TECHNICAL NOTE - TN-2023-02-28, FEBRUARY 2023. 1 Abstract This short technical note points out an erroneous claim about a new rule of combination of basic belief assignments presented recently by Kenn et al. in [1], referred as Kenn's rule of combination (or just as KRC for short). We prove thanks a very simple counter-example that Kenn's rule is not associative. Consequently, the results of the method proposed by Kenn et al. highly depends on the ad-hoc sequential order chosen for the fusion process as proposed by the authors. This serious problem casts in doubt the interest of this method and its real ability to provide trustful results and to make good decisions to help for precise breast cancer therapy. Recently a paper devoted to the Breast Cancer Precision Therapy by Kenn et al. [1] attracted our attention for two main reasons: 1) this application of information fusion is very interesting and important; 2) Kenn's et al. method is based on a new rule of combination of basic belief assignments (BBAs).

Mountain river torrents and snow avalanches generate human and material damages with dramatic consequences. Knowledge about natural phenomenona is often lacking and expertise is required for decision and risk management purposes using multi-disciplinary quantitative or qualitative approaches. Expertise is considered as a decision process based on imperfect information coming from more or less reliable and conflicting sources. A methodology mixing the Analytic Hierarchy Process (AHP), a multi-criteria aid-decision method, and information fusion using Belief Function Theory is described. Fuzzy Sets and Possibilities theories allow to transform quantitative and qualitative criteria into a common frame of discernment for decision in Dempster-Shafer Theory (DST ) and Dezert-Smarandache Theory (DSmT) contexts. Main issues consist in basic belief assignments elicitation, conflict identification and management, fusion rule choices, results validation but also in specific needs to make a difference between importance and reliability and uncertainty in the fusion process.

We present in this paper some examples of how to compute by hand the PCR5 fusion rule for three sources, so the reader will better understand its mechanism. We also take into consideration the importance of sources, which is different from the classical discounting of sources.

In this paper the behavior of three combinational rules for temporal/sequential attribute data fusion for target type estimation are analyzed. The comparative analysis is based on: Dempster's fusion rule proposed in Dempster-Shafer Theory; Proportional Conflict Redistribution rule no. 5 (PCR5), proposed in Dezert-Smarandache Theory and one alternative class fusion rule, connecting the combination rules for information fusion with particular fuzzy operators, focusing on the t-norm based Conjunctive rule as an analog of the ordinary conjunctive rule and t-conorm based Disjunctive rule as an analog of the ordinary disjunctive rule. The way how different t-conorms and t-norms functions within TCN fusion rule influence over target type estimation performance is studied and estimated.

Comments on ``A new combination of evidence based on compromise'' by K. Yamada

Martin and Osswald \cite{Martin07} have recently proposed many generalizations of combination rules on quantitative beliefs in order to manage the conflict and to consider the specificity of the responses of the experts. Since the experts express themselves usually in natural language with linguistic labels, Smarandache and Dezert \cite{Li07} have introduced a mathematical framework for dealing directly also with qualitative beliefs. In this paper we recall some element of our previous works and propose the new combination rules, developed for the fusion of both qualitative or quantitative beliefs.

The management and combination of uncertain, imprecise, fuzzy and even paradoxical or high conflicting sources of information has always been, and still remains today, of primal importance for the development of reliable modern information systems involving artificial reasoning. In this introduction, we present a survey of our recent theory of plausible and paradoxical reasoning, known as Dezert-Smarandache Theory (DSmT), developed for dealing with imprecise, uncertain and conflicting sources of information. We focus our presentation on the foundations of DSmT and on its most important rules of combination, rather than on browsing specific applications of DSmT available in literature. Several simple examples are given throughout this presentation to show the efficiency and the generality of this new approach.

In this paper, we propose in Dezert-Smarandache Theory (DSmT) framework, a new probabilistic transformation, called DSmP, in order to build a subjective probability measure from any basic belief assignment defined on any model of the frame of discernment. Several examples are given to show how the DSmP transformation works and we compare it to main existing transformations proposed in the literature so far. We show the advantages of DSmP over classical transformations in term of Probabilistic Information Content (PIC). The direct extension of this transformation for dealing with qualitative belief assignments is also presented.

This paper deals with enriched qualitative belief functions for reasoning under uncertainty and for combining information expressed in natural language through linguistic labels. In this work, two possible enrichments (quantitative and/or qualitative) of linguistic labels are considered and operators (addition, multiplication, division, etc) for dealing with them are proposed and explained. We denote them $qe$-operators, $qe$ standing for "qualitative-enriched" operators. These operators can be seen as a direct extension of the classical qualitative operators ($q$-operators) proposed recently in the Dezert-Smarandache Theory of plausible and paradoxist reasoning (DSmT). $q$-operators are also justified in details in this paper. The quantitative enrichment of linguistic label is a numerical supporting degree in $[0,\infty)$, while the qualitative enrichment takes its values in a finite ordered set of linguistic values. Quantitative enrichment is less precise than qualitative enrichment, but it is expected more close with what human experts can easily provide when expressing linguistic labels with supporting degrees. Two simple examples are given to show how the fusion of qualitative-enriched belief assignments can be done.

In this paper we extend the new family of (quantitative) Belief Conditioning Rules (BCR) recently developed in the Dezert-Smarandache Theory (DSmT) to their qualitative counterpart for belief revision. Since the revision of quantitative as well as qualitative belief assignment given the occurrence of a new event (the conditioning constraint) can be done in many possible ways, we present here only what we consider as the most appealing Qualitative Belief Conditioning Rules (QBCR) which allow to revise the belief directly with words and linguistic labels and thus avoids the introduction of ad-hoc translations of quantitative beliefs into quantitative ones for solving the problem.