Two-stage Bayesian models—application to ZEDB project

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Abstract

A well-known mathematical tool to analyze plant specific reliability data for nuclear power facilities is the two-stage Bayesian model. Such two-stage Bayesian models are standard practice nowadays, for example in the German ZEDB project or in the Swedish T-Book, although they may differ in their mathematical models and software implementation. In this paper, we review the mathematical model, its underlying assumptions and supporting arguments. Reasonable conditional assumptions are made to yield tractable and mathematically valid form for the failure rate at plant of interest, given failures and operational times at other plants in the population. The posterior probability of failure rate at plant of interest is sensitive to the choice of hyperprior parameters since the effect of hyperprior distribution will never be dominated by the effect of observation. The methods of Pörn and Jeffrey for choosing distributions over hyperparameters are discussed. Furthermore, we will perform verification tasks associated with the theoretical model presented in this paper. The present software implementation produces good agreement with ZEDB results for various prior distributions. The difference between our results and those of ZEDB reflect differences that may arise from numerical implementation, as that would use different step size and truncation bounds.

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 i=1n[P(Xi|λi)P(λi|q)dλi] 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

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