Semigroup Structure of Singleton Dempster–Shafer Evidence Accumulation

Dempster-Shafer theory is one of the main tools for reasoning about data obtained from multiple sources, subject to uncertain information. In this work abstract algebraic properties of the Dempster-Shafer set of mass assignments are investigated and compared with the properties of the Bayes set of probabilities. The Bayes set is a special case of the Dempster-Shafer set, where all non-singleton masses are fixed at zero. The language of semigroups is used, as appropriate subsets of the Dempster-Shafer set, including the Bayes set and the singleton Dempster-Shafer set, under either a mild restriction or a slight extension, are semigroups with respect to the Dempster-Shafer evidence combination operation. These two semigroups are shown to be related by a semigroup homomorphism, with elements of the Bayes set acting as images of disjoint subsets of the Dempster-Shafer set. Subsequently, an inverse mapping from the Bayes set onto the set of these subsets is identified and a procedure for computing certain elements of these subsets, acting as subset generators, is obtained. The algebraic relationship between the Dempster-Shafer and Bayes evidence accumulation schemes revealed in the investigation elucidates the role of uncertainty in the Dempster-Shafer theory and enables direct comparison of results of the two analyses.