On the Validity of Dempster's Fusion Rule and its Interpretation as a Generalization of Bayesian Fusion Rule

Abstract

The purpose of this paper is to focus on and discuss both: (i) the validity of the Dempster's rule of combination and foundations of Dempster–Shafer theory and (ii) the real compatibility (or not) of the Dempster's rule with the Bayes fusion rule. We analyze and explain, on the basis of a generic example, the inconsistent behavior of the Dempster's rule of combination, introduced by Shafer in his mathematical theory of evidence, as a valid method to combine sources of information. We identify the cause and the effect of the dictatorial power behavior of this rule and of its impossibility to manage the conflicts between the sources no matter of their level. For comparison purpose, we present the respective solutions obtained by the more efficient proportional conflict redistribution rule number 5 proposed originally in Dezert–Smarandache theory framework. The inherent contradiction of Dempster–Shafer theory foundations is identified and proved. Then, a deep analysis of the compatibility of the Dempster's fusion rule with the Bayes fusion rule (from a fusion standpoint) is made on the basis of proposed new interesting formulation of the Bayes rule. We prove that the Dempster's rule does not behave as the Bayes fusion rule in general, because both methods deal very differently with the prior information when it is really informative (not uniform). Only in the very particular case where the basic belief assignments to combine are Bayesian and when the prior information is uniform (or vacuous), the Dempster's rule remains consistent with the Bayes fusion rule. It is proved, that in more general cases, the Dempster's rule is incompatible with the Bayes rule and it is not a generalization of the Bayes fusion rule.