Abstract
Probability, evidence, and possibility theories share the same idea of normalizing the total certainty, or belief, and their fundamental difference should be positioned on the way this total certainty is distributed. In probability theory, this total certainty is fractioned between different alternatives, whereas in evidence theory this total certainty is fractioned between all subsets of alternatives. Finally, in possibility theory, the total certainty is not fractioned but is positioned at the level of each alternative. This chapter presents a brief review of the fundamental definitions and concepts used in monotone measures theory. A brief recall of probability, evidence, and possibility theories is then given. Finally, major bridges allowing to transform one uncertainty model into other models are detailed.
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Notes
- 1.
From the preface of the (2009) Wang and Klir’s book entitled Generalized Measure Theory, here is why we will be using monotone measures rather than fuzzy measures in the rest of this chapter. “…However, it should be emphasized that we made a deliberate decision to abandon the central term of our previous book (fuzzy measure theory), the term ‘fuzzy measure.’ We judge this term to be highly misleading. Indeed, the so-called fuzzy measures do not involve any fuzziness. They are just special set functions that are defined on specified classes of classical sets, not on classes of fuzzy sets. Since the primary characteristic of such functions is monotonicity, we deemed it reasonable to call these set functions monotone measures rather than fuzzy measures.”
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Solaiman, B., Bossé, É. (2019). The Interrelated Uncertainty Modeling Theories. In: Possibility Theory for the Design of Information Fusion Systems. Information Fusion and Data Science. Springer, Cham. https://doi.org/10.1007/978-3-030-32853-5_5
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DOI: https://doi.org/10.1007/978-3-030-32853-5_5
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