# General Combination Rules For Qualitative And Quantitative Beliefs1

## 1 I. Introduction

Many fusion theories have been studied for the combination of the experts opinions expressed either quantitatively or qualitatively such as voting rules [11], [31], possibility theory [6], [35], and belief functions theory [2], [17]. All these fusion approaches can be divided basically into four steps: modelization, *parameters estimation* (depending on the model, not always necessary)), *combination* and *decision*. The most difficult step is presumably the first one which depends highly on the problem and application we have to cope with. However, it is only at the combination step that we can take into account useful information such as the conflict (partial or total) between the experts and/or the specificity of the expert's response.

The voting rules are not adapted to the modelization of conflict between experts [31]. Although both possibility and probability-based theories can model imprecise and uncertain data at the same time, in many applications, the experts are only able to express their "certainty" (or belief) only from their partial knowledge, experience and from their own perception of the reality. In such context, the belief function-based theories provide an appealing general mathematical framework for dealing with quantitative and qualitative beliefs.

In this paper we present the most recent advances in belief functions theory for managing the conflict between the sources of evidence/experts and their specificity. For the first time in the literature both the quantitative and qualitative aspects of beliefs are presented in a unified mathematical framework. This paper actually extends two papers [13], [15] presented during the 10th International Conference on Information Fusion (Fusion 2007) in Qu´ebec City, Canada on July 9-12, 2007 in the session "Combination in Evidence Theory".

Section II briefly recalls the basis of belief functions theories, *i.e.* the Mathematical Theory of Evidence or Dempster-Shafer theory (DST) developed by Shafer in 1976 [2], [17], and its natural extension called Dezert-Smarandache Theory (DSmT) [18],
[19]. We introduce in this section the notion of quantitative and qualitative beliefs and the operators on linguistic labels for dealing directly with qualitative beliefs. Section III presents the main classical quantitative combination rules used so far, *i.e.*
Dempster's rule, Yager's rule, Dubois-Prade's rule and the recent Proportional Conflict Redistribution rules (PCR) proposed by Smarandache and Dezert [22] and extended by Martin and Osswald in [19]. Some examples are given to illustrate how these rules work. Section IV explains through different examples how all the classical quantitative combination rules can be directly and simply translated/extended into the qualitative domain in order to combine easily any qualitative beliefs expressed in natural language by linguistic labels. Section V proposes new general quantitative rules of combination which allow to take into account both the discounting of the sources (if any) and the proportional conflict redistribution. The direct extension of these general rules into the qualitative domain is then presented in details on several examples in section VI.

1Manuscript received September 24, 2007; released for publication August 5, 2008.

Refereeing of this contribution was handled by Fabio Roli.