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• #### Appendix M: Acronyms 157-158

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Appendix G On the Quantification of Uncertainty and Enhancing Probabilistic Risk Analysis Nozer D. Singpurwalla Professor, Department of Statistics George Washington University, Washington, D.C. PREAMBLE (or the objective chances) of all possible outcomes (also known as consequences) of an action, and the utilities of This appendix consists of two parts. In Part 1, we over- each outcome. Probabilities and chances are ways to quan- view some commonly used approaches for quantifying tify uncertainty (i.e., the possibility mentioned above), and uncertainty. The overview is necessarily terse, but adequate quantification is a necessary step for invoking the logical references are provided. Herein we introduce the notions argument. Utilities are numerical values of the consequences of chance, probability, likelihood, belief, and plausibility, of each outcome, on a zero to one scale. Indeed, utilities terms that commonly arise in the context of risk analysis. are probabilities and must therefore obey the rules (or the Also mentioned here are the notions of consequences and calculus) of probability (cf. Lindley, 1985, p. 56). They utilities, both of which are germane to risk analysis and quantify one’s preferences between consequences. Thus the risk management. Part 1 can serve as a supplement to the modern notion of risk entails the twin notions of probability “Lexicon of Probabilistic Risk Assessment Terms” given in (or chance) and utility. Its computation via the sum of prod- Appendix A of this report. ucts rule mentioned above (cf. Morgeson et al. [2006] for a In Part 2 we put forth some thoughts and ideas for enhanc- detailed application of this principle to terrorist risk assess- ing PRA (Probabilistic Risk Analysis) with some statistical ment) is a consequence of the calculus of probability. The and decision theoretic methodologies that are available in the quantification of uncertainty by probability is, according to literature, and which could be advantageously invoked. We de Finetti (1972) and Lindley (1982), the only satisfactory close this section by alluding to the possibility of some new way. Alternatives to probability, like Zadeh’s (1979) possibil- research in PRA, namely, the development of an architecture ity, do not lead to a prescription for the quantification of risk; for adversarial risk analysis and decision making in vague this is one of its biggest drawbacks. (or fuzzy) environments. It is our hope that this appendix will fill in any gaps of interpretation of the Lexicon that is given in the text, so that Chance and Probability: Metrics for this appendix and the Lexicon of Appendix A are linked. To Quantifying Uncertainty better facilitate a broad based appreciation of the material The use of probability as a metric for quantifying uncer- presented here, this appendix has been deliberately cast in tainty dates back to the 16th century. However, discussions a conversational style. That is, mathematical notation has about its meaning and interpretation continue until today. been avoided. The distinction between chance and probability (cf. Good, 1990) is a consequence of such debates and discussions. In PART 1.  APPROACHES TO QUANTIFYING his review article, Kolmogorov (1969) wholeheartedly sub- UNCERTAINTY scribes to probability as an objective chance that is agreed upon by all even though it can never be observed. It is defined Introduction as the limit of a relative frequency; the operational word being “limit.” To Kolmogorov, chance and probability were From a layperson’s point of view, the term “risk” connotes synonymous, and thus the word chance does not appear in his the possibility that an undesirable event will occur. However, writings. To de Finetti (1976) and others, like Savage (1972), the modern technical meaning of the term is different. Here, probability is subjective and personal, and encapsulates ones risk is the sum of the product of one’s personal probabilities disposition to a two-sided bet. De Finetti (1972) goes further 111

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