The contradiction between belief functions: Its description, measurement, and correction based on generalized credal sets

Abstract

In the paper, we formalize the notion of contradiction between belief functions: we argue that belief functions are not contradictory if they provide non-contradictory models for decision-making. To elaborate on this idea, we take the decision rule from imprecise probabilities and show that sources of information described by belief functions are not contradictory iff the intersection of corresponding credal sets is not empty. We demonstrate that evidential conjunctive and disjunctive rules fit with this idea and they are justified in a probabilistic setting. In the case of contradictory sources of information, we analyze possible conjunctions and show how the result can be described by generalized credal sets. Based on generalized credal sets, we propose a measure of contradiction between information sources and find its axiomatics. We show how the contradiction correction can be produced based on generalized credal sets and how it can be done on sets of surely desirable gambles.