Belief functions with nonstandard values

Abstract

The notions of basic probability assignment and belief function, playing the basic role in the Dempster-Shafer model of uncertainty quantification and processing often called Dempster-Shafer theory, are generalized in such a way that their values are not numbers from the unit interval of reals, but rather infinite sequences of real numbers including those greater than one and the negative ones. Within this extended space it is possible to define inverse probability assignments and, consequently, to define the dual operation to the Dempster combination rule, also to assignments ascribing, to the whole space of discourse, the degree of belief “smaller than any positive real number” or “quasi-zero”, in a sense; the corresponding inverse assignments than take “quasi-infinite” values. This approach extends the space of invertible, or non-dogmatic, in the sense introduced by Ph. Smets, basic probability assignments and belief functions, when compared with the other approaches suggested till now.