Compatibility of Quantitative and Qualitative Representations of Belief

Abstract

The compatibility of quantitative and qualitative representations of beliefs was studied extensively in probability theory. It is only recently that this important topic is considered in the context of belief functions. In this paper, the compatibility of various quantitative belief measures and qualitative belief structures is investigated. Four classes of belief measures considered are: the probability function, the monotonic belief function, Shafer's belief function, and Smets' generalized belief function. The analysis of their individual compatibility with different belief structures not only provides a sound basis for these quantitative measures, but also alleviates some of the difficulties in the acquisition and interpretation of numeric belief numbers. It is shown that the structure of qualitative probability is compatible with monotonic belief functions. Moreover, a belief structure slightly weaker than that of qualitative belief is compatible with Smets' generalized belief functions.