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Matrix equations for normalizing factors in local a posteriori inference of truth estimates in algebraic Bayesian networks

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Abstract

A posteriori inference is one of three kinds of inference that underlie the processing of knowledge patterns with probabilistic uncertainty in intelligent decision making systems using algebraic Bayesian networks (ABNs). In this paper, the key terms and formulations of theorems describing local a posteriori inference in algebraic Bayesian networks are given in a matrix-vector language. The main result is that matrix equations are constructed for normalizing factors appearing in the formulas of a posteriori probabilities of proposition-quanta and ideals of conjuncts. The matrix equations of local a priori inference formulated in general not only simplify the preparation of specifications of appropriate inference algorithms and make their implementation more transparent, but also open a possibility for application of classical mathematical techniques to the analysis of the properties of inference results.

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References

  1. R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960; Nauka, Moscow, 1969).

    Google Scholar 

  2. V. I. Gorodetskii, “Algebraic Bayesian networks–new paradigm of expert systems,” in Anniversary Collection of Proceedings of Institutes of the Russian Academy of Sciences Department of Informatics, Computer Science and Automation, Vol. 2 (Ross. Akad. Nauk, Moscow, 1993) [in Russian].

    Google Scholar 

  3. V. I. Gorodetskii and A. L. Tulup’ev, “Formation of noncontradicting knowledge bases with an uncertainty,” Izv. Ross. Akad. Nauk, Ser. Teor. Sist. Upr. 5, 33–42 (1997).

    MathSciNet  Google Scholar 

  4. A. V. Sirotkin, “Computational complexity of local posteriori inference algorithms in algebraic Bayesian networks,” Tr. SPIIRAN, No. 3(18), 188–214 (2011).

    Google Scholar 

  5. A. V. Sirotkin and A. L. Tulup’ev, “Local a priori inference in algebraic Bayesian network: A set of basic algorithms,” Tr. SPIIRAN, No. 5, 100–111 (2007).

    Google Scholar 

  6. A. L. Tulup’ev, “Algebraic Bayesian networks: a local probabilistic logic inference,” in Handbook on Elements of Soft Computing (Anatoliya, St. Petersburg, 2007) [in Russian].

    Google Scholar 

  7. A. L. Tulup’ev, Algebraic Bayesian Networks: a Probabilistic Logic Approach to the Modelling of Knowledge Bases with an Uncertainty (SPIIRAN, St. Petersburg, 2000) [in Russian].

    Google Scholar 

  8. A. L. Tulup’ev, Bayesian Networks: A Probabilistic Logic Inference in Cycles (S.-Peterb. Gos. Univ., St. Petersburg, 2008) [in Russian].

    Google Scholar 

  9. A. L. Tulup’ev and A. V. Sirotkin, “Algebraic Bayesian networks: Principle of decomposition and probabilistic logic inference under uncertainty,” Inf.-Izmer. Upravl. Sist. 6 (10), 85–87 (2008).

    Google Scholar 

  10. A. L. Tulup’ev and A. V. Sirotkin, “Matrix equations for local probabilistic logic inference of truth values of elements in algebraic Bayesian networks,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 3, 63–72 (2012).

    Google Scholar 

  11. A. L. Tulup’ev, S. I. Nikolenko, and A. V. Sirotkin, Bayesian Networks: A Probabilistic Logic Approach (Nauka, St. Petersburg, 2006) [in Russian].

    Google Scholar 

  12. A. L. Tulup’ev, A. V. Sirotkin, and S. I. Nikolenko, Bayesian Belief Networks: Probabilistic Logic Inference in Acyclic Directed Graphs (S.-Peterb. Gos. Univ., St. Petersburg, 2009) [in Russian].

    Google Scholar 

  13. A. A. Fil’chenkov, “Hierarchy of algebraic Bayesian network global structures as a system of graphs and hypergraphs,” Nauchno-Tekhn. Vestn. Inf. Tekhnol., Mekh. Opt. 13 (1), 75–81 (2013).

    Google Scholar 

  14. N. J. Nilsson, “Probabilistic Logic,” Artif. Intell. 47, 71–87 (1986).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia

    A. L. Tulupyev, A. V. Sirotkin & A. A. Zolotin

  2. St. Petersburg Institute for Informatics and Automation, Russian Academy of Sciences, 14-ya liniya V.O. 39, St. Petersburg, 199178, Russia

    A. L. Tulupyev

Authors
  1. A. L. Tulupyev
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  2. A. V. Sirotkin
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  3. A. A. Zolotin
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Corresponding author

Correspondence to A. L. Tulupyev.

Additional information

Original Russian Text © A.L. Tulupyev, A.V. Sirotkin, A.A. Zolotin, 2015, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2015, No. 3, pp. 391–398.

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Tulupyev, A.L., Sirotkin, A.V. & Zolotin, A.A. Matrix equations for normalizing factors in local a posteriori inference of truth estimates in algebraic Bayesian networks. Vestnik St.Petersb. Univ.Math. 48, 168–174 (2015). https://doi.org/10.3103/S1063454115030073

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  • DOI: https://doi.org/10.3103/S1063454115030073

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