General Fusion Operators from Cox’s Postulates

This chapter presents new important links between the most important theories developed in literature for managing uncertainties (i.e. probability, fuzzy sets and evidence theories). The Information fusion introduces special operators ◦ in the probability theory, in the fuzzy set theory and in the theory of evidence. The mathematical theory of evidence and the fuzzy set theory often replace probabilities in medicine, economy and automatics. The choice between these three quite distinct theories depends on the intrinsic nature of the data to combine. This chapter shows that same four postulates support actually these apparently distinct theories. We unify these three theories from the four following postulates: non-contradiction, continuity, universality, context dependence and prove that a same functional equation is supported by probability theory, evidence theory and fuzzy set theories. In other words, the same postulates applied on confidences, under different conditions, either in the dependence or independence situation, imply the same foundation for the various modern theories of information fusion in the framework of uncertainty by using deductions that we have unified. The independence between elementary confidences have not to be understood in the sense of probabilistic meaning.