Probability Elicitation in Influence Diagram Modeling by Using Interval Probability

ABSTRACT

In decision modeling with influence diagrams, the most challenging task is probability elicitation from domain experts. It is usually very difficult for experts to directly assign precise probabilities to chance nodes. In this paper, we propose an approach to elicit probability effectively by using the concept of interval probability (IP). During the elicitation process, a group of experts assign intervals to probabilities instead of assigning exact values. Then the intervals are combined and converted into the point valued probabilities. The detailed steps of the elicitation process are given and illustrated by constructing the influence diagram for employee recruitment decision for a China’s IT Company. The proposed approach provides a convenient and comfortable way for experts to assess probabilities. It is useful in influence diagrams modeling as well as in other subjective probability elicitation situations.

In decision modeling with influence diagrams, the most challenging task is probability elicitation from domain experts. It is usually very difficult for experts to directly assign precise probabilities to chance nodes. In this paper, we propose an approach to elicit probability effectively by using the concept of interval probability (IP). During the elicitation process, a group of experts assign intervals to probabilities instead of assigning exact values. Then the intervals are combined and converted into the point valued probabilities. The detailed steps of the elicitation process are given and illustrated by constructing the influence diagram for employee recruitment decision for a China’s IT Company. The proposed approach provides a convenient and comfortable way for experts to assess probabilities. It is useful in influence diagrams modeling as well as in other subjective probability elicitation situations.

Cite this paper

X. Hu, H. Luo and C. Fu, "Probability Elicitation in Influence Diagram Modeling by Using Interval Probability,"*International Journal of Intelligence Science*, Vol. 2 No. 4, 2012, pp. 89-95. doi: 10.4236/ijis.2012.24012.

X. Hu, H. Luo and C. Fu, "Probability Elicitation in Influence Diagram Modeling by Using Interval Probability,"

References

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[2] C. Bielza, M. Gómez, S. Ríos-INsua, and J. A. F. D. Pozo, “Structural, Elicitation and Computational Issues Faced When Solving Complex Decision Making Problems with Influence Diagrams,” Computers & Operations Research, Vol. 27, No. 7-8, 2000, pp. 725-740.

[3] C. Bielza, M. Gómez and P. P. Shenoy, “Modeling Challenges with Influence Diagrams: Constructing Probability and Utility Models,” Decision Support Systems, Vol. 49, No. 4, 2010, pp. 354-364.

[4] C. Bielza, M. Gómez and P. P. Shenoy, “A Review of Representation Issues and Modeling Challenges with Influence Diagrams,” Omega, Vol. 39, No. 3, 2010, pp. 227-241.

[5] G. F. Cooper and E. Herskovits, “A Bayesian Method for the Induction of Probabilistic Networks from Data,” Machine Learning, Vol. 9, 1992, pp. 309-348.

[6] S. L. Lauritzen, “The EM Algorithm for Graphical Association Models with Missing Data,” Computational Statistics & Data Analysis, Vol. 19, No. 2, 1995, pp. 191-201.

[7] D. Heckerman, “Bayesian Networks for Data Mining,” Data Mining and Knowledge Discovery, Vol. 1, No. 1, 1997, pp. 79-119.

[8] S. Renooij and C. Witteman, “Talking Probabilities: Communicating Probabilistic Information with Words and Numbers,” International Journal of Approximate Reasoning, Vol. 22, 1999, pp. 169-194.

[9] C. S. Spetzler and C. S. S. von Hostein, “Probability Encoding in Decision Analysis,” Manage Science, Vol. 22, No. 3 1975, pp. 340-358.

[10] H. Wang, D. Dash and M. J. Druzdzel, “A Method for Evaluating Elicitation Schemes for Probabilistic Models,” IEEE Transactions On Systems Man And Cybernetics, Part B—Cybernetics, Vol. 32, No. 1, 2002, pp. 38-43.

[11] S. Monti and G. Carenini, “Dealing with the Expert Inconsistency in Probability Elicitation,” IEEE Transactions on Knowledge and Data Engineering, Vol. 12, No. 3, 2000, pp. 499-508.

[12] D. A. Wiegmann, “Developing a Methodology for Eliciting subjective Probability Estimates during Expert evaluations of Safety Interventions: Application for Bayesian Belief Networks,” Aviation Human Factors Division, University of Illinois at Urbana-Champaign, 2005. http://www.humanfactors.uiuc.edu/Reports&PapersPDFs/TechReport/05-13.pdf

[13] A. Cano and S. Moral, “Using Probability Trees to Compute Marginals with Imprecise Probabilities,” International Journal of Approximate Reasoning, Vol. 29, No. 1, 2002, pp. 1-46.

[14] L. M. de Campos, J. F. Huete and S. Mora, “Probability Intervals: A Tool for Uncertain Reasoning,” International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, Vol. 2, No. 2, 1994, pp. 167-196.

[15] R. R. Yager and V. Kreinovich, “Decision Making under Interval Probabilities,” International Journal of Approximate Reasoning, Vol. 22, 1999, pp. 195-215.

[16] P. Guo and H. Tanaka, “Decision Making with Interval Probabilities,” European Journal of Operational Research, Vol. 203, 2010, pp. 444-454.

[17] K. Weichselberger, “The Theory of Interval-Probability as a Unifying Concept for Uncertainty,” International Journal of Approximate Reasoning, Vol. 24, 2000, pp. 149-170.

[18] J. W. Hall, D. I. Blockley and J. P. Davis, “Uncertain Inference Using Interval Probability Theory,” International Journal Of Approximate Reasoning, Vol. 19, 1998, pp. 247-264.

[19] U. B. Kjaerulff and A. L. Madsen, “Bayesian Networks and influence Diagrams: A Guide to Construction and Analysis,” Springer, Berlin, 2007.

[20] R. D. Shachter, “Evaluating Influence Diagrams,” Operations Research, Vol. 34, No. 6, 1986, pp. 871-882.

[21] F. Jensen, F. V. Jensen and S. L. Dittmer, “From Influence Diagrams to Junction Trees,” Proceeding of the Tenth Conference on Uncertainty in Artificial Intelligence, Seattle, pp. 367-373.

[22] N. L. Zhang, “Probabilistic Inference in Influence Dia- grams,” Proceedings of the 4th Conference on Uncertainty in Artificial Intelligence, Madison, Wisconsin, 1998, pp. 514-522.

[23] R. F. Nau, “The Aggregation of Imprecise Probabilities,” Journal Of Statistical Planning and Inference, Vol. 105, 2002, pp. 265-282.

[24] S. Mora and J. D. Sagrado, “Aggregation of Imprecise Probabilities,” In: B. Bouchon-Meunier, Ed., Aggregation and Fusion of Imperfect Information, Physica-Verlag, Heidelberg, 1997, pp. 162-168.

[25] F. G. Cozman, “Graphical Models for Imprecise Probabilities,” International Journal of Approximate Reasoning, Vol. 39, No. 2-3, 2005, pp. 167-184.

[26] P. Walley, “Towards a Unified Theory of Imprecise Probability,” The 1st International Symposium on Imprecise Probabilities and Their Applications, Ghent, Belgium, 1999.

[27] T. Augustin, “Generalized Basic Probability Assignments,” International Journal of General Systems, Vol. 34, No. 4, 2005, pp.451-463.

[28] R. T. Clemen and R. L. Winkler, “Combining Probability Distributions from Experts in Risk Analysis,” Risk Analysis, Vol. 19, No. 2, 1999, pp. 187-203.

[29] V. Kreinovich, “Maximum Entropy and Interval Computations,” Reliable Computing, Vol. 2, No. 1, 1996, pp. 63-79.

[1] R. A. Howard and J. E. Matheson, “Influence diagrams” In: R. A. Howard and E. M. James, Ed., Readings on the Principles and Applications of Decision Analysis, Menlo Park, 1983, pp. 719-763.

[2] C. Bielza, M. Gómez, S. Ríos-INsua, and J. A. F. D. Pozo, “Structural, Elicitation and Computational Issues Faced When Solving Complex Decision Making Problems with Influence Diagrams,” Computers & Operations Research, Vol. 27, No. 7-8, 2000, pp. 725-740.

[3] C. Bielza, M. Gómez and P. P. Shenoy, “Modeling Challenges with Influence Diagrams: Constructing Probability and Utility Models,” Decision Support Systems, Vol. 49, No. 4, 2010, pp. 354-364.

[4] C. Bielza, M. Gómez and P. P. Shenoy, “A Review of Representation Issues and Modeling Challenges with Influence Diagrams,” Omega, Vol. 39, No. 3, 2010, pp. 227-241.

[5] G. F. Cooper and E. Herskovits, “A Bayesian Method for the Induction of Probabilistic Networks from Data,” Machine Learning, Vol. 9, 1992, pp. 309-348.

[6] S. L. Lauritzen, “The EM Algorithm for Graphical Association Models with Missing Data,” Computational Statistics & Data Analysis, Vol. 19, No. 2, 1995, pp. 191-201.

[7] D. Heckerman, “Bayesian Networks for Data Mining,” Data Mining and Knowledge Discovery, Vol. 1, No. 1, 1997, pp. 79-119.

[8] S. Renooij and C. Witteman, “Talking Probabilities: Communicating Probabilistic Information with Words and Numbers,” International Journal of Approximate Reasoning, Vol. 22, 1999, pp. 169-194.

[9] C. S. Spetzler and C. S. S. von Hostein, “Probability Encoding in Decision Analysis,” Manage Science, Vol. 22, No. 3 1975, pp. 340-358.

[10] H. Wang, D. Dash and M. J. Druzdzel, “A Method for Evaluating Elicitation Schemes for Probabilistic Models,” IEEE Transactions On Systems Man And Cybernetics, Part B—Cybernetics, Vol. 32, No. 1, 2002, pp. 38-43.

[11] S. Monti and G. Carenini, “Dealing with the Expert Inconsistency in Probability Elicitation,” IEEE Transactions on Knowledge and Data Engineering, Vol. 12, No. 3, 2000, pp. 499-508.

[12] D. A. Wiegmann, “Developing a Methodology for Eliciting subjective Probability Estimates during Expert evaluations of Safety Interventions: Application for Bayesian Belief Networks,” Aviation Human Factors Division, University of Illinois at Urbana-Champaign, 2005. http://www.humanfactors.uiuc.edu/Reports&PapersPDFs/TechReport/05-13.pdf

[13] A. Cano and S. Moral, “Using Probability Trees to Compute Marginals with Imprecise Probabilities,” International Journal of Approximate Reasoning, Vol. 29, No. 1, 2002, pp. 1-46.

[14] L. M. de Campos, J. F. Huete and S. Mora, “Probability Intervals: A Tool for Uncertain Reasoning,” International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, Vol. 2, No. 2, 1994, pp. 167-196.

[15] R. R. Yager and V. Kreinovich, “Decision Making under Interval Probabilities,” International Journal of Approximate Reasoning, Vol. 22, 1999, pp. 195-215.

[16] P. Guo and H. Tanaka, “Decision Making with Interval Probabilities,” European Journal of Operational Research, Vol. 203, 2010, pp. 444-454.

[17] K. Weichselberger, “The Theory of Interval-Probability as a Unifying Concept for Uncertainty,” International Journal of Approximate Reasoning, Vol. 24, 2000, pp. 149-170.

[18] J. W. Hall, D. I. Blockley and J. P. Davis, “Uncertain Inference Using Interval Probability Theory,” International Journal Of Approximate Reasoning, Vol. 19, 1998, pp. 247-264.

[19] U. B. Kjaerulff and A. L. Madsen, “Bayesian Networks and influence Diagrams: A Guide to Construction and Analysis,” Springer, Berlin, 2007.

[20] R. D. Shachter, “Evaluating Influence Diagrams,” Operations Research, Vol. 34, No. 6, 1986, pp. 871-882.

[21] F. Jensen, F. V. Jensen and S. L. Dittmer, “From Influence Diagrams to Junction Trees,” Proceeding of the Tenth Conference on Uncertainty in Artificial Intelligence, Seattle, pp. 367-373.

[22] N. L. Zhang, “Probabilistic Inference in Influence Dia- grams,” Proceedings of the 4th Conference on Uncertainty in Artificial Intelligence, Madison, Wisconsin, 1998, pp. 514-522.

[23] R. F. Nau, “The Aggregation of Imprecise Probabilities,” Journal Of Statistical Planning and Inference, Vol. 105, 2002, pp. 265-282.

[24] S. Mora and J. D. Sagrado, “Aggregation of Imprecise Probabilities,” In: B. Bouchon-Meunier, Ed., Aggregation and Fusion of Imperfect Information, Physica-Verlag, Heidelberg, 1997, pp. 162-168.

[25] F. G. Cozman, “Graphical Models for Imprecise Probabilities,” International Journal of Approximate Reasoning, Vol. 39, No. 2-3, 2005, pp. 167-184.

[26] P. Walley, “Towards a Unified Theory of Imprecise Probability,” The 1st International Symposium on Imprecise Probabilities and Their Applications, Ghent, Belgium, 1999.

[27] T. Augustin, “Generalized Basic Probability Assignments,” International Journal of General Systems, Vol. 34, No. 4, 2005, pp.451-463.

[28] R. T. Clemen and R. L. Winkler, “Combining Probability Distributions from Experts in Risk Analysis,” Risk Analysis, Vol. 19, No. 2, 1999, pp. 187-203.

[29] V. Kreinovich, “Maximum Entropy and Interval Computations,” Reliable Computing, Vol. 2, No. 1, 1996, pp. 63-79.