Interval valued entropies for Dempster–Shafer structures

摘要

We first describe the basic ideas associated with the Dempster–Shafer belief structure and note its role in modeling uncertainty that manifests both imprecision and randomness. The inherent imprecision requires the use of intervals in expressing the associated probabilities. The ideas of plausibility and belief are introduced to provide upper and lower bounds on these interval-valued probabilities. We turn to the question of formalizing the entropy for these belief structures. As was the case with the associated probabilities the imprecision mandates the use of an interval-valued formulations for entropy. We suggest both Shannon type and Gini type formulations of interval-valued entropy for D–S belief structures. Here we make use of the classic scalar formulations of the D–S entropy introduced by Hoehle and Yager to provide the bounds for this interval-valued formulation. We discuss a more general type of belief structure in which our underlying uncertainty, while still manifesting imprecision, is based on a measure instead of a probability distribution. We show how to formulate entropy for these measure-based belief structures.