A Simple Proportional Conflict Redistribution Rule

TL;DR

This paper introduces the Proportional Conflict Redistribution rule (PCR1), a new method for combining evidence that generalizes and extends existing rules like WAO, handling degenerate cases and open-world scenarios.

Contribution

The paper proposes PCR1, a novel evidence combination rule that extends WAO to degenerate and open-world cases, providing a more comprehensive conflict redistribution method.

Findings

PCR1 generalizes WAO for all cases.

PCR1 handles degenerate cases with zero column sums.

PCR1 manages open-world scenarios with conflicting mass.

Abstract

One proposes a first alternative rule of combination to WAO (Weighted Average Operator) proposed recently by Josang, Daniel and Vannoorenberghe, called Proportional Conflict Redistribution rule (denoted PCR1). PCR1 and WAO are particular cases of WO (the Weighted Operator) because the conflicting mass is redistributed with respect to some weighting factors. In this first PCR rule, the proportionalization is done for each non-empty set with respect to the non-zero sum of its corresponding mass matrix - instead of its mass column average as in WAO, but the results are the same as Ph. Smets has pointed out. Also, we extend WAO (which herein gives no solution) for the degenerate case when all column sums of all non-empty sets are zero, and then the conflicting mass is transferred to the non-empty disjunctive form of all non-empty sets together; but if this disjunctive form happens to be empty, then one considers an open world (i.e. the frame of discernment might contain new hypotheses) and thus all conflicting mass is transferred to the empty set. In addition to WAO, we propose a general formula for PCR1 (WAO for non-degenerate cases).