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A Review on Distinct Methods and Approaches to Perform Triangulation for Bayesian Networks

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Advances in Probabilistic Graphical Models

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 213))

Abstract

Triangulation of a Bayesian network (BN) is somehow a necessary step in order to perform inference in a more efficient way, either if we use a secondary structure as the join tree (JT) or implicitly when we try to use other direct techniques on the network. If we focus on the first procedure, the goodness of the triangulation will affect on the simplicity of the join tree and therefore on a quicker and easier inference process.

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References

  1. Amir, E. (2001) Efficient approximation for triangulation of minimum treewidth. In:Proceedings of the 17th Conference on Uncertainty in Artificial Intelligene (UAI-01), 7–15

    Google Scholar 

  2. Becker A. and Geiger D. (1996) A sufficiently fast algorithm for finding close to optimal junction trees. In: Proceedings of the 12h Annual Conference on Uncertainty in Artificial Intelligence (UAI–96), 81–89

    Google Scholar 

  3. Berry A., Blair J.R.S. and Heggernes P. (2002) Maximum Cardinality Search for Computing Minimal Triangulations. WG ’02: Revised Papers from the 28th International Workshop on Graph-Theoretic Concepts in Computer Science. In: Lecture Notes in Computer Science, 2573, 1–12, Springer Verlag

    Google Scholar 

  4. Berry A., Blair J.R.S., Heggernes P. and Peyton B.W. (2004) Maximum Cardinality Search for Computing Minimal Triangulations of Graphs. Algorithmica 39(4): 287–298

    Article  MathSciNet  Google Scholar 

  5. Berry A., Bordat J-P., Heggernes P., Simonet G. and Villanger Y. A wide-range algorithm for minimal triangulation from an arbitrary ordering. To appear in Journal of Algorithms.

    Google Scholar 

  6. Blair J.R.S, Heggernes P. and Telle J.A. (2001) A practical algorithm for making filled graphs minimal. Theoretical Computer Science 205(1-2):125–141

    Article  MathSciNet  Google Scholar 

  7. Bodlaender H.L., Koster A.M. et al (2001) Pre-processing for triangulation of probabilistic networks. In: Proceedings of the 17th Conference on Uncertainty in Artificial Intelligene (UAI-01), 32–39

    Google Scholar 

  8. Cano A. and Moral S. (1994) Heuristic algorithms for the triangulation of graphs. In: Proceedings of the 5th International Conference on Information Processing and Management of Uncertainty in Knowledge Based Systems (IPMU), 1, 166–171, Paris (France)

    Google Scholar 

  9. Cano A., Moral S. and Salmerón A. (2000) Penniless propagation. International Journal of Intelligent Systems, 15:1027–1059

    Article  Google Scholar 

  10. Cano A., Moral S. and Salmern A. (2002) Lazy evaluation in Penniless propagation over join trees. Networks, 39:175–185

    Article  MathSciNet  Google Scholar 

  11. Darwiche A. and Hopkins M. (2001) Using Recursive Decomposition to Construct Elimination Orders, Jointrees, and Dtrees. Lecture Notes in Computer Science, 2143, 180–190, Springer Verlag.

    Google Scholar 

  12. David L. (1987) Genetic algorithms and simulated annealing.

    Google Scholar 

  13. Dechter R. (2003) Constraint Processing. Morgan Kaufmann, 2003

    Google Scholar 

  14. Dorigo M. and Gambardella L. (1997) Ant Colony System: a Cooperative Learning Approach to the Traveling Salesman Problem. IEEE Transactions on Evolutionary Computation, 1:53–66

    Article  Google Scholar 

  15. Dorigo M. and Stützle T. Ant colony optimization. MIT Press, 2004.

    Google Scholar 

  16. Even S. and Tarjan R.E. (1975) Network flow and testing graph connectivity. SIAM Journal on Computing, 4 :507–518

    Article  MathSciNet  Google Scholar 

  17. Flores M.J. and Gámez J.A. (2003) Triangulation of Bayesian networks by retriangulation. International Journal of Intelligent Systems 18(2):153–164

    Article  Google Scholar 

  18. Flores M.J., Gámez J.A. and Olesen K.G. (2003) Incremental Compilation of Bayesian networks. In: Proceedings of the 19th Annual Conference on Uncertainty in Artificial Intelligence (UAI–03), Morgan Kaufmann, 233–240

    Google Scholar 

  19. Flores M.J. (2005) Bayesian networks inference: Advanced algorithms for triangulation and partial abduction. PhD thesis, Departamento de Sistemas Informaticos (Computing Systems Department), University of Castilla - La Mancha UCLM, Spain

    Google Scholar 

  20. de Campos L.M., Gámez J.A., and Moral S.

    Google Scholar 

  21. de Campos L.M., Gámez J.A., and Moral S. (2002) On the problem of performing exact partial abductive inference in Bayesian belief networks using junction trees. In: B. Bouchon-Meunier, J. Gutierrez, L. Magdalena, and R.R. Yager, editors, Technologies for Constructing Intelligent Systems 2: Tools, 289–302 Springer Verlag

    Google Scholar 

  22. Gámez J.A. and Puerta J.M. (2002) Searching for the best elimination sequence in Bayesian networks by using ant colony optimization. Pattern Recognition Letters., 23:261–277

    Article  Google Scholar 

  23. Gogate V. and Dechter R. (2004) A Complete Anytime Algorithm for Treewidth. In: Proceedings of the Twentieth Annual Conference on Uncertainty in Artificial Intelligence (UAI–04), pp. 201–208, AUAI Press

    Google Scholar 

  24. Goldberg D.E. (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley

    Google Scholar 

  25. Heggernes P. and Villanger Y. (2002) Efficient Implementation of a Minimal Triangulation Algorithm. In: ESA ’02: Proceedings of the 10th Annual European Symposium on Algorithms. Lecture Notes in Computer Science, 2461, 550–561, Springer Verlag

    Google Scholar 

  26. Hernández L.D. (1995) Dise no y validación de nuevos algoritmos para el tratamiento de grafos de dependencias (Validation and design of new algorithms to dependency graph processing.) Doctoral thesis, Dpto. de Ciencias de la Computación e I.A. Universidad de Granada, Spain

    Google Scholar 

  27. HUGIN Expert A/S. API manual for the Hugin Decision Engine V6.3. http://developer.hugin.com/documentation/API_Manuals

    Google Scholar 

  28. Jensen F.V., Lauritzen S.L. and Olesen K.G. (1990) Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269–282

    MathSciNet  MATH  Google Scholar 

  29. Jensen F.V. and Jensen F. (1994) Optimal junction trees. In: Proceedings of the Tenth Annual Conference on Uncertainty in Artificial Intelligence (UAI–94), 360–366, Morgan-Kaufmann

    Google Scholar 

  30. Kjærulff U. (1990) Triangulation of graphs - algorithms giving small total space. Technical Report R 90-09, Department of Mathematics and Computer Science. Institute of Electronic Systems. Aalborg University

    Google Scholar 

  31. Kjærulff U. (1992) Optimal decomposition of probabilistic networks by simulated annealing. Statistics and Computing, 2:7–17

    Article  Google Scholar 

  32. Larra naga P., Kuijpers C.M., Poza M. and Murga R.H. (1997) Decomposing Bayesian networks: triangulation of the moral graph with genetic algorithms. Statistics and Computing, 7:19–34

    Article  Google Scholar 

  33. Lauritzen S.L., Speed T.P. and Vijayan K. (1984) Decomposable graphs and hypergraphs. Journal of the Australian Mathematical Society Series A 36: 12–29

    Article  MathSciNet  Google Scholar 

  34. Lauritzen S.L. and Spiegelhalter D.J. (1988) Local computations with probabilities on graphical structures and their application to expert systems. J.R. Statistics Society. Series B, 50(2):157–224

    MathSciNet  MATH  Google Scholar 

  35. Leimer H.G. (1993) Optimal decomposition by clique separators. Discrete Mathematics, 113:99-123

    Article  MathSciNet  Google Scholar 

  36. Madsen A.L. and Jensen F.V. (1999) Lazy Propagation: A Junction Tree Inference Algorithm based on Lazy Evaluation. Artificial Intelligence, 113 (1-2):203–245, Elsevier Science Publishers, North-Holland

    Article  MathSciNet  Google Scholar 

  37. Michalewicz Z. (1996) Genectic Algorithms + Data Structures = Evolution Programs. Springer-Verlag

    Google Scholar 

  38. Olesen K.G. and Madsen A.L. (2002) Maximal prime subgraph decomposition of bayesian networks. IEEE Transactions on Systems, Man and Cybernetics, Part B:(32), 21–31

    Article  Google Scholar 

  39. Park J.D. and Darwiche A. (2003): Solving MAP Exactly using Systematic Search. In: Proceedings of the 19th Annual Conference on Uncertainty in Artificial Intelligence (UAI–03), Morgan Kaufmann, 459–468

    Google Scholar 

  40. Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo

    MATH  Google Scholar 

  41. Peyton B.W. (2001) Minimal Orderings Revisited. SIAM Journal on Matrix Analysis and Applications, 23(1):271–294

    Article  MathSciNet  Google Scholar 

  42. Rose D., Tarjan R.E. and Lueker G.S. (1976) Algorithmic aspects of vertex elimination graphs. SIAM Journal on Computing, 5:266–283

    Article  MathSciNet  Google Scholar 

  43. Selvakkumaran N. and Karypis G. Multi-Objective Hypergraph Partitioning Algorithms for Cut and Maximum Subdomain Degree Minimization IEEE Transactions on Computer Aided Design (in press)

    Google Scholar 

  44. Rose D. (1972) A graph theoretic study of the numerical solution of sparse positive definite systems of linear equations. In: R. Reed ed. Graph Theory and Computing, 183–217, Academic Press, New York

    Google Scholar 

  45. Shafer G.R. and Shenoy P.P. (1990) Probability propagation. Annals of Mathematics and Artificial Intelligence, 2:327–352

    Article  MathSciNet  Google Scholar 

  46. Shafer G.R. and Shenoy P.P. (1990) Axioms for probability and belief-function propagation. In: R.D. Shachter, T.S. Levitt, L.N. Kanal and J.F. Lemmer (eds.), Uncertainty in Artificial Intelligence, 4, 169–198, Elsevier Science Publishers B.V. (North-Holland)

    Google Scholar 

  47. Shoikhet K. and Geiger D. (1997) Practical algorithm for finding optimal triangulations. In: Proceedings of the National Conference on Artificial Intelligence, AAAI, USA, 185–190

    Google Scholar 

  48. Tarjan R.E. and Yannakakis M. (1984) Simple linear-time algorithms to test chordality of graps, text acyclicity of hypergraphs and selectively reduce acyclic hypergraphs. SIAM Journal on Computing, 13:566–579

    Article  MathSciNet  Google Scholar 

  49. Tarjan R.E. (1985) Decomposition by clique separators. Discrete Mathematics, 55:221–232

    Article  MathSciNet  Google Scholar 

  50. Villanger Y. (2004) LEX M versus MCS-M. Technical Report Reports in Informatics 261, University of Bergen, Norway

    Google Scholar 

  51. Wen, W. X. (1990) Decomposing belief networks by simulated annealing. In: (C. P. Tsang, ed.) Proceedings of the Australian Joint Conference on Artificial Intelligence, 103–118

    Google Scholar 

  52. Wen, W.X. (1991) Optimal decomposition of belief networks. In: P.P. Bonissone, M. Henrion, L.N. Kanal and J.F. Lemmer (eds.), Uncertainty in Artificial Intelligence, 6, 209–224 North-Holland

    Google Scholar 

  53. Kirkpatrick S., Gelatt C.D. and Vecchi M.P. (1983) Optimization by simulated annealing. Science, 220:671–680

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Sistemas Informáticos & SIMD - i3 A, Universidad de Castilla-La Mancha, Campus Universitario s/n., 02071, Albacete, Spain

    M. Julia Flores & José A. Gámez

Authors
  1. M. Julia Flores
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  2. José A. Gámez
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Editor information

Editors and Affiliations

  1. Institute for Computing and Information Sciences, Radboud University Nijmegen, Toernooiveld 1, 6525 ED, Nijmegen, The Netherlands

    Peter Lucas Dr.

  2. Department of Computer Science and Artificial Intelligence, University of Granada, c/. Daniel Saucedo Aranda, s/n, 18071, Granada, Spain

    José A. Gámez Dr.

  3. Department of Statistics and Applied Mathematics, The University of Almería, La Cañada de San Urbano s/n, 04120, Almería, Spain

    Antonio Salmerón Dr.

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Flores, M.J., Gámez, J.A. (2007). A Review on Distinct Methods and Approaches to Perform Triangulation for Bayesian Networks. In: Lucas, P., Gámez, J.A., Salmerón, A. (eds) Advances in Probabilistic Graphical Models. Studies in Fuzziness and Soft Computing, vol 213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68996-6_6

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  • DOI: https://doi.org/10.1007/978-3-540-68996-6_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68994-2

  • Online ISBN: 978-3-540-68996-6

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