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Comparison of Shades and Hiddenness of Conflict

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12897))

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Abstract

Conflict, dissonance, inconsistency, entropy. There are many notions related to one phenomenon. When working with uncertainty, there can be different sources of information, and often they are in some level of mutual disagreement. When working with belief functions, one of the approaches how to measure conflict is closely connected with a belief mass assigned by the non-normalized conjunctive rule to the empty set. Recently, we have observed and presented cases where a conflict of belief functions is hidden (there is a zero mass assigned to the empty set by the non-normalized conjunctive rule). Above that, we distinguish several degrees of such a hiddeness. In parallel, Pichon et al. introduced a new family of conflict measures of different strengths, the so-called shades of conflict. In this paper, we compare both approaches not only from the theoretical point of view but also by examples.

This work was supported by the institutional support RVO: 67985807 (first author) and grant GA ČR 19-04579S (second author).

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Notes

  1. 1.

    In this comparison, we present just the notions necessary for understanding of both the compared approaches, for more detail including motivations see [9] and [13]. For irrelevance of dependence/independence of belief sources for conflicts see also [9].

  2. 2.

    Note, that this union is called a core of a belief function and denoted by \(\mathcal {C} \) by Shafer in [14] ; on the other hand Cuzzolin uses a completely different (conjunctive) core defined as \(\mathcal {C} = \bigcap _{A \in \mathcal {F} } A\) in [2].

  3. 3.

    More precisely, Martin et al.do not use , but \(\oplus \) sign, which we use as Shafer in his normalized approach [14] for Demspter’s rule, where always \(m(\emptyset )=0\), \((m_i \oplus m_j)(\emptyset )=0\) and \((\bigoplus _1^k m)(\emptyset ) = 0\).

  4. 4.

    There are focal elements \(m',m''\) substituted by some of \(m^{+}, m^{++}, m^{*}, m^{**} \) in the following extensions; analogously focal elements \(m^{i}, m^{ii} \) are substituted by \(m^{xi} \) and/or \(m^{xii} \) later in extensions of the Little Angel example.

  5. 5.

    Note, that denotes here always the result of non-normalized conjunctive combination of a couple of bbas corresponding to the example extension in question, thus it varies and we can see its precise definition from the context.

  6. 6.

    Let us notice, that the example from Fig. 4 (iii) is, in fact, discounting of that from 4 (i), hence conflict decreasing while the number of defined shades of conflict is increased from 3 to 8; and analogously 3 (iii) is discounting of 3 (i) with four more shades.

  7. 7.

    There are interesting open issues: what is max number of shades of conflict related to a hidden conflict for this example; in general on \(\varOmega _5\), and in full generality on \(\varOmega _n\)? And analogously, a number of shades related to a hidden conflict of the 2-nd degree and k-th degree?.

  8. 8.

    More correctly, there are \(2^n-1\) measures \(\kappa _N\) for BFs on \(\varOmega _n\), where only \(|{\mathcal F_{12} } |\) values are defined as N-conflict of corresponding BFs \(m_1\) and \(m_2\) by Pichon et al. [13]. Nevertheless, on the other hand, a degree of auto-conflict is not limited, thus there exist values of \(\kappa _N(m_1,m_2)\) for all \(N \le 2^n - 1\) regardless the definitions from Sect. 4, originally from [13] : ; in fact there exist values for any finite N and limit is either 0 for totally non-conflicting BFs or 1 if there is any kind of conflict.

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Authors and Affiliations

  1. Institute of Computer Sciences, Czech Academy of Sciences, Prague, Czechia

    Milan Daniel

  2. Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czechia

    Václav Kratochvíl

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  1. Milan Daniel
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  2. Václav Kratochvíl
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Correspondence to Václav Kratochvíl .

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  1. Institute of Information Theory and Automation, Prague, Czech Republic

    Jiřina Vejnarová

  2. Insight Centre for Data Analytics, School of Computer Science and IT University College Cork, Cork, Ireland

    Nic Wilson

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Daniel, M., Kratochvíl, V. (2021). Comparison of Shades and Hiddenness of Conflict. In: Vejnarová, J., Wilson, N. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2021. Lecture Notes in Computer Science(), vol 12897. Springer, Cham. https://doi.org/10.1007/978-3-030-86772-0_23

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