Modeling and updating uncertain evidence within belief function theory

We propose a framework that enhances the expressiveness of the evidential and credal interpretations of Belief Function Theory while remaining within its scope. It allows uncertain evidence to be represented “as is” by associating meaningful intervals of  or  to focal elements, providing an intrinsic justification for belief values. This improves the modeling and manipulation of knowledge. From a credal perspective, the framework enables the accurate representation of non-maximal credal sets, when their extrema are belief and plausibility functions.
We introduce three update operations that extend Dempster's, geometric, and Bayesian conditioning to uncertain evidence. These updates are expressed in terms of transfer of evidence, ensuring linear complexity relative to the number of focal elements. This approach provides clear evidential semantics to Bayesian conditioning, resolves several of its anomalies by making it tractable and commutative, and explains its apparent dilation effect. Most importantly, it accurately yields the updated credal set, rather than merely providing its bounds.