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International Conference of the Italian Association for Artificial Intelligence

AIxIA 2020: AIxIA 2020 – Advances in Artificial Intelligence pp 335–355Cite as

From Contrastive to Abductive Explanations and Back Again

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12414))

Abstract

Explanations of Machine Learning (ML) models often address a question. Such explanations can be related with selecting feature-value pairs which are sufficient for the prediction. Recent work has investigated explanations that address a question, i.e. finding a change of feature values that guarantee a change of prediction. Given their goals, these two forms of explaining predictions of ML models appear to be mostly unrelated. However, this paper demonstrates otherwise, and establishes a rigorous formal relationship between and explanations. Concretely, the paper proves that, for any given instance, explanations are minimal hitting sets of explanations and vice-versa. Furthermore, the paper devises novel algorithms for extracting and enumerating both forms of explanations.

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Notes

  1. 1.

    There is also a recent XAI service offered by Google: https://cloud.google.com/explainable-ai/, inspired on similar ideas [28].

  2. 2.

    In contrast with recent work [39], which studies the relationship between global model-based (abductive) explanations and adversarial examples.

  3. 3.

    A local abductive (resp. contrastive) explanation is a minimal hitting set of the set of all local contrastive (resp. abductive) explanations.

  4. 4.

    \({\mathcal {M}}\) is referred to as the (formal) model of the ML model \(\mathbb {M}\). The use of FOL is not restrictive, with fragments of FOL being used in recent years for modeling ML models in different settings. These include NNs [38] and Bayesian Network Classifiers [76], among others.

  5. 5.

    This alternative notation is used for simplicity and clarity with respect to earlier work [38, 39, 75]. Furthermore, defining \({\mathcal {M}}\) as a predicate allows for multiple predictions for the same point in feature space. Nevertheless, such cases are not considered in this paper.

  6. 6.

    By definition of prime implicant, abductive explanations are sufficient reasons for the prediction. Hence the names used in recent work: abductive explanations [38], PI-explanations [75, 76] and sufficient reasons [15, 16].

  7. 7.

    The definitions in this section are often presented for the propositional case, but the extension to the first-order case is straightforward.

  8. 8.

    The choice of a decision tree aims only at keeping the example(s) presented in the paper as simple as possible. The ideas proposed in the paper apply to any ML model that can be represented with FOL. This encompasses any existing ML model, with minor adaptations in case the ML model keeps state.

  9. 9.

    The abbreviations used relate with the names in the decision tree, and serve for saving space.

  10. 10.

    Depending on the ML problem, more expressive fragments of FOL logic could be considered [47]. Well-known examples include real, integer and integer-real arithmetic, but also nonlinear arithmetic [47].

  11. 11.

    Which in this case are used as propositional variables.

  12. 12.

    Although in general not the case, in Example 5 and Example 6 an MUS of size 1 is also an MCS of size 1.

  13. 13.

    The prototype and the experimental setup are available at https://github.com/alexeyignatiev/xdual.

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Acknowledgments

This work is supported by the AI Interdisciplinary Institute ANITI (Artificial and Natural Intelligence Toulouse Institute), funded by the French program “Investing for the Future – PIA3” under Grant agreement no ANR-19-PI3A-0004.

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Authors and Affiliations

  1. Monash University, Melbourne, Australia

    Alexey Ignatiev

  2. VMware Research, Palo Alto, CA, USA

    Nina Narodytska

  3. IRIT, CNRS, Toulouse, France

    Nicholas Asher & Joao Marques-Silva

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  1. Alexey Ignatiev
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  2. Nina Narodytska
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Correspondence to Nina Narodytska .

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  1. Università degli Studi di Torino, Turin, Italy

    Matteo Baldoni

  2. Department of Informatics, Systems and C, University of Milano-Bicocca, Milan, Italy

    Stefania Bandini

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Ignatiev, A., Narodytska, N., Asher, N., Marques-Silva, J. (2021). From Contrastive to Abductive Explanations and Back Again. In: Baldoni, M., Bandini, S. (eds) AIxIA 2020 – Advances in Artificial Intelligence. AIxIA 2020. Lecture Notes in Computer Science(), vol 12414. Springer, Cham. https://doi.org/10.1007/978-3-030-77091-4_21

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