Abstract
This chapter considers aleatory models that are used frequently in probabilistic modeling situations typical of PRA. These three most commonly used aleatory models are the binomial, Poisson, and exponential distributions. For each of these three distributions, we demonstrate the Bayesian inference process for three general categories of prior distribution: conjugate, noninformative, and nonconjugate prior distributions. Lastly, we describe how prior distributions may be specified, including some cautions for developing an informative prior, and we introduce the concept of a Bayesian p-value for checking the predictions of the model against the observed data.
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- 1.
Because the posterior credible intervals obtained from updating the “zero–zero” prior do not provide adequate coverage, we do not advocate the routine use of this prior as a replacement for the Jeffreys prior. The zero–zero prior is systematically biased low, in the anti-conservative direction. A Bayesian making supposedly fair bets would not come out even in the long run. Also, choosing a prior after examining the data violates the spirit of Bayesian inference.
- 2.
Note that in a reliability analysis, as opposed to a PRA, the unknown parameter of interest might be q = 1−p, in which case q would be close to one for highly reliable components.
- 3.
Note that the parameters of the logistic-normal distribution are not related to the mean and error factor in as simple a fashion as was the case for the lognormal distribution; the mean and higher moments of the logistic-normal distribution must be calculated numerically. Thus, for simplicity, we use the median and error factor, for which the relations are algebraic, as shown in the script
- 4.
As discussed for the binomial distribution earlier, we do not advocate the routine use of the “zero–zero” gamma prior as a replacement for the Jeffreys prior.
- 5.
The “information provided” represents the analyst’s state of knowledge for the system or component being evaluated and must be independent from any data to be used in updating the prior distribution.
- 6.
We use a spreadsheet tool in what follows, but a more accurate alternative is the Parameter Solver software, developed by the M. D. Anderson Cancer Center. It can be downloaded free of charge from https://biostatistics.mdanderson.org/SoftwareDownload/ProductDownloadFiles/ParameterSolver_V2.3_WithFX1.1.exe
Reference
Siu NO, Kelly DL (1998) Bayesian parameter estimation in probabilistic risk assessment. Reliab Eng Syst Saf 62:89–116
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© 2011 Springer-Verlag London Limited
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Kelly, D., Smith, C. (2011). Bayesian Inference for Common Aleatory Models. In: Bayesian Inference for Probabilistic Risk Assessment. Springer Series in Reliability Engineering. Springer, London. https://doi.org/10.1007/978-1-84996-187-5_3
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DOI: https://doi.org/10.1007/978-1-84996-187-5_3
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