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Combined analysis of unique and repetitive events in quantitative risk assessment. (English) Zbl 1352.62030
Summary: For risk assessment to be a relevant tool in the study of any type of system or activity, it needs to be based on a framework that allows for jointly analyzing both unique and repetitive events. Separately, unique events may be handled by predictive probability assignments on the events, and repetitive events with unknown/uncertain frequencies are typically handled by the probability of frequency (or Bayesian) approach. Regardless of the nature of the events involved, there may be a problem with imprecision in the probability assignments. Several uncertainty representations with the interpretation of lower and upper probability have been developed for reflecting such imprecision. In particular, several methods exist for jointly propagating precise and imprecise probabilistic input in the probability of frequency setting. In the present position paper we outline a framework for the combined analysis of unique and repetitive events in quantitative risk assessment using both precise and imprecise probability. In particular, we extend an existing method for jointly propagating probabilistic and possibilistic input by relaxing the assumption that all events involved have frequentist probabilities; instead we assume that frequentist probabilities may be introduced for some but not all events involved, i.e. some events are assumed to be unique and require predictive – possibly imprecise – probabilistic assignments, i.e. subjective probability assignments on the unique events without introducing underlying frequentist probabilities for these. A numerical example related to environmental risk assessment of the drilling of an oil well is included to illustrate the application of the resulting method.

##### MSC:
 62C86 Statistical decision theory and fuzziness 62A01 Foundations and philosophical topics in statistics 68T37 Reasoning under uncertainty in the context of artificial intelligence
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##### References:
 [1] Aguirre, F.; Destercke, S.; Dubois, D.; Sallak, M.; Jacob, C., Inclusion-exclusion principle for belief functions, Int. J. Approx. Reason., 55, 8, 1708-1727, (2014) · Zbl 1433.68437 [2] Aven, T., On the need for restricting the probabilistic analysis in risk assessments to variability, Risk Anal., 30, 3, 354-360, (2010), with discussion, pp. 381-384 [3] Aven, T., A risk concept applicable for both probabilistic and non-probabilistic perspectives, Saf. Sci., 49, 1080-1086, (2011) [4] Aven, T., On when to base event trees and fault trees on probability models and frequentist probabilities in quantitative risk assessments, Int. J. Perform. Eng., 8, 3, 311-320, (2012) [5] Aven, T., Foundations of risk analysis, (2012), Wiley Chichester · Zbl 1237.91001 [6] Aven, T.; Reniers, G., How to define and interpret a probability in a risk and safety setting, Saf. Sci., 51, 1, 223-231, (2013) [7] Aven, T.; Renn, O., The role of quantitative risk assessments for characterizing risk and uncertainty and delineating appropriate risk management options, with special emphasis on terrorism risk, Risk Anal., 29, 4, 587-600, (2009) [8] Aven, T.; Zio, E., Some considerations on the treatment of uncertainties in risk assessment for practical decision making, Reliab. Eng. Syst. Saf., 96, 1, 64-74, (2011) [9] T. Aven, P. Baraldi, R. Flage, E. Zio, Uncertainty in risk assessment: the representation and treatment of uncertainties by probabilistic and non-probabilistic methods, Wiley-Blackwell, Chichester. [10] Bernardo, J. M.; Smith, A. F.M., Bayesian theory, (1994), Wiley Chichester · Zbl 0796.62002 [11] Baudrit, C.; Couso, I.; Dubois, D., Joint propagation of probability and possibility in risk analysis: towards a formal framework, Int. J. Approx. Reason., 45, 1, 82-105, (2007) · Zbl 1123.68123 [12] Baudrit, C.; Dubois, D.; Guyonnet, D., Joint propagation and exploitation of probabilistic and possibilistic information in risk assessment, IEEE Trans. Fuzzy Syst., 14, 5, 593-608, (2006) [13] Cooper, J. A.; Ferson, S.; Ginzburg, L., Hybrid processing of stochastic and subjective uncertainty data, Risk Anal., 16, 6, 785-791, (1996) [14] De Cooman, G.; Aeyels, D., Supremum preserving upper probabilities, Inf. Sci., 118, 173-212, (1999) · Zbl 0952.60009 [15] De Finetti, B., La prévision: ses lois logiques, ses sources subjectives, Ann. Inst. Henri Poincaré, 7, 1-68, (1937) · JFM 63.1070.02 [16] Dempster, A. P., Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Stat., 38, 325-339, (1967) · Zbl 0168.17501 [17] Dubois, D., Possibility theory and statistical reasoning, Comput. Stat. Data Anal., 51, 47-69, (2006) · Zbl 1157.62309 [18] Dubois, D., Representation, propagation and decision issues in risk analysis under incomplete probabilistic information, Risk Anal., 30, 361-368, (2010) [19] Dubois, D.; Kerre, E.; Mesiar, R.; Prade, H., Fuzzy interval analysis, (Dubois, D.; Prade, H., Fundamentals of Fuzzy Sets, Handb. Fuzzy Sets Ser., (2000), Kluwer Boston, MA), 483-581 · Zbl 0988.26020 [20] Dubois, D.; Foulloy, L.; Mauris, G.; Prade, H., Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities, Reliab. Comput., 10, 273-297, (2004) · Zbl 1043.60003 [21] Dubois, D.; Prade, H., Random sets and fuzzy interval analysis, Fuzzy Sets Syst., 42, 1, 87-101, (1991) · Zbl 0734.65041 [22] Dubois, D.; Prade, H., Formal representations of uncertainty, (Bouyssou, D.; Dubois, D.; Pirlot, M.; Prade, H., Decision-Making Process - Concepts and Methods, (2009), ISTE London & Wiley), 85-156, Chap. 3 [23] Dubois, D.; Prade, H.; Smets, P., Representing partial ignorance, IEEE Trans. Syst. Man Cybern., 26, 3, 361-377, (1996) [24] Flage, R., Contributions to the treatment of uncertainty in risk assessment and management, (2010), University of Stavanger Norway, PhD Thesis UiS no. 100 [25] Guyonnet, D.; Bourgine, B.; Dubois, D.; Fargier, H.; Côme, B.; Chilès, J. P., Hybrid approach for addressing uncertainty in risk assessments, J. Environ. Eng., 129, 68-78, (2003) [26] Hacking, I., Logic of statistical inference, (1965), Cambridge University Press Cambridge, UK · Zbl 0133.41604 [27] Jacob, C.; Dubois, D.; Cardoso, J., Uncertainty handling in quantitative BDD-based fault-tree analysis by interval computation, (Benferhat, S.; Grant, J., Int. Conf. on Scalable Uncertainty Management, SUM 2011, Dayton, Ohio, Lect. Notes Comput. Sci., vol. 6929, (2011), Springer), 205-218 [28] Kaplan, S., Will the real probability please stand up?, Reliab. Eng. Syst. Saf., 23, 4, 285-292, (1988) [29] Kaplan, S., The words of risk analysis, Risk Anal., 17, 407-417, (1997) [30] Kaplan, S.; Garrick, B. J., On the quantitative definition of risk, Risk Anal., 1, 1, 11-27, (1981) [31] Limbourg, P.; Savic, R.; Petersen, J.; Kochs, H.-D., Fault tree analysis in an early design stage using the Dempster-Shafer theory of evidence, (Aven, T.; Vinnem, J. E., Risk, Reliability and Societal Safety, (2007), Taylor & Francis Group London), 713-722 [32] Lindley, D. V., The philosophy of statistics, Statistician, 49, 3, 293-337, (2000) [33] Lindley, D. V., Understanding uncertainty, (2006), Wiley Hoboken, NJ · Zbl 1135.60002 [34] Lindley, D. V.; Phillips, L. D., Inference for a Bernoulli process (a Bayesian view), Am. Stat., 30, 3, 112-119, (1976) · Zbl 0335.62025 [35] Lodwick, W. A., Fundamentals of interval analysis and linkage to fuzzy set theory, (Pedrycz, W.; Skowron, A.; Kreinovich, V., Handbook of Granular Computing, (2008), John Wiley & Sons Ltd.) [36] Mauris, G., Inferring a possibility distribution from very few measurements, (Dubois, D.; etal., Soft Methods for Handling Variability and Imprecision, SMPS 2008, Adv. Soft Comput., (2008), Springer), 92-99 [37] Montgomery, V., New statistical methods in risk assessment by probability bounds, (2009), Durham University, PhD thesis [38] Moore, R., Methods and applications of interval analysis, SIAM Stud. Appl. Math., (1979), Philadelphia [39] Ross, S. M., Stochastic processes, (1996), Wiley · Zbl 0888.60002 [40] Shafer, G., A mathematical theory of evidence, (1976), Princeton University Press · Zbl 0359.62002 [41] (2011), Interpretations of probability, in: Stanford Encyclopedia Philosophy (SEP) [42] Singpurwalla, N. D., Reliability and risk: A Bayesian perspective, (2006), Wiley Chichester · Zbl 1152.62070 [43] Singpurwalla, N. D.; Wilson, A. G., Probability, chance and the probability of chance, IIE Trans., 41, 12-22, (2009) [44] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall London · Zbl 0732.62004 [45] Weichselberger, K., The theory of interval-probability as a unifying concept for uncertainty, Int. J. Approx. Reason., 24, 149-170, (2000) · Zbl 0995.68123 [46] Zadeh, L. A., The concept of a linguistic variable and its application to approximate reasoning. I, Inf. Sci., 8, 199-249, (1975) · Zbl 0397.68071
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