Basic Elements of Dempster-Shafer Theory

The greatest part of works dealing with the fundamentals of Dempster-Shafer theory is conceived either on the combinatoric, or on the axiomatic, but in both the cases on a very abstract level. The first approach begins by the assumption that S is a nonempty finite set, that m is a mapping which ascribes to each A ⊂ S a real number m(A) from the unit interval [0,1] in such a way that ∑A⊂ S m(A) = 1 (m is called a basic probability assignment on S), and that the (normalized) belief function induced by m is the mapping bel m : P(S) → [0,1] defined, for each A ⊂ S, by bel m (A) = (1 - m(o))-1 ∑o≠ B⊂A m(B), if m(o) < 1, bei m being undefined otherwise Shafer (1976) and elsewhere). The other (axiomatic) approach begins with the idea that belief function on a finite nonempty set S is a mapping bel: P(S) → [0,1], satisfying certain conditions (obeying certain axioms, in other terms). If these conditions (axioms) are strong and reasonable enough, it can be proved that it is possible to define uniquely a basic probability assignment m on S such that the belief function induced by m is identical with the original belief function defined by axioms, so that both the approaches meet each other and yield the same notion of belief function (Smets (1994)). The problems how to understand and obtain the probability distribution m over P(S) in the first case, or how to justify the particular choice of the demands imposed to belief functions in the other case, are put aside or are “picked before brackets” and they are not taken as a part of Dempster-Shafer theory in its formalized setting.