Publication: The effect of block parameter perturbations in Gaussian Bayesian networks: Sensitivity and robustness
Full text at PDC
Advisors (or tutors)
Elsevier Science Inc
n this work we study the effects of model inaccuracies on the description of a Gaussian Bayesian network with a set of variables of interest and a set of evidential variables. Using the Kullback-Leibler divergence measure, we compare the output of two different networks after evidence propagation: the original network, and a network with perturbations representing uncertainties in the quantitative parameters. We describe two methods for analyzing the sensitivity and robustness of a Gaussian Bayesian network on this basis. In the sensitivity analysis, different expressions are obtained depending on which set of parameters is considered inaccurate. This fact makes it possible to determine the set of parameters that most strongly disturbs the network output. If all of the divergences are small, we can conclude that the network output is insensitive to the proposed perturbations. The robustness analysis is similar, but considers all potential uncertainties jointly. It thus yields only one divergence, which can be used to confirm the overall sensitivity of the network. Some practical examples of this method are provided, including a complex, real-world problem
M. Bednarski, W. Cholewa, W. Frid, Identification of sensitivities in Bayesian networks, Engineering Applications of Artificial Intelligence 17 (2004) 327–335. K. Burns, Bayesian inference in disputed authorship: a case study of cognitive errors and a new system for decision support, Information Sciences 176 (2006) 1570–1589. E. Castillo, U. Kjærulff, Sensitivity analysis in Gaussian Bayesian networks using a symbolic-numerical technique, Reliability Engineering and System Safety 79 (2003) 139–148. H. Chan, A. Darwiche, Sensitivity analysis in Bayesian networks: form single to multiple parameters, in: En Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, AUAI Press, Arlington, Virginia, USA, 2004, pp. 67–75. H. Chan, A. Darwiche, A distance measure for bounding probabilistic belief change, International Journal of Approximate Reasoning 38 (2) (2005) 149–174. V.M.H. Coupé, L.C. van der Gaag, J.D.F. Habbema, Sensitivity analysis: an aid for belief-network quantification, The Knowledge Engineering Review 15 (3) (2000) 215–232. V.M.H. Coupé, L.C. van der Gaag, Properties of sensitivity analysis of Bayesian belief networks, Annals of Mathematics and Artificial Intelligence 36 (2002) 323–356. R.G. Cowell, A.P. Dawid, S.L. Lauritzen, D.J. Spiegelhalter, Probabilistic Networks and Expert Systems, Springer, Barcelona, 1999. M.A. Gómez-Villegas, P. Main, R. Susi, Sensitivity analysis in Gaussian Bayesian networks using a divergence measure, Communications in Statistics: Theory and Methods 36 (3) (2007) 523–539. M.A. Gómez-Villegas, P. Main, R. Susi, Extreme inaccuracies in Gaussian Bayesian networks, Journal of Multivariate Analysis 99 (2008) 1929–1940. M.A. Gómez-Villegas, P. Main, H. Navarro, R. Susi, Evaluating the difference between graph structures in Gaussian Bayesian networks, Expert Systems with Applications 38 (2011) 12409–12414. E.R. Hansen, R.R. Smith, Interval arithmetic in matrix computations, SIAM Journal of Numerical Analysis 4 (1994) 1–9. J. Horn, Positive definiteness and stability of interval matrices, SIAM Journal of Matrix Analysis and Applications 15 (1994) 175–184. J. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 2005. G.B. Huang, D.H. Wang, Y. Lan, Extreme learning machines: a survey, International Journal of Machine Learning and Cybernetics 2 (2) (2011) 107–122. F.V. Jensen, T. Nielsen, Bayesian Networks and Decision Graphs, sixth ed., Springer, New York, 2007. L.X. Jiang, H. Zhang, Z.H. Cai, A novel Bayes model: hidden Nadve Bayes, IEEE Transactions on Knowledge and Data Engineering 21 (10) (2009) 1361–1371. R.A. Johnson, D.W. Wichern, Applied Multivariate Statistical Analysis, sixth ed., Prentice-Hall, 2007. U. Kjærulff, L.C. van der Gaag, Making sensitivity analysis computationally efficient, in: Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco, CA, USA, 2000, pp. 315–325. S. Kullback, R.A. Leibler, On information and sufficiency, Annals of Mathematical Statistics 22 (1951) 79–86. K.B. Laskey, Sensitivity analysis for probability assessments in Bayesian networks, IEEE Transactions on Systems, Man and Cybernetics 25 (1995) 901–909. S.L. Lauritzen, Graphical Models, Clarendon Press, Oxford, 1996. P. Main, H. Navarro, Analyzing the effect of introducing a kurtosis parameter in Gaussian Bayesian networks, Reliability Engineering and System Safety 94 (2009) 922–926. R.E. McCulloch, Local model influence, Journal American Statistical Association 84 (1989) 473–478. S.L. Normand, D. Tritchler, Parameter updating in Bayes network, Journal American Statistical Association 87 (1992) 1109–1115. J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann, San Mateo, CA, 1988. R Development Core Team, 2011. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. <http://www.R-project.org/>. S. Renooij. Bayesian network sensitivity to arc-removal, In: Proceedings of the Fifth European Workshop on Probabilistic Graphical Models, Helsinki, 2010. R. Shachter, C. Kenley, Gaussian influence diagrams, Management Science 35 (1989) 527–550. L.C. Van der Gaag, S. Renooij, V.M.H. Coup é, Sensitivity analysis of probabilistic networks, in: Advances in Probabilistic Graphical Models, Studies in Fuzziness and Soft Computing, vol. 213, Springer, Berlin, 2007, pp. 103–124. X.Z. Wang, C.R. Dong, Improving generalization of fuzzy if–then rules by maximizing fuzzy entropy, IEEE Transactions on Fuzzy Systems 17 (3) (2009) 556–567. X.Z. Wang, C.R. Dong, J.H. Yan, Maximum ambiguity based sample selection in fuzzy decision tree induction, IEEE Transactions on Knowledge and Data Engineering, 2011. 10.1109/TKDE.2011.67. J. Whittaker, Graphical Models in Applied Multivariate Statistics, Wiley, England, 1990. L. Xu, J. Pearl, Structuring causal tree models with continuous variables, Uncertainty in Artificial Intelligence 3 (1989) 209–222.