Uncertain Evidence: Extending Belief Function Theory... Within the Scope of Belief Function Theory!

Abstract

We propose a framework enhancing the expressiveness of the evidential and credal interpretations of Belief Function Theory while remaining within its scope. It allows for expressing uncertain evidence "as is" by associating intervals to focal elements. It facilitates the modeling and manipulation of knowledge by allowing for questionable evidence or uncertain random sets.Credally, belief functions delimit maximal ("convex") and homogeneous credal sets.The proposed framework enables the representation of non-maximal, non-homogeneous credal sets, admitting belief and plausibility functions as extrema.These refined credal sets reflect the uncertain reliability of evidence---maximal homogeneous credal sets are recovered when evidence is known to be reliable.We propose three update operations extending Dempster’s, geometric, and Bayesian conditioning, expressed in terms of transfer hence with linear complexity relative to the size of the evidence.This resolves several anomalies of traditional Bayesian conditioning, making it tractable and commutative, and explains its apparent dilation effect. Most importantly, it accurately yields the updated credal set rather than providing its bounds only.

Keywords: Belief functions, Interpretation, Evidence, Credal set, Conditioning, Dilation