster‐Shafer theory of evidence is firmly rooted in classical probability theory.