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Representations of Uncertainty in Artificial Intelligence: Probability and Possibility

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Abstract

Due to its major focus on knowledge representation and reasoning, artificial intelligence was bound to deal with various frameworks for the handling of uncertainty: probability theory, but more recent approaches as well: possibility theory, evidence theory, and imprecise probabilities. The aim of this chapter is to provide an introductive survey that lays bare specific features of two basic frameworks for representing uncertainty: probability theory and possibility theory, while highlighting the main issues that the task of representing uncertainty is faced with. This purpose also provides the opportunity to position related topics, such as rough sets and fuzzy sets, respectively motivated by the need to account for the granularity of representations as induced by the choice of a language, and the gradual nature of natural language predicates. Moreover, this overview includes concise presentations of yet other theoretical representation frameworks such as formal concept analysis, conditional events and ranking functions, and also possibilistic logic, in connection with the uncertainty frameworks addressed here. The next chapter in this volume is devoted to more complex frameworks: belief functions and imprecise probabilities.

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Notes

  1. 1.

    In fact, in this chapter, v denotes an ill-known entity that may be for instance a random variable in a probabilistic setting, or rather an imprecisely known entity but which does not vary strictly speaking.

  2. 2.

    Apparent, because the mathematical settings proposed by Kolmogorov and De Finetti (1974) are different, especially for the notion of conditioning, even if the Kolmogorov setting seems to be overwhelmingly adopted by mathematicians.

  3. 3.

    We come back to the logic of conditional events at the end of this section.

  4. 4.

    If \(L = \mathbb {N}\), the conventions are opposite: 0 means possible and \(\infty \) means impossible.

  5. 5.

    The Bayesian-like rule in terms of necessity measures, \(N(A \cap B) = \min (N(A\mid B), N(B))\), is trivial. Its least specific solution, minimizing necessity degrees, is \(N(A\mid B) = N(A \cap B) = \min (N(A), N(B))\), which defines in turn \(\varPi (A\mid B) = \varPi (\overline{B} \cup A)\). It comes down to interpreting a conditional event as a material implication.

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Authors and Affiliations

  1. Université de Technologie de Compiègne, CNRS, Heudiasyc (UMR 7253), Paris, France

    Thierry Denœux

  2. IRIT, CNRS and Université Paul Sabatier, Toulouse, France

    Didier Dubois & Henri Prade

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  1. Thierry Denœux
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  2. Didier Dubois
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  3. Henri Prade
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Correspondence to Thierry Denœux .

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  1. CRIL-CNRS, Université d'Artois and Institut Universitaire de France, Lens, France

    Pierre Marquis

  2. Aix Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France

    Odile Papini

  3. IRIT, CNRS and Université Paul Sabatier, Toulouse, France

    Henri Prade

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Denœux, T., Dubois, D., Prade, H. (2020). Representations of Uncertainty in Artificial Intelligence: Probability and Possibility. In: Marquis, P., Papini, O., Prade, H. (eds) A Guided Tour of Artificial Intelligence Research. Springer, Cham. https://doi.org/10.1007/978-3-030-06164-7_3

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