Why Dempster´s fusion rule is not a generalization of Bayes fusion rule

In this paper, we analyze Bayes fusion rule in details from a fusion standpoint, as well as the emblematic Dempster´s rule of combination introduced by Shafer in his Mathematical Theory of evidence based on belief functions. We propose a new interesting formulation of Bayes rule and point out some of its properties. A deep analysis of the compatibility of Dempster´s fusion rule with Bayes fusion rule is done. We show that Dempster´s rule is compatible with Bayes fusion rule only in the very particular case where the basic belief assignments (bba´s) to combine are Bayesian, and when the prior information is modeled either by a uniform probability measure, or by a vacuous bba. We show clearly that Dempster´s rule becomes incompatible with Bayes rule in the more general case where the prior is truly informative (not uniform, nor vacuous). Consequently, this paper proves that Dempster´s rule is not a generalization of Bayes fusion rule.