Menu

Publications

Editor-in-Chief

**Nikiforov**

Vladimir O.

D.Sc., Prof.

Vladimir O.

D.Sc., Prof.

Partners

DOI: 10.17586/2226-1494-2015-15-4-676-684

DOI: 10.17586/2226-1494-2015-15-4-676-684

# MATRIX-VECTOR ALGORITHMS OF LOCAL POSTERIORI INFERENCE IN ALGEBRAIC BAYESIAN NETWORKS ON QUANTA PROPOSITIONS

**Read the full article**';

**Article in**Russian

**For citation:**Zolotin A.A., Tulupyev A.L., Sirotkin A.V. Matrix-vector algorithms of local posteriori inference in algebraic bayesian networks on quanta propositions.

*Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2015, vol.15, no. 4, pp. 676–684.

**Abstract**

Posteriori inference is one of the three kinds of probabilistic-logic inferences in the probabilistic graphical models theory and the base for processing of knowledge patterns with probabilistic uncertainty using Bayesian networks. The paper deals with a task of local posteriori inference description in algebraic Bayesian networks that represent a class of probabilistic graphical models by means of matrix-vector equations. The latter are essentially based on the use of tensor product of matrices, Kronecker degree and Hadamard product. Matrix equations for calculating posteriori probabilities vectors within posteriori inference in knowledge patterns with quanta propositions are obtained. Similar equations of the same type have already been discussed within the confines of the theory of algebraic Bayesian networks, but they were built only for the case of posteriori inference in the knowledge patterns on the ideals of conjuncts. During synthesis and development of matrix-vector equations on quanta propositions probability vectors, a number of earlier results concerning normalizing factors in posteriori inference and assignment of linear projective operator with a selector vector was adapted. We consider all three types of incoming evidences - deterministic, stochastic and inaccurate - combined with scalar and interval estimation of probability truth of propositional formulas in the knowledge patterns. Linear programming problems are formed. Their solution gives the desired interval values of posterior probabilities in the case of inaccurate evidence or interval estimates in a knowledge pattern. That sort of description of a posteriori inference gives the possibility to extend the set of knowledge pattern types that we can use in the local and global posteriori inference, as well as simplify complex software implementation by use of existing third-party libraries, effectively supporting submission and processing of matrices and vectors when programming in Java, C++ or C#.

**Keywords:**Bayesian networks, posteriori inference, inference algorithms, evidence propagation, knowledge pattern over quantapropositions.

**Acknowledgements.**The research was partially supported by RFBR Grant No 15-01-09001-а.

**References**

1. Arsene O., Dumitrache L., Mihu I. Expert system for medicine diagnosis using software agents. Expert Systems with Applications, 2015, vol. 42, no. 4, pp. 1825–1834. doi: 10.1016/j.eswa.2014.10.026

2. Tulup'ev A.L., Sirotkin A.V., Nikolenko S.I. Baiesovskie Seti Doveriya: Logiko-Veroyatnostnyi Vyvod v Atsiklicheskikh Napravlennykh Grafakh [Bayesian Belief Networks: Logical-Probabilistic Inference in the Acyclic Directed Graph]. St. Petersburg, SPbSU Publ., 2009, 400 p.

3. Tulup'ev A.L., Nikolenko S.I., Sirotkin A.V. Baiesovskie Seti: Logiko-Veroyatnostnyi Podkhod [Bayesian Networks: Logical-Probabilistic Approach]. St. Petersburg, Nauka Publ., 2006, 607 p.

4. Shang J.D., Wang Z.L., Huang Q. A robust algorithm for joint sparse recovery in presence of impulsive noise. IEEE Signal Processing Letters, 2015, vol. 22, no. 8, pp. 1166–1170. doi: 10.1109/LSP.2014.2387435

5. Peng P., Tian Y., Wang Y., Li J., Huang T. Robust multiple cameras pedestrian detection with multi-view Bayesian network. Pattern Recognition, 2015, vol. 48, no. 5, pp. 1760–1772. doi: 10.1016/j.patcog.2014.12.004

6. Dabrowski J.J., Villiers J.P. Maritime piracy situation modelling with dynamic Bayesian networks. Information Fusion, 2015, vol. 23, pp. 116–130. doi: 10.1016/j.inffus.2014.07.001

7. Leu S.S., Chang C.M. Bayesian-network-based fall risk evaluation of steel construction projects by fault tree transformation. Journal of Civil Engineering and Management, 2015, vol. 21, pp. 334–342.

8. Mahboob Q., Schone E., Maschek U., Trinckauf J. Investment into human risks in railways and decision optimization. Human and Ecological Risk Assessment, 2015, vol. 21, no. 5, pp. 1299–1313. doi: 10.1080/10807039.2014.958375

9. Shin J., Son H., Khalil Ur R., Heo G. Development of a cyber security risk model using Bayesian networks. Reliability Engineering System Safety, 2015, vol. 134, pp. 208–217. doi: 10.1016/j.ress.2014.10.006

10. Arangio S., Bontempi F. Structural health monitoring of a cable-stayed bridge with Bayesian neural networks. Structure and Infrastructure Engineering, 2015, vol. 11, no. 4, pp. 575–587. doi: 10.1080/15732479.2014.951867

11. Dawson P., Gailis R., Meehan A. Detecting disease outbreaks using a combined Bayesian network and

particle filter approach. Journal of Theoretical Biology, 2015, vol. 370, pp. 171–183. doi:

10.1016/j.jtbi.2015.01.023

12. Chang Y.-S., Fan C.-T., Lo W.-T., Hung W.-C., Yuan S.-M. Mobile cloud-based depression diagnosis using an ontology and a Bayesian network. Future Generation Computer Systems, 2015, vol. 43–44, pp. 87–98. doi: 10.1016/j.future.2014.05.004

13. Rafiq M.I., Chryssanthopoulos M.K., Sathananthan S. Bridge condition modelling and prediction using dynamic Bayesian belief networks. Structure and Infrastructure Engineering, 2015, vol. 11, no. 1, pp. 38–50.

doi: 10.1080/15732479.2013.879319

14. Hamilton S.H., Pollino C.A., Jakeman A.J. Habitat suitability modelling of rare species using Bayesian networks: model evaluation under limited data. Ecological Modelling, 2015, vol. 299, pp. 64–78. doi: 10.1016/j.ecolmodel.2014.12.004

15. Liu K.F.-R., Kuo J.-Y., Yeh K., Chen C.-W., Liang H.-H., Sun Y.-H. Using fuzzy logic to generate conditional probabilities in Bayesian belief networks: a case study of ecological assessment. International Journal of Environmental Science and Technology, 2015, vol. 12, no. 3, pp. 871–884. doi: 10.1007/s13762-013-0459-x

16. Sirotkin A.V. Algebraicheskie Baiesovskie Seti: Vychislitel'naya Slozhnost' Algoritmov Logiko- Veroyatnostnogo Vyvoda v Usloviyakh Neopredelennosti. Dis. kand. fiz.-mat. nauk [Algebraic Bayesian Networks: Computational Complexity of Algorithms of Logic-Probabilistic Inference Under Uncertainty. Diss. Ph.D. in Phys.-Math. Sci.]. St. Petersburg, SPbGU, 2011, 218 p.

17. Tulup'ev A.L. Algebraicheskie Baiesovskie Seti: Logiko-Veroyatnostnaya Graficheskaya Model' Baz Fragmentov Znanii s Neopredelennost'yu. Dis. dr. fiz.-mat. nauk [Algebraic Bayesian Networks: Logical- Probabilistic Graphical Model Database Fragments of Knowledge with Uncertainty. Diss. Dr. in Phys.-Math. Sci.]. St. Petersburg, SPbGU, 2009, 670 p.

18. Tulup'ev A.L., Sirotkin A.V. Matrichnye uravneniya lokal'nogo logiko-veroyatnostnogo vyvoda otsenok istinnosti elementov v algebraicheskikh baiesovskikh setyakh [Matrix-Vector Equations for Local Probabilistic-Logic Inference in Algebraic Bayesian Networks]. Vestnik of the St. Petersburg University: Mathematics, 2012, no. 3, pp. 63–72.

19. Tulup'ev A.L. Baiesovskie Seti: Logiko-Veroyatnostnyi Vyvod v Tsiklakh [Bayesian Networks: Logic- Probabilistic Inference in Cycles]. St. Petersburg, SPbGU, 2008, 140 p.

20. Tulup'ev A.L. Aposteriornye otsenki veroyatnostei v algebraicheskikh baiesovskikh setyakh [A posteriori probabilistic estimates in algebraic Bayesian networks]. Vestnik of the St. Petersburg University: Seriya 10: Prikladnaya Matematika. Informatika. Protsessy Upravleniya, 2012, no. 2, pp. 51–59.

21. Zolotin A.A., Tulupyev A.L., Sirotkin A.V. Matrix-vector algorithms for normalizing factors in algebraic Bayesian networks local posteriori inference. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 1, pp. 78–85 (in Russian)

22. Tulup'ev A.L. Algebraicheskie Baiesovskie Seti: Lokal'nyi Logiko-Veroyatnostnyi Vyvod [Algebraic Bayesian Networks: a Local Logic-Probabilistic Inference]. St. Petersburg, SPbGU Publ., Anatoliya, 2007, 80 p.

23. Bellman R. Introduction to Matrix Analysis. NY, McGraw-Hill, 1970.