Abstract
The relational probabilistic conditional logic FO-PCL employs the principle of maximum entropy (ME). We show that parametric uniformity of an FO-PCL knowledge base \(\mathcal R\) can be exploited for solving the optimization problem required for ME reasoning more efficiently. The original ME optimization problem containing a large number of linear constraints, one for each ground instance of a conditional, can be replaced by an optimization problem containing just one linear constraint for each conditional. We show that both optimization problems have the same ME distribution as solution. An implementation employing Generalized Iterative Scaling illustrates the benefits of our approach
The research reported here was partially supported by the DFG (grant BE 1700/7-2).
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References
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)
Darroch, J.N., Ratcliff, D.: Generalized iterative scaling for log-linear models. Annals of Mathematical Statistics 43(5), 1470–1480 (1972)
de Salvo Braz, R., Amir, E., Roth, D.: Lifted first-order probabilistic inference. In: Kaelbling, L.P., Saffiotti, A. (eds.) Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence, IJCAI 2005, pp. 1319–1325. Professional Book Center (2005)
Delgrande, J.: On first-order conditional logics. Artificial Intelligence 105, 105–137 (1998)
Finthammer, M.: An iterative scaling algorithm for maximum entropy reasoning in relational probabilistic conditional logic. In: Hüllermeier, E. (ed.) SUM 2012. LNCS, vol. 7520, pp. 351–364. Springer, Heidelberg (2012)
Finthammer, M., Beierle, C.: Using equivalences of worlds for aggregation semantics of relational conditionals. In: KI 2012: Proceedings of 35th Annual German Conference on Advances in Artificial Intelligence AI, Saarbrücken, Germany, September 24-27. LNCS (LNAI), Springer (to appear, 2012)
Finthammer, M., Thimm, M.: An integrated development environment for probabilistic relational reasoning. Logic Journal of the IGPL (to appear, 2012)
Fisseler, F.: Learning and Modeling with Probabilistic Conditional Logic. Dissertations in Artificial Intelligence, vol. 328. IOS Press (2010)
Fisseler, J.: First-order probabilistic conditional logic and maximum entropy. Logic Journal of the IGPL (to appear, 2012)
Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6, 721–741 (1984)
Halpern, J.: Reasoning About Uncertainty. MIT Press (2005)
Jaimovich, A., Meshi, O., Friedman, N.: Template based inference in symmetric relational markov random fields. In: Proc. of the 23rd Conference on Uncertainty in Artificial Intelligence. AUAI Press (2007)
Kern-Isberner, G.: Characterizing the principle of minimum cross-entropy within a conditional-logical framework. Artificial Intelligence 98, 169–208 (1998)
Kern-Isberner, G.: Conditionals in Nonmonotonic Reasoning and Belief Revision. LNCS (LNAI), vol. 2087. Springer, Heidelberg (2001)
Kern-Isberner, G., Lukasiewicz, T.: Combining probabilistic logic programming with the power of maximum entropy. Artificial Intelligence, Special Issue on Nonmonotonic Reasoning 157(1-2), 139–202 (2004)
Kern-Isberner, G., Thimm, M.: Novel semantical approaches to relational probabilistic conditionals. In: Lin, F., Sattler, U., Truszczyński, M. (eds.) Proc. of the 12th Int’l. Conference on the Principles of Knowledge Representation and Reasoning (KR 2010), May 2010, pp. 382–392. AAAI Press (May 2010)
Krämer, A., Beierle, C.: On Lifted Inference for a Relational Probabilistic Conditional Logic with Maximum Entropy Semantics. In: Lukasiewicz, T., Sali, A. (eds.) FoIKS 2012. LNCS, vol. 7153, pp. 224–243. Springer, Heidelberg (2012)
Milch, B., Zettlemoyer, L., Kersting, K., Haimes, M., Kaelbling, L.P.: Lifted probabilistic inference with counting formulas. In: Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence, AAAI 2008, Chicago, Illinois, USA, July 13-17. AAAI Press (2008)
Paris, J.: The uncertain reasoner’s companion – A mathematical perspective. Cambridge University Press (1994)
Poole, D.: First-order probabilistic inference. In: Gottlob, G., Walsh, T. (eds.) Proc. IJCAI 2003, pp. 985–991. Morgan Kaufmann (2003)
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Finthammer, M., Beierle, C. (2012). How to Exploit Parametric Uniformity for Maximum Entropy Reasoning in a Relational Probabilistic Logic. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds) Logics in Artificial Intelligence. JELIA 2012. Lecture Notes in Computer Science(), vol 7519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33353-8_15
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