On the KL Divergence of Probability Mixtures for Belief Contraction

  • Kinzang ChhogyalEmail author
  • Abhaya Nayak
  • Abdul Sattar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9324)


Probabilistic belief change is an operation that takes a probability distribution representing a belief state along with an input sentence representing some information to be accommodated or removed, and maps it to a new probability distribution. In order to choose from many such mappings possible, techniques from information theory such as the principle of minimum cross-entropy have previously been used. Central to this principle is the Kullback-Leibler (KL) divergence. In this short study, we focus on the contraction of a belief state P by a belief a, which is the process of turning the belief a into a non-belief. The contracted belief state \(P^-_a\) can be represented as a mixture of two states: the original belief state P, and the resultant state \(P^*_{\lnot a}\) of revising P by \(\lnot a\). Crucial to this mixture is the mixing factor \(\epsilon \) which determines the proportion of P and \(P^*_{\lnot a}\) that are to be used in this process. We show that once \(\epsilon \) is determined, the KL divergence of \(P^-_a\) from P is given by a function whose only argument is \(\epsilon \). We suggest that \(\epsilon \) is not only a mixing factor but also captures relevant aspects of P and \(P^*_{\lnot a}\) required for computing the KL divergence.


Belief Revision Belief State Belief Change Argumentation Framework Probability Mixture 
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  1. Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: Partial meet contraction and revision functions. J. Symb. Log. 50(2), 510–530 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chhogyal, K., Nayak, A.C., Zhuang, Z., Sattar, A.: Probabilistic belief contraction using argumentation. In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI 2015, Buenos Aires, Argentina, pp. 2854–2860, July 25–31, 2015Google Scholar
  3. Gärdenfors, P.: Knowledge in Flux. Modelling the Dymanics of Epistemic States. MIT Press (1988)Google Scholar
  4. Jaynes, E.T.: Information theory and statistical mechanics. Physical Review 106(4), 620 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Kern-Isberner, G.: Linking iterated belief change operations to nonmonotonic reasoning. In: Principles of Knowledge Representation and Reasoning: Proceedings of the Eleventh International Conference, KR 2008, pp. 166–176 (2008)Google Scholar
  6. Kullback, S., Leibler, R.A.: On information and sufficiency. The Annals of Mathematical Statistics, 79–86 (1951)Google Scholar
  7. Potyka, N., Beierle, C., Kern-Isberner, G.: Changes of relational probabilistic belief states and their computation under optimum entropy semantics. In: Timm, I.J., Thimm, M. (eds.) KI 2013. LNCS, vol. 8077, pp. 176–187. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  8. Ramachandran, R., Ramer, A., Nayak, A.C.: Probabilistic belief contraction. Minds and Machines 22(4), 325–351 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Griffith UniversityBrisbaneAustralia
  2. 2.Macquarie UniversitySydneyAustralia

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