Compatibility of Quantitative and Qualitative Representations of Belief

The compatibility of quantitative and quali­ tative representations of beliefs was studied extensively in probability theory. It is only recently that this important topic is consiJ­ ered in the context of belief functions. In this paper, the compatibility of various quan­ titative belief measures and qualitative belief structures is investigated. Four classes of be­ lief measures considered are: the probabil­ ity function, the monotonic belief function, Shafer's belief function, and Smets' general­ ized belief function. The analysis of their individual compatibility with Jiff ercut belief structures not only provides a sound basis for these quantitative measures, but also allevi­ ates some of the difficulties in the acquisition and interpretation of numeric belief numbers. It is shown that the structure of qualilati·ve probability is compatible with monotonic be­ lief functions. Moreover, a belief structure slightly weaker than that of q1lalitative belief is compatible with Smets' generalized belief functions.