Abstract
Probability plays a central role in quantifying risk and uncertainty. In contrast to the laws of probability, which are well established, there are several different interpretations of probability. This article traces the major positions of the frequency, subjective, and logical schools, and summarizes the implications of these approaches for risk professionals. The article concludes that despite profoundly different interpretations and consequences, current industry demands require professionals to be facile in using different approaches to probability.
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Further Reading
There are many wonderful sources, too numerous to mention here, on probability and inference, in addition to those already cited. von Mises (23) attempts to establish a mathematical foundation for frequency probability with infinite sequences or “collectives”. Classical texts on mathematical probability are those by Feller (24, 25) and Breiman (26). References (27, 28) are standard introductions to frequentist statistical inference. The books by Jeffreys (29) and de Finetti (30, 31) are landmark texts for subjective Bayesian probability. Kyburg et al. (32) provide a collection of historical, influential articles on subjective probability. Fishburn (33) provides a brief overview of the axioms of subjective probability. Early texts on Bayesian inference are by Box et al. (34), DeGroot (35), Ferguson (36), and Zellner (37). For theoretically inclined readers, two of the finest, modern texts on Bayesian inference are by Berger (38) and Bernardo et al. (39), and the work of Gelman et al. (40) and Congdon (41) should appeal to applied researchers. Cowell et al. (42) and Jensen (43) provide good introductions to Bayes nets.
The internet holds a wealth of material. In particular, Hájek (44) provides a scholarly account, and Fonseca (45) summarizes the axioms of rationality. Lenk (46) provides mathematical notes on Bayesian inference using Markov chain Monte Carlo. Free statistical software, mostly frequentist, but also Bayesian, is available through http://www.r-project.org/; free Bayesian software can be obtained at http://www.mrc-bsu.cam.ac.uk/bugs/, where one can also join an online community of Bayesians.
Encyclopedia of Quantitative Risk Analysis and Assessment
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