Abstract
Similarity among worlds plays a pivotal role in providing the semantics for different kinds of belief change. Although similarity is, intuitively, a context-sensitive concept, the accounts of similarity presently proposed are, by and large, context blind. We propose an account of similarity that is context sensitive, and when belief change is concerned, we take it that the epistemic input provides the required context. We accordingly develop and examine two accounts of probabilistic belief change that are based on such evidence-sensitive similarity. The first switches between two extreme behaviors depending on whether or not the evidence in question is consistent with the current knowledge. The second gracefully changes its behavior depending on the degree to which the evidence is consistent with current knowledge. Finally, we analyze these two belief change operators with respect to a select set of plausible postulates.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In general, we write \(B^*_\alpha \) to mean the (the result of) revision of B with \(\alpha \) by application of operator \(*\).
- 2.
Similar axioms of distance have been adopted in mathematics and psychology for a long time.
- 3.
The term separability has been defined differently by different authors.
- 4.
Other interpretations of expansion in the probabilistic setting may be considered in the future.
References
Lewis, D.: Probabilities of conditionals and conditional probabilities. Philos. Rev. 85(3), 297–315 (1976)
Ramachandran, R., Nayak, A.C., Orgun, M.A.: Belief erasure using partial imaging. In: Li, J. (ed.) AI 2010. LNCS (LNAI), vol. 6464, pp. 52–61. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17432-2_6
Chhogyal, K., Nayak, A., Schwitter, R., Sattar, A.: Probabilistic belief revision via imaging. In: Pham, D.-N., Park, S.-B. (eds.) PRICAI 2014. LNCS (LNAI), vol. 8862, pp. 694–707. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13560-1_55
Mishra, S., Nayak, A.: Causal basis for probabilistic belief change: distance vs. closeness. In: Sombattheera, C., Stolzenburg, F., Lin, F., Nayak, A. (eds.) MIWAI 2016. LNCS (LNAI), vol. 10053, pp. 112–125. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49397-8_10
Rens, G., Meyer, T., Casini, G.: On revision of partially specified convex probabilistic belief bases. In: Kaminka, G., Fox, M., Bouquet, P., Dignum, V., Dignum, F., Van Harmelen, F. (eds.) Proceedings of the Twenty-Second European Conference on Artificial Intelligence (ECAI-2016), The Hague, The Netherlands, pp. 921–929. IOS Press, September 2016
Gärdenfors, P.: Knowledge in Flux: Modeling the Dynamics of Epistemic States. MIT Press, Cambridge (1988)
Ashby, F.G., Ennis, D.M.: Similarity measures. Scholarpedia 2(12), 4116 (2007)
Choi, S.S., Cha, S.H., Tappert, C.: A survey of binary similarity and distance measures. Syst. Cybern. Inform. 8(1), 43–48 (2010)
Shepard, R.: Toward a universal law of generalization for psychological science. Science 237(4820), 1317–1323 (1987)
Jäkel, F., Schölkopf, B., Wichmann, F.: Similarity, kernels, and the triangle inequality. J. Math. Psychol. 52(5), 297–303 (2008). http://www.sciencedirect.com/science/article/pii/S0022249608000278
Yearsley, J.M., Barque-Duran, A., Scerrati, E., Hampton, J.A., Pothos, E.M.: The triangle inequality constraint in similarity judgments. Prog. Biophys. Mol. Biol. 130(Part A), 26–32 (2017). http://www.sciencedirect.com/science/article/pii/S0079610716301341. Quantum information models in biology: from molecular biology to cognition
Dubois, D., Moral, S., Prade, H.: Belief change rules in ordinal and numerical uncertainty theories. In: Dubois, D., Prade, H. (eds.) Belief Change, vol. 3, pp. 311–392. Springer, Dordrecht (1998). https://doi.org/10.1007/978-94-011-5054-5_8
Voorbraak, F.: Probabilistic belief change: expansion, conditioning and constraining. In: Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence, UAI 1999, San Francisco, CA, USA, pp. 655–662. Morgan Kaufmann Publishers Inc. (1999). http://dl.acm.org/citation.cfm?id=2073796.2073870
Lehmann, D., Magidor, M., Schlechta, K.: Distance semantics for belief revision. J. Symb. Log. 66(1), 295–317 (2001)
Jaynes, E.: Where do we stand on maximum entropy? In: The Maximum Entropy Formalism, pp. 15–118. MIT Press (1978)
Paris, J., Vencovská, A.: In defense of the maximum entropy inference process. Int. J. Approx. Reason. 17(1), 77–103 (1997). http://www.sciencedirect.com/science/article/pii/S0888613X97000145
Kern-Isberner, G.: Revising and updating probabilistic beliefs. In: Williams, M.A., Rott, H. (eds.) Frontiers in Belief Revision, Applied Logic Series, vol. 22, pp. 393–408. Kluwer Academic Publishers/Springer, Dordrecht (2001). https://doi.org/10.1007/978-94-015-9817-0_20
Beierle, C., Finthammer, M., Potyka, N., Varghese, J., Kern-Isberner, G.: A framework for versatile knowledge and belief management. IFCoLog J. Log. Appl. 4(7), 2063–2095 (2017)
Chan, H., Darwiche, A.: On the revision of probabilistic beliefs using uncertain evidence. Artif. Intell. 163, 67–90 (2005)
Grove, A., Halpern, J.: Updating sets of probabilities. In: Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence, UAI 1998, San Francisco, CA, USA, pp. 173–182. Morgan Kaufmann (1998). http://dl.acm.org/citation.cfm?id=2074094.2074115
Mork, J.C.: Uncertainty, credal sets and second order probability. Synthese 190(3), 353–378 (2013). https://doi.org/10.1007/s11229-011-0042-2
Boutilier, C.: On the revision of probabilistic belief states. Notre Dame J. Form. Log. 36(1), 158–183 (1995)
Makinson, D.: Conditional probability in the light of qualitative belief change. Philos. Log. 40(2), 121–153 (2011)
Chhogyal, K., Nayak, A., Sattar, A.: Probabilistic belief contraction: considerations on epistemic entrenchment, probability mixtures and KL divergence. In: Pfahringer, B., Renz, J. (eds.) AI 2015. LNCS (LNAI), vol. 9457, pp. 109–122. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26350-2_10
Zhuang, Z., Delgrande, J., Nayak, A., Sattar, A.: A unifying framework for probabilistic belief revision. In: Bacchus, F. (ed.) Proceedings of the Twenty-fifth International Joint Conference on Artificial Intelligence (IJCAI 2017), pp. 1370–1376. AAAI Press, Menlo Park (2017). https://doi.org/10.24963/ijcai.2017/190
Alchourrón, C., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Log. 50(2), 510–530 (1985)
Acknowledgements
Gavin Rens was supported by a Clause Leon Foundation postdoctoral fellowship while conducting this research. This research has been partially supported by the Australian Research Council (ARC), Discovery Project: DP150104133. This work is based on research supported in part by the National Research Foundation of South Africa (Grant number UID 98019). Thomas Meyer has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agr. No. 690974.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Rens, G., Meyer, T., Kern-Isberner, G., Nayak, A. (2018). Probabilistic Belief Revision via Similarity of Worlds Modulo Evidence. In: Trollmann, F., Turhan, AY. (eds) KI 2018: Advances in Artificial Intelligence. KI 2018. Lecture Notes in Computer Science(), vol 11117. Springer, Cham. https://doi.org/10.1007/978-3-030-00111-7_29
Download citation
DOI: https://doi.org/10.1007/978-3-030-00111-7_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-00110-0
Online ISBN: 978-3-030-00111-7
eBook Packages: Computer ScienceComputer Science (R0)