Bayesian parameter estimation in probabilistic risk assessment

doi:10.1016/S0951-8320(97)00159-2

Reliability Engineering & System Safety

Volume 62, Issues 1–2, October–November 1998, Pages 89-116

https://doi.org/10.1016/S0951-8320(97)00159-2 Get rights and content

Abstract

Bayesian statistical methods are widely used in probabilistic risk assessment (PRA) because of their ability to provide useful estimates of model parameters when data are sparse and because the subjective probability framework, from which these methods are derived, is a natural framework to address the decision problems motivating PRA. This paper presents a tutorial on Bayesian parameter estimation especially relevant to PRA. It summarizes the philosophy behind these methods, approaches for constructing likelihood functions and prior distributions, some simple but realistic examples, and a variety of cautions and lessons regarding practical applications. References are also provided for more in-depth coverage of various topics.

Introduction

A probabilistic risk assessment (PRA) is an exercise aimed at estimating the probability and consequences of accidents for the facility or process being studied. Although not required by this definition, PRA often involves the analysis of low-probability events for which few data are available. Bayesian parameter estimation techniques are useful because, unlike classical statistical methods, they are able to incorporate a wide variety of information types, e.g. expert judgement as well as statistical data, in the estimation process.

In addition to their ability to deal with sparse data, Bayesian techniques are appropriate for use in PRA because they are derived from the framework of subjective probability. This framework, which holds that probability is a subjective (internal) measure of event likelihood, is an integral part of current theories on decision-making under uncertainty (e.g. [1]). A further, practical advantage of the subjective probability framework in PRA applications is that propagation of uncertainties through complex models is relatively simple. On the other hand, it is very difficult, and intractable in `real' problems, to propagate classical statistical confidence intervals through PRA models to estimate a confidence interval for a composite result of interest (e.g.reactor core damage frequency).

For these reasons, Bayesian techniques, first championed for use in PRA in the late 1970s 2, 3, have become widespread in that field. This paper presents a tutorial on Bayesian techniques especially relevant to PRA. It outlines the motivation behind the Bayesian approach, summarizes the general parameter estimation problem, presents methods for constructing likelihood functions and prior distributions, discusses some useful results, and ends with a variety of cautions and comments regarding practical applications. The intent is to provide an expository review with simple but realistic examples. A number of references are provided for readers interested in exploring various topics in greater detail.

Section snippets

Motivation and framework

The following discussion provides an introduction to the issues motivating the use of Bayesian methods in PRA and outlines the parameter estimation problem that is the focus of this paper. More detailed discussions can be found in [4] and [5].

Bayes's theorem

According to the theory of subjective probability, Bayes's theorem is the only way in which a coherent analyst, i.e. an analyst whose probabilities behave according to the laws of probability [8], can update his/her state of knowledge. This section describes the general features of Bayes's theorem and introduces a simple application of it. Additional material, both introductory and advanced, can be found in numerous texts (e.g. [9], [10]).

The likelihood function

The construction of an appropriate likelihood function requires engineering/scientific knowledge specific to the process being modeled, as well as probabilistic modeling expertise. This section raises a number of general points on modeling considerations and provides some functions widely used in PRAs. A special topic relevant to the construction of the likelihood function, i.e. the treatment of uncertain data, is discussed in Section 7.

Prior distributions

Development of an appropriate prior distribution is, sometimes with just cause, the most controversial part of a Bayesian analysis. It can often be the most resource-intensive as well. This section discusses some of the issues to be considered when creating an informative prior distribution (i.e. a prior distribution that at least partly reflects the analyst's state of knowledge) as well as a number of so-called `objective', noninformative prior distributions.

A numerical example

We now illustrate the concepts presented in the preceding sections with a simplified example. Assume that our population of interest consists of 10 emergency diesel generators (EDGs), similar to those used by commercial nuclear power plants in the USA as sources of emergency power. Assume that our task is to estimate φ₁, the probability that the diesel at our plant (designated EDG no.1) will fail to start on demand. At our plant, there have been 140 successful demands (i.e. no failures). The

Implementation notes and other special topics

With the development of a prior distribution and a likelihood function, the estimation process is fairly straightforward for problems of common interest in PRA; it often requires only the solution (analytical or numerical) of Bayes's theorem for a single unknown parameter. This section briefly discusses a number of topics relevant to Eq. (5), including some common pitfalls. It also introduces an advanced topic relevant to current PRA applications: the treatment of uncertain data.

Concluding remarks

To many PRA analysts, `Bayesian analysis' can mean a mechanical updating of either a roughly defined informative prior distribution (i.e. a prior distribution which only roughly approximates the analyst's belief concerning the parameter of interest) or some form of `objective' prior distribution (e.g. a noninformative or maximum entropy prior). While Bayes's theorem is straightforward in principle, and while practical Bayesian analysis for PRA applications is often a fairly simple process,

Acknowledgements

The authors gratefully acknowledge the helpful comments of G. Apostolakis, A. Mosleh, T. Leahy and the anonymous referees, and the assistance of D. Siu in preparing the manuscript. This paper was supported by the Idaho National Engineering Laboratory Center for Reliability and Risk Assessment under the US Department of Energy (DOE) Idaho Field Office Contract DE-AC07-76ID01570. The opinions, findings, conclusions and recommendations expressed herein are those of the authors and do not

References (54)

A. Mosleh
Hidden sources of uncertainty: judgment in the collection and analysis of data
Nuclear Engineering and Design
(1986)
G. Apostolakis et al.
Data specialization for plant specific risk studies
Nuclear Engineering and Design
(1980)
W.E. Vesely
More comprehensive approaches for determining risk sensitivities to component aging
Nuclear Engineering and Design
(1988)
K. Pörn
The two-stage Bayesian method used for the T-Book application
Reliability Engineering and System Safety
(1996)
C.L. Atwood
Constrained non-informative priors in risk assessment
Reliability Engineering and System Safety
(1996)
C.A. Clarotti et al.
Response: On the `true' value of a failure rate and related statistical issues
Reliability Engineering and System Safety
(1989)
N. Siu
A Monte Carlo method for multiple parameter estimation in the presence of uncertain data
Reliability Engineering and System Safety
(1990)
Keeney, R. L. and Raiffa, H., Decisions with Multiple Objectives. Cambridge University Press,...
G. Apostolakis
Probability and risk assessment: the subjectivistic viewpoint and some suggestions
Nuclear Safety
(1978)
G.W. Parry et al.
Characterization and evaluation of uncertainty in probabilistic risk analysis
Nuclear Safety
(1981)

G. Apostolakis

The concept of probability in safety assessments of technological systems

Science

(1990)

S. Kaplan et al.

On the quantitative definition of risk

Risk Analysis

(1981)

Apostolakis, G., A commentary on model uncertainty. In Model Uncertainty: its Characterization and Quantification, eds...

De Finetti, B., Theory of Probability. Wiley, New York,...

Lindley, D. V., Introduction to Probability and Statistics from a Bayesian Viewpoint, Part 2: Inference. Cambridge...

Winkler, R. L. and Hays, W. L., Statistics: Probability, Inference, and Decision, 2nd edn. Holt, Rinehart and Winston,...

Kahneman, D., Slovic, P. and Tversky, A., Judgment under Uncertainty: Heuristics and Biases. Cambridge University...

Jeffreys, H., Theory of Probability, 3rd edn. Clarendon Press, Oxford,...

Papoulis, A., Probability, Random Variables, and Stochastic Processes. McGraw-Hill, New York,...

G. Apostolakis et al.

Expert opinion and statistical evidence: an application to reactor core melt frequency

Nuclear Science and Engineering

(1979)

West, M., Modelling expert opinion. In Bayesian Statistics 3, eds J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A....

A. Mosleh et al.

The assessment of probability distributions from expert opinions with an application to seismic fragility curves

Risk Analysis

(1986)

H.F. Martz et al.

A statistical model for combining biased expert opinions

IEEE Transactions on Reliability R-

(1984)

S. Kaplan

On a `two-stage' Bayesian procedure for determining failure rates

IEEE Transactions on Power Apparatus and Systems PAS-

(1983)

J.J. Deely et al.

Bayes empirical Bayes

Journal of the American Statistical Association

(1986)

Dang, V. N., Chu, T.-L. and Siu, N., Lognormal extension of the two-stage Bayesian method: applications from the Surry...

Kazarians, M. and Apostolakis, G., Modeling rare events: the frequencies of fires in nuclear power plants. In Low...

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¹: The views and conclusions in this paper are those of the authors and should not be interpreted as necessarily representing the views or official policies, either expressed or implied, of the US Nuclear Regulatory Commission or the US Department of Energy.

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Bayesian parameter estimation in probabilistic risk assessment1

Abstract

Introduction

Section snippets

Motivation and framework

Bayes's theorem

The likelihood function

Prior distributions

A numerical example

Implementation notes and other special topics

Concluding remarks

Acknowledgements

Nuclear Engineering and Design

Nuclear Engineering and Design

Nuclear Engineering and Design

Reliability Engineering and System Safety

Reliability Engineering and System Safety

Reliability Engineering and System Safety

Reliability Engineering and System Safety

Probability and risk assessment: the subjectivistic viewpoint and some suggestions

Nuclear Safety

Characterization and evaluation of uncertainty in probabilistic risk analysis

Nuclear Safety

The concept of probability in safety assessments of technological systems

Science

On the quantitative definition of risk

Risk Analysis

Expert opinion and statistical evidence: an application to reactor core melt frequency

Nuclear Science and Engineering

The assessment of probability distributions from expert opinions with an application to seismic fragility curves

Risk Analysis

A statistical model for combining biased expert opinions

IEEE Transactions on Reliability R-

On a `two-stage' Bayesian procedure for determining failure rates

IEEE Transactions on Power Apparatus and Systems PAS-

Bayes empirical Bayes

Journal of the American Statistical Association

Bayesian parameter estimation in probabilistic risk assessment¹