Conditions for the Existence of Belief Functions Corresponding to Intervals of Belief

John F. Lemmer, Henry E. Kyburg, Jr.

While every Shafer belief function corresponds to a set of interval beliefs on the atoms of the frame of discernment, an arbitrarily specified set of intervals of belief may not correspond to any belief function, even when it does correspond to bounds imposed by sets of probability functions. This paper proves necessary and sufficient conditions which must be met by a set of belief intervals over atoms if a corresponding belief function exists. The sufficiency is proved via an an O(n) algorithm which will always construct an corresponding belief function, if one exits, for a specific set of intervals.

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