Abstract
Although undirected cycles in directed graphs of Bayesian belief networks have been thoroughly studied, little attention has so far been given to a systematic analysis of directed (feedback) cycles. In this paper we propose a way of looking at those cycles; namely, we suggest that a feedback cycle represents a family of probabilistic distributions rather than a single distribution (as a regular Bayesian belief network does). A non-empty family of distributions can be explicitly represented by an ideal of conjunctions with interval estimates on the probabilities of its elements. This ideal can serve as a probabilistic model of an expert’s uncertain knowledge pattern; such models are studied in the theory of algebraic Bayesian networks. The family of probabilistic distributions may also be empty; in this case, the probabilistic assignment over cycle nodes is inconsistent. We propose a simple way of explicating the probabilistic relationships an isolated directed cycle contains, give an algorithm (based on linear programming) of its consistency checking, and establish a lower bound of the complexity of this checking.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Castillo, E., Gutierrez, J.M., Hadi, A.S.: Modeling Probabilistic Networks of Discrete and Continuous Variables. Journal of Multivariate Analysis 64, 48–65 (1998)
Fagin, R., Halpern, J.Y., Megiddo, N.: A Logic for Reasoning about Probabilities. Report RJ 6190(60900) 4/12/88, pp. 1–41
Fagin, R., Halpern, J.Y.: Uncertainty, Belief, and Probability. In: Proc. of the 11th International Joint Conference on Artificial Intelligence, pp. 1161–1167 (1989)
Fagin, R., Halpern, J.Y.: Uncertainty, Belief, and Probability-2. In: Proc. of the IEEE Simposium on Logic and Computer Science, vol. 7, pp. 160–173 (1991)
Gorodetski, V.: Adaptation Problems in Expert System. International Journal of Adaptive Control and Signal Processing 6, 201–210 (1992)
Gorodetski, V.: Expert Systems Adaptation. Transactions of the Russian Academy of Sciences ”Technical Cybernetics” 5, 101–110 (1993) (in Russian)
Gorodetski, V.: Interval-valued Probabilistic Measure of Uncertainty in Knowledge Engineering. In: Yusupov, R.M. (ed.) Theoretical Basis and Applied Intelligent Information Technologies. Russian Academy of Science, SPIIRAS, pp. 44–58 (1998) (in Russian)
Gorodetski, V.I., Nesterov, V.M.: Interval probabilities and knowledge engineering. In: Alefeld, G., Trejo, R.A. (eds.) Interval computations and its applications to reasoning under uncertainty, knowledge representation and control theory, Mexico City, pp. 15–20 (1998)
Gorodetski, V., Skormin, V., Popyack, L.: Data Mining Technology for Failure Prognostics of Avionics. IEEE Transactions on Aerospace and Electronic Systems 38(2), 388–403 (2002)
Gorodetskii, V.I., Tulupyev, A.L.: Generating Consistent Knowledge Bases with Uncertainty. Journal of Computer and Systems Sciences International 5, 33–42 (1997)
Heckerman, D., Chickering, D., Meek, C., Rounthwaite, R., Kadie, C.: Dependency Networks for Density Estimation, Collaborative Filtering, and Data Visualization. Journal of Machine Learning Research 1, 49–75 (2000); Also appears as Technical Report MSR-TR-00-16, Microsoft Research (February 2000)
Hulten, G., Chickering, D.M., Heckerman, D.: Learning Bayesian Networks from Dependency Networks: A Preliminary Study. In: Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, Key West, FL (2003)
Jensen, F.V.: Bayesian networks and decision graphs. Springer, Heidelberg (2002)
Mihajlovic, V., Petkovic, M.: Dynamic Bayesian Networks: a state of the art. Technical report, no. TR-CTIT-01-34, Centre for Telematics and Information Technology, University of Twente (2001)
Neapolitan, R.E.: Learning Bayesian Networks. Pearson Prentice Hall, London (2004)
Nesterov, V.M., Gorodetski, V.I.: Interval algorithm for checking the consistency of the algebraic Bayesian net. In: Alefeld, G., Trejo, R.A. (eds.) Interval computations and its applications to reasoning under uncertainty, knowledge representation and control theory, Mexico City, pp. 28–29 (1998)
Nikolenko, S.I., Tulupyev, A.L.: Edge flipping as a technique for dealing with directed cycles in Bayesian networks. In: MEPhI-2005 Science Session Proceedings. Intellectual systems and technologies, vol. 3, pp. 176–178, Moscow, MEPhI (2005) (in Russian)
Nilsson, N.J.: Probabilistic Logic. Artificial Intelligence 28, 71–87 (1986)
Pearl, J.: How to do with probabilities what people say you can’t. In: Quaglini, S., Barahona, P., Andreassen, S. (eds.) AIME 2001. LNCS (LNAI), vol. 2101, pp. 283–292. Springer, Heidelberg (2001)
Pearl, J.: Probabilistic reasoning using graphs. In: Bouchon, B., Yager, R.R. (eds.) Uncertainty in Knowledge-Based Systems, pp. 201–202. Springer, Heidelberg (1987)
Pearl, J.: Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, San Mateo (1988)
Sprites, P.: Directed cyclic graphical representations of feedback models. Uncertainty in Artificial Intelligence 12, 454–461 (1996)
Studený, M., Bouckaert, R.: On chain graph models for description of conditional independence structures. The Annals of Statistics 26(4), 1434–1495 (1998)
Tulupyev, A.L.: Algebraic Bayesian Networks: Theoretical Background and Consistency, St.-Petersburg (1995) (in Russian)
Tulupyev, A.L.: Algebraic Bayesian Networks: a Probabilistic Logic Approach to Modeling Knowledge Bases with Uncertainties, St.-Petersburg (2000) (in Russian)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tulupyev, A.L., Nikolenko, S.I. (2005). Directed Cycles in Bayesian Belief Networks: Probabilistic Semantics and Consistency Checking Complexity. In: Gelbukh, A., de Albornoz, Á., Terashima-Marín, H. (eds) MICAI 2005: Advances in Artificial Intelligence. MICAI 2005. Lecture Notes in Computer Science(), vol 3789. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11579427_22
Download citation
DOI: https://doi.org/10.1007/11579427_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29896-0
Online ISBN: 978-3-540-31653-4
eBook Packages: Computer ScienceComputer Science (R0)