Evidential reasoning rule for evidence combination

This paper aims to establish a unique Evidential Reasoning (ER) rule to combine multiple pieces of independent evidence conjunctively with weights and reliabilities. The novel concept of Weighted Belief Distribution (WBD) is proposed and extended to WBD with Reliability (WBDR) to characterise evidence in complement of Belief Distribution (BD) introduced in Dempster–Shafer (D–S) theory of evidence. The implementation of the orthogonal sum operation on WBDs and WBDRs leads to the establishment of the new ER rule. The most important property of the new ER rule is that it constitutes a generic conjunctive probabilistic reasoning process, or a generalised Bayesian inference process. It is shown that the original ER algorithm is a special case of the ER rule when the reliability of evidence is equal to its weight and the weights of all pieces of evidence are normalised. It is proven that Dempsterʼs rule is also a special case of the ER rule when each piece of evidence is fully reliable. The ER rule completes and enhances Dempsterʼs rule by identifying how to combine pieces of fully reliable evidence that are highly or completely conflicting through a new reliability perturbation analysis. The main properties of the ER rule are explored to facilitate its applications. Several existing rules are discussed and compared with the ER rule. Numerical and simulation studies are conducted to show the features of the ER rule.