Abstract
This paper deals with belief graphical models and probability-possibility transformations. It first analyzes some properties of transforming a credal network into a possibilistic one. In particular, we are interested in satisfying some properties of probability-possibility transformations like dominance and order preservation. The second part of the paper deals with using probability-possibility transformations in order to perform MAP inference in credal networks. This problem is known for its high computational complexity in comparison with MAP inference in Bayesian and possibilistic networks. The paper provides preliminary experimental results comparing our approach with both exact and approximate inference in credal networks.
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Notes
- 1.
It is important to note that the number of extreme points can reach N! where N is the number of interpretations [16].
- 2.
- 3.
Let \(\pi '\) and \(\pi ''\) be two possibility distributions, \(\pi '\) is more specific than \(\pi ''\) iff \(\forall {{\omega }_{i}}{\in }{\varOmega }\), \(\pi '(\omega _{i})\le \pi ''(\omega _{i})\).
- 4.
In the rest of this paper, TR denotes an interval-based probability-possibility transformation satisfying the following principles:
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Dominance: The possibility distribution \(\pi \) obtained from the IPD IP by TR dominates every probability distribution p compatible with IP, namely \(\forall {\phi } \subseteq \varOmega \), \(\pi (\phi ) \ge p(\phi )\).
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Order preservation: Given two interpretations \(\omega _i \in {\varOmega }\) and \(\omega _j \in {\varOmega }\), \(\pi (\omega _i) < \pi (\omega _j)\) iff \(\overline{p}(\omega _i) < \underline{p}(\omega _j)\).
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- 5.
The permutation property of probability-possibility transformations is discussed in [14].
- 6.
- 7.
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Acknowledgements
This work is done with the support of a CNRS funded project PEPS FaSciDo 2015 - MAPPOS.
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Benferhat, S., Levray, A., Tabia, K. (2015). Probability-Possibility Transformations: Application to Credal Networks. In: Beierle, C., Dekhtyar, A. (eds) Scalable Uncertainty Management. SUM 2015. Lecture Notes in Computer Science(), vol 9310. Springer, Cham. https://doi.org/10.1007/978-3-319-23540-0_14
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