Bayesian inference in probabilistic risk assessment—The current state of the art

doi:10.1016/j.ress.2008.07.002

Reliability Engineering & System Safety

Volume 94, Issue 2, February 2009, Pages 628-643

https://doi.org/10.1016/j.ress.2008.07.002 Get rights and content

Abstract

Markov chain Monte Carlo (MCMC) approaches to sampling directly from the joint posterior distribution of aleatory model parameters have led to tremendous advances in Bayesian inference capability in a wide variety of fields, including probabilistic risk analysis. The advent of freely available software coupled with inexpensive computing power has catalyzed this advance. This paper examines where the risk assessment community is with respect to implementing modern computational-based Bayesian approaches to inference. Through a series of examples in different topical areas, it introduces salient concepts and illustrates the practical application of Bayesian inference via MCMC sampling to a variety of important problems.

Introduction

The advent of Markov chain Monte Carlo (MCMC) sampling has proliferated Bayesian inference throughout the world, across a wide array of disciplines. The freely available software package known as Bayesian inference using Gibbs sampling (BUGS) has been in the vanguard of this proliferation since the mid-1990s [1]. However, more recent advances in this software, leading first to WinBUGS and now to an open-source version (OpenBUGS), including interfaces to the open-source statistical package R [2], have brought MCMC to a wider audience. Problems which would have been intractable a decade ago can now be solved in short order with these software packages. We will use this software to illustrate several applications from the field of risk assessment. While our examples are from applications of risk assessment to technology, it should be noted that there have also been many applications of Bayesian inference in behavioral science [3], finance [4], human health [5], process control [6], and ecological risk assessment [7]. We mention these other applications to indicate the degree to which Bayesian inference is being used in the wider community, including other branches of risk assessment; however, our principal focus is on applications of risk assessment to engineered systems (including non-nuclear technology). Within this focus, we will explore specific Bayesian topics and examples in greater detail, including

•
Hierarchical modeling of variability
•
Modeling of time-dependent reliability (with and without repair)
•
Modeling of random durations, such as the time to suppress a fire or recover power
•
Treatment of uncertain and missing data
•
Regression models
•
Model selection and validation

Siu and Kelly's earlier work [8] is the starting point for this paper, as it presented a tutorial on Bayesian inference for probabilistic risk assessment, and the elementary portions of that paper remain vital today. For the reader needing a more basic introduction to Bayesian inference, that paper is a good starting point, and it also provides a good point of entry to the vast (and vastly expanding) literature on Bayesian inference. However, advances have been made since the publication of that paper in 1998, and it is to these advances that the current paper is devoted. Additional relevant information on the Bayesian approach may be found in [9] and [10] has become a popular practical reference for the PRA community.

Section snippets

Background

First, we provide a brief overview of key concepts relevant to later discussion. In general, we have a goal of performing inference calculations, starting with data. “Data” are the observed values of a physical process and may be subject to uncertainties, such as imprecision in measurement, censoring, and interpretation errors. Next, at a higher level from data, we have “information,” which is the result of evaluating, manipulating, or organizing data and other information in a way that adds to

Hierarchical Bayesian modeling of variability

As discussed by Siu and Kelly [8], treatment of variability that can exist among sources of data is important if total uncertainty, including population variability, is to be properly represented by the resulting posterior distribution. Siu and Kelly [8] discusses hierarchical Bayesian analysis briefly, but (due to software and computer limitations that existed at that time) focuses on an older approach of Kaplan [14] called “two-stage Bayes,” and on (parametric) empirical Bayes. As pointed out

Modeling time trends

It is sometimes the case that the usual Poisson and binomial models are rendered invalid because the parameter of interest (λ or p, respectively) is not constant over time. In the US for example, recent data analysis in [18] has suggested decreasing values of λ and p for several important components. As an example, let us examine valve leakage data from [19]. These data are shown in Table 5.

We first carry out a qualitative check to see if there appears to be any systematic time trend in p. To

Modeling random durations

There are many instances in which time is the random variable of interest. For example, a facility may be threatened by fire, and the time of interest is the time needed to suppress the fire, which must be shorter than the time to damage vital equipment (which is also a random variable). Another important application for many modern facilities is inference on the time needed to restore ac power once it has been lost. And of course such models can be used for the renewal process described above.

Treatment of missing and uncertain data

It is not uncommon in risk applications to encounter situations in which the observed data, which would normally enter into Bayes’ theorem via the likelihood function, are either missing or the exact values are not known with certainty. For example, in the simple Poisson aleatory model used for initiating event frequencies and failure rates, one may not know the exposure time, t, accurately. Sometimes only an interval estimate is available. This is especially the case for failure rates, less

Bayesian regression models

Regression models are commonplace in statistics, and have been applied in risk analysis, as well. An example discussed earlier is the use of a logit or loglinear model for trend in a binomial or Poisson parameter. In these models, time was the predictor variable. In human reliability analysis (HRA), regression models have been used to estimate human error probabilities. The original SLIM (see [34]) and its derivatives are examples. We will examine a simple example from the published literature.

Bayesian model validation

The frequentist approach to model checking or validation typically involves comparing the observed value of a test statistic to percentiles of the sampling distribution for that statistic. Given that the null hypothesis is true, we would not expect to see “extreme” values of the test statistic. One Bayesian approach to model checking involves calculating the posterior probability of the various hypotheses and choosing the one that is most likely. One can also use summary statistics derived from

Other models

A variety of other “models of the world” are amenable to inference via Bayes’ Theorem, including Bayesian belief networks (BBN), influence diagrams, and fault trees. Graph models such as BBNs and influence diagrams are very similar to the directed acyclic graphs discussed earlier. In fact, an influence diagram is just a directed acyclic graph to which decision nodes have been added, as discussed in [47]. An example of these types of models is shown in Fig. 11, where we list six possible

Conclusions

In the scientific and engineering communities, we rely on mathematical models of reality, both deterministic and aleatory. These models contain parameters—whose values are estimated from information—of which data are a subset. Uncertain parameters (in the epistemic sense) are inputs to the models used to infer the values of future observables, leading to an increase in scientific knowledge. Further, these parameters may be known to high precision and thus have little associated epistemic

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View full text

Bayesian inference in probabilistic risk assessment—The current state of the art

Abstract

Introduction

Section snippets

Background

Hierarchical Bayesian modeling of variability

Modeling time trends

Modeling random durations

Treatment of missing and uncertain data

Bayesian regression models

Bayesian model validation

Other models

Conclusions

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Int J Approx Reason

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Reliab Eng

WinBUGS—a Bayesian modelling framework: concepts, structure, and extensibility

Stat Comput

Bayesian methods: a social and behavioral sciences approach

Contemporary Bayesian econometrics and statistics

Statistical methods in spatial epidemiology

Bayesian process monitoring, control, and optimization

Hierarchical modelling for the environmental sciences

Probability theory—the logic of science

Handbook of parameter estimation for probabilistic risk assessment

An introduction to Bayesian inference and decision

The concept of probability in safety assessments of technological systems

Science

On a two-stage Bayesian procedure for determining failure rates

IEEE Trans Power Apparatus Syst

Probabilistic reasoning in intelligent systems: networks of plausible inference

Inference and monitoring convergence

Industry-average performance for components and initiating events at US commercial nuclear power plants

Repairable systems reliability