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From Possibilistic Rule-Based Systems to Machine Learning - A Discussion Paper

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Scalable Uncertainty Management (SUM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12322))

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Abstract

This paper is a plea for developing possibilistic learning methods that would be consistent with if-then rule-based reasoning. The paper first recall the possibility theory-based handling of cascading sets of parallel if-then rules. This is illustrated by an example describing a classification problem. It is shown that the approach is both close to a possibilistic logic handling of the problem and can also be put under the form of a max-min-based matrix calculus describing a function underlying a structure somewhat similar to a max-min neural network. The second part of the paper discusses how possibility distributions can be obtained from precise or imprecise statistical data, and then surveys the few existing works on learning in a possibilistic setting. A final discussion emphasizes the interest of handling learning and reasoning in a consistent way.

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Notes

  1. 1.

    For the \(\max \)-\(\min \) composition, \(\begin{bmatrix} \pi (q) \\ \pi (\lnot q) \end{bmatrix} = \begin{bmatrix} \varPi (p \wedge q) &{} \ \varPi (\lnot p \wedge q) \\ \varPi (p \wedge \lnot q) &{} \ \varPi (\lnot p\wedge \lnot q) \end{bmatrix} \otimes \begin{bmatrix} \pi (p) \\ \pi (\lnot p)\end{bmatrix} \ \ \ (1')\), taking advantage of the monotonicity and of the max-decomposability of possibility measures. In contrast with (1), there are inequality constraints between matrix terms and \((\pi (p), \pi (\lnot p))\).

  2. 2.

    By duality the inequality (2) writes \(\begin{bmatrix} \pi (q) \\ \pi (\lnot q) \end{bmatrix} \le \begin{bmatrix} \varPi (p \wedge q) &{} \ \varPi (\lnot p \wedge q) \\ \varPi (p \wedge \lnot q) &{} \ \varPi (\lnot p \wedge \lnot q) \end{bmatrix} \boxdot \begin{bmatrix} \pi (\lnot p) \\ \pi (p)\end{bmatrix}\ \ \ (2')\) where \(\boxdot \) is the \(\min -\max \) product. Note that (2’) provides an upper bound for \(\begin{bmatrix} \pi (q) \\ \pi (\lnot q) \end{bmatrix}\) obtained by (1’). It can be checked that this upper bound coincides with this latter vector if \(\varPi (p \wedge q) = \varPi (\lnot p \wedge \lnot q)=1\) or if \(\varPi (\lnot p \wedge q)=\varPi (p \wedge \lnot q)=1\).

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Authors and Affiliations

  1. IRIT, CNRS & Univ. Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Didier Dubois & Henri Prade

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  1. Didier Dubois
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Correspondence to Henri Prade .

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  1. KU Leuven, Heverlee, Belgium

    Jesse Davis

  2. Artois University, Lens, France

    Karim Tabia

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Dubois, D., Prade, H. (2020). From Possibilistic Rule-Based Systems to Machine Learning - A Discussion Paper. In: Davis, J., Tabia, K. (eds) Scalable Uncertainty Management. SUM 2020. Lecture Notes in Computer Science(), vol 12322. Springer, Cham. https://doi.org/10.1007/978-3-030-58449-8_3

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