Characterizing and Reasoning about Probabilistic and Non-Probabilistic Expectation

Some alternatives to probability in the literature include sets of probability measure [Huber 1981; Walley 1991], Dempster-Shafer belief functions [Shafer 1976] and the closely related nonadditive measures [Schmeidler 1989], and possibility measures [Dubois and Prade 1990]. In this paper, we consider the notion of expectation for all these representations of uncertainty. We do not take a stand here on what the "right" way is to represent uncertainty; we simply investigate characterizations of expectation and reasoning about expectation, both for probability and for other representations of uncertainty. It is well known that a probability measure determines a unique expectation function that is linear (i.e., E (aX + bY) = aE (X) + bE (Y)), monotone (i.e., X Y implies E ( X) E (Y)), and maps constant functions to their value. Conversely, given an expectation function E (that is, a function from random variables to the reals) that is linear, monotone, and maps constant functions to their value, there is a unique probability measure µ such that E = E µ. That is, there is a 1-1 mapping from probability measures to (probabilistic) expectation functions. One of the goals of this paper is to provide similar characterizations of expectation for other representations of uncertainty. Some work along these lines has already been done, particulary with regard to sets of probability measures [Huber 1981; Walley 1991; 1981]. 1 However, there seems to be surprisingly little work on characterizing expectation in the context of other measures of uncertainty, such as belief functions [Shafer 1976] and possibility measures [Dubois and Prade 1990].