Uncertainty, Belief, and Probability

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. Thus, the measure of belief in an event can 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 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. 1 Introduction Dealing with uncertainty is a fundamental issue for Al. The most widely-used approach to dealing with uncertainty is undoubtedly the Bayesian approach. It has the advantage of relying on well-understood techniques from probability theory, as well as some philosophical justification on the grounds that a "rational'* agent must assign uncertainties to events in a way that satisfies the axioms of probability [Cox46, Sav54]. On the other hand, the Bayesian approach has been widely criticized for requiring an agent to assign a subjective probability to every event. While this can be done in principle by having the agent play a suitable betting game [Jef83], 1 it does have a number of drawbacks. Among others, there is the computational difficulty of arriving at the probability. There is also the issue of whether it is reasonable 'This idea is due to Ramsey [Ram3l] and was rediscovered by von Neumann and Morgenstern [vNM47]; a clear exposition can be found in [LR57]. to describe confidence by a single point rather than a range. While an agent might be prepared to agree that the probability of an event lies within a given range, say between 1/3 and 1/2, he might not be prepared to say that it is precisely .435. Not surprisingly, there has been a great deal of debate regarding the Bayesian approach (see [Che85] and …