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On the consistency of PCR6 with the averaging rule and its application to probability estimation

Since the development of belief function theory introduced by Shafer in seventies, many combination rules have been proposed in the literature to combine belief functions specially (but not only) in high conflicting situations because the emblematic Dempster's rule generates counter-intuitive and unacceptable results in practical applications. Many attempts have been done during last thirty years to propose better rules of combination based on different frameworks and justifications. Recently in the DSmT (Dezert-Smarandache Theory) framework, two interesting and sophisticate rules (PCR5 and PCR6 rules) have been proposed based on the Proportional Conflict Redistribution (PCR) principle. These two rules coincide for the combination of two basic belief assignments, but they differ in general as soon as three or more sources have to be combined altogether because the PCR used in PCR5 and in PCR6 are different. In this paper we show why PCR6 is better than PCR5 to combine three or more sources of evidence and we prove the coherence of PCR6 with the simple Averaging Rule used classically to estimate the probability based on the frequentist interpretation of the probability measure. We show that such probability estimate cannot be obtained using Dempster-Shafer (DS) rule, nor PCR5 rule.