Two-stage Bayesian models—application to ZEDB project
Introduction
ZEDB [1] is the major German effort to collect data from nuclear facilities. The goal of the project is to create a reliability database, which contains all major plant events: failure events, operational experience, and maintenance actions. As a mathematical tool to analyse ZEDB data, a two-stage Bayesian model was chosen. Firstly, we identify the standard conditional independence assumptions and derive the general form of the posterior distribution for failure rate λ0 at plant of interest 0, given failures and observation times at plants 0,1,…,n. Any departure for the derived mathematical form necessarily entails a departure from the conditional independence assumptions. Vaurio's one-stage empirical Bayes model is discussed as an alternative to the two-stage model [2]. Hofer [3], [4], [5] criticized the standard two-stage model and proposed an alternative, which is also discussed. Finally, the methods of Pörn and Jeffrey for choosing a non-informative hyperprior distribution are discussed.
Section snippets
Bayesian two-stage hierarchical models
Bayesian two-stage or hierarchical models are widely employed in a number of areas. The common theme of these applications is the assimilation of data from different sources, as shown in Fig. 1. The data from agent i is characterized by an exposure Ti and a number of events Xi. The exposure Ti is not considered stochastic, as it can usually be observed with certainty. The number of events for a given exposure follows a fixed distribution type, in this case Poisson. The parameter(s) of this
ZEBD software verification
Three datasets are used to check the concordance with the results from [16]. Differences between our results and those of ZEDB reflect differences that may arise from an independent implementation based on public information. Although ZEDB recommends the lognormal model, both the lognormal and gamma models are supported, and both are benchmarked here. The datasets are presented in Table 2, Table 3, Table 4 (the underlined field is the plant of interest).
Truncation
Using a gamma prior, the method of truncation seems to have a large influence on the posterior distribution of λ. It has been shown in Section 2.3 that the likelihood in α and β has no maximum, but it is asymptotically maximal along a ridge. Cooke et al. [8] showed that different choices of truncation ranges can affect the median and the 95% quantile by a factor 5. In (4), the term cannot be calculated analytically when we have a lognormal distribution as prior for λ
Conclusions
Two-stage models provide a valid method for assimilating data from other plants. The conditional independence assumptions are reasonable and yield a tractable and mathematically valid form for the failure rate a plant of interest, given failures and operational times at other plants in the population. However, the choice of hyperprior must be defensible since improper hyperpriors do not always become proper after observations. The lognormal model enjoys a significant advantage over the gamma
References (20)
On two-stage Bayesian modeling of initiating event frequencies and failure rates
Reliab Eng Syst Saf
(1999)- et al.
ZEDB—a practical application of a 2-stage Bayesian analysis code, ESREL 1998 conference, Trøndheim, Norway
On analytic empirical Bayes estimation of failure rates
Risk Anal
(1987)- et al.
On the solution approach for Bayesian modeling of initiating event frequencies and failure rates
Risk Anal
(1997) - et al.
Bayesian modeling of failure rates and initiating event frequencies, ESREL 1999, Munich, Germany
- Pörn K. On empirical Bayesian inference applied to Poisson probability models. Linköping studies in science and...
- et al.
Bayesian modeling of initiating event frequencies at nuclear power plants
Risk Anal
(1990) - Cooke RM, Dorrepaal J, Bedford TJ. Review of SKI data processing methodology SKI report;...
- Cooke RM, Bunea C, Charitos T, Mazzuchi TA. Mathematical review of ZEDB two-stage Bayesian models. Report 02-45,...
Real analysis
(1968)
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