A new distance between BPAs based on the power-set-distribution pignistic probability function

In Dempster-Shafer evidence theory, the pignistic probability function is used to transform the basic probability assignment (BPA) into pignistic probabilities. Since the transformation is from the power set of the frame of discernment to the set itself, it may cause some information loss. The distance between betting commitments is constructed on the basis of the pignistic probability function and is used to measure the dissimilarity between two BPAs. However, it is a pseudo-metric and it may bring unreasonable results in some cases. To solve such problem, we propose a power-set-distribution (PSD) pignistic probability function based on the new explanation of the non-singleton focal elements in the BPA. The new function is directly operated on the power set, so it takes more information contained in the BPA than the pignistic probability function does. Based on the new function, the distance between PSD betting commitments which can better measure the dissimilarity between two BPAs is also proposed, and the proof that it is a metric is provided. In order to demonstrate the performance of the new distance, numerical examples are given to compare it with three existing dissimilarity measures. Moreover, its applications in combining the conflicting BPAs are also presented through two examples.