The Generalization of the Conjunctive Rule for Aggregating Contradictory Sources of Information Based on Generalized Credal Sets

In the paper we consider the generalization of the conjunctive rule in the theory of imprecise probabilities. Let us remind that the conjunction rule, produced on credal sets, gives their intersection and it is not defined if this intersection is empty. In the last case the sources of information are called contradictory1. Meanwhile, in the Dempster-Shafer theory it is possible to use the conjunctive rule for contradictory sources of information having as a result a nonnormalized belief function that can be greater than zero at empty set. In the paper we try to exploit this idea and introduce into consideration so called generalized credal sets allowing to model imprecision (non-specificity), conflict, and contradiction in information. Based on generalized credal sets the conjunctive rule is well defined for contradictory sources of information and it can be conceived as the generalization of the conjunctive rule for belief functions. We also show how generalized credal sets can be used for modeling information when the avoiding sure loss condition is not satisfied, and consider coherence conditions and natural extension based on generalized credal sets.