Semigroup Structure of Singleton Dempster-Shafer
Evidence Accumulation
November 2007
Andrzej K. Brodzik, The MITRE Corporation
Robert H. Enders, The MITRE Corporation
ABSTRACT
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 identi ed 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.
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Additional Search Keywords
data fusion, evidence accumulation, Dempster-Shafer theory, Dempster-Shafer
mass, Bayes inference, uncertainty, semigroup, semigroup homomorphism
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