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2011 | Buch

Introduction and Motivation Kapitel lesen Erstes Kapitel lesen

Bayesian Inference for Probabilistic Risk Assessment

A Practitioner's Guidebook

verfasst von: Dana Kelly, Curtis Smith

Verlag: Springer London

Buchreihe : Springer Series in Reliability Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Bayesian Inference for Probabilistic Risk Assessment provides a Bayesian foundation for framing probabilistic problems and performing inference on these problems. Inference in the book employs a modern computational approach known as Markov chain Monte Carlo (MCMC). The MCMC approach may be implemented using custom-written routines or existing general purpose commercial or open-source software. This book uses an open-source program called OpenBUGS (commonly referred to as WinBUGS) to solve the inference problems that are described. A powerful feature of OpenBUGS is its automatic selection of an appropriate MCMC sampling scheme for a given problem. The authors provide analysis “building blocks” that can be modified, combined, or used as-is to solve a variety of challenging problems.

The MCMC approach used is implemented via textual scripts similar to a macro-type programming language. Accompanying most scripts is a graphical Bayesian network illustrating the elements of the script and the overall inference problem being solved. Bayesian Inference for Probabilistic Risk Assessment also covers the important topics of MCMC convergence and Bayesian model checking.

Bayesian Inference for Probabilistic Risk Assessment is aimed at scientists and engineers who perform or review risk analyses. It provides an analytical structure for combining data and information from various sources to generate estimates of the parameters of uncertainty distributions used in risk and reliability models.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction and Motivation
Abstract
This chapter introduces and describes, at a high level, the topic of Bayesian inference. The overall motivation and background for a formal method of logical inference using probability is discussed. Key terms used to describe the Bayesian inference approach are also defined.
Dana Kelly, Curtis Smith
Chapter 2. Introduction to Bayesian Inference
Abstract
This chapter describes the interpretation of the components of Bayes’ Theorem. The relevant parts of the theorem are described, and a simple example is demonstrated using both a discrete and continuous prior distribution.
Dana Kelly, Curtis Smith
Chapter 3. Bayesian Inference for Common Aleatory Models
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.
Dana Kelly, Curtis Smith
Chapter 4. Bayesian Model Checking
Abstract
This chapter discusses the Bayesian approach to checking the reasonableness of the model, which consists of both the aleatory model describing the occurrence of observable quantities, and the prior distribution for the parameters of the aleatory model. It focuses on the use of posterior predictive checks on the putative model. Approaches include direct use of the posterior predictive distribution and summary statistics derived from this distribution. Both graphical checks and quantitative checks are covered. The latter utilize a so-called Bayesian p-value.
Dana Kelly, Curtis Smith
Chapter 5. Time Trends for Binomial and Poisson Data
Abstract
In this chapter, we will see how to develop models in which p and λ are explicit functions of time, relaxing the assumptions of constant p and constant λ in the binomial and Poisson distribution, respectively. This introduces new unknown parameters and makes the Bayesian inference significantly more complicated mathematically. However, modern tools such as OpenBUGS make this analysis no less tractable than the single-parameter cases analyzed earlier.
Dana Kelly, Curtis Smith
Chapter 6. Checking Convergence to Posterior Distribution
Abstract
One issue with any Monte Carlo sampling technique, and especially Markov chain Monte Carlo, is convergence. Before samples can be used for parameter estimation, the analyst must have reasonable assurance that the Markov chain(s) used to generate the samples has converged to the posterior distribution. This chapter presents qualitative and quantitative convergence checks that an analyst can use to obtain this assurance and avoid pitfalls caused by lack of convergence.
Dana Kelly, Curtis Smith
Chapter 7. Hierarchical Bayes Models for Variability
Abstract
This chapter discusses the Bayesian framework for expanding common likelihood functions introduced in earlier chapters to include additional variability. This variability can be over time, among sources, etc.
Dana Kelly, Curtis Smith
Chapter 8. More Complex Models for Random Durations
Abstract
This chapter considers aleatory models that allow for a non-constant rate. Such models are often used in risk assessment for recovery and repair. Three commonly used distributions are treated: Weibull, lognormal, and gamma. Bayesian model checking is covered using posterior predictive checks and information criteria based on a penalized likelihood function. Also covered is the impact of parameter uncertainty on derived quantities, such as nonrecovery probabilities; failure to consider parameter uncertainty can lead to nonconservatively low estimates of such quantities, and thus to overall risk metrics that are nonconservative.
Dana Kelly, Curtis Smith
Chapter 9. Modeling Failure with Repair
Abstract
In earlier chapters we analyzed times to occurrence of an event of interest. Such an event could be failure of a component or system. If the failure is not repaired, and the component or system is replaced following failure, then the earlier analysis methods are applicable. However, in this chapter, we consider the case in which the failed component or system is repaired and placed back into service.
Dana Kelly, Curtis Smith
Chapter 10. Bayesian Treatment of Uncertain Data
Abstract
So far we have only analyzed cases where the observed data were known with certainty and completeness. Reality is messier in a number of ways with respect to observed data. For example, the number of binomial demands may not have been recorded and may have to be estimated. Similarly, the exposure time in the Poisson distribution may have to be estimated. One may not always be able to tell the exact number of failures that have occurred, because of imprecision in the failure criterion. When observing times at which failures occur (i.e., random durations), various types of censoring can occur. For example, a number of components may be placed in test, but the test is terminated before all the components have failed. This produces a set of observed data consisting of the recorded failure times for those components that have failed. For the components that did not fail before the test was terminated, all we know is that the failure time was longer than the duration of the test. As another example, in recording fire suppression times, the exact time of suppression may not be known; in some cases, all that may be available is an interval estimate (e.g., between 10 and 20 min). In this chapter, we will examine how to treat all of these cases, which we refer to generally as uncertain data.
Dana Kelly, Curtis Smith
Chapter 11. Bayesian Regression Models
Abstract
Sometimes a parameter in an aleatory model, such as p in the binomial distribution or λ in the Poisson distribution, can be affected by observable quantities such as pressure, mass, or temperature. For example, in the case of a pressure vessel, very high pressure and high temperature may be leading indicators of failures. In such cases, information about the explanatory variables can be used in the Bayesian inference paradigm to inform the estimates of p or λ. We have already seen examples of this in Chap. 5, where we modeled the influence of time on p and λ via logistic and loglinear regression models, respectively. In this chapter, we extend this concept to more complex situations, such as a Bayesian regression approach that estimates the probability of O-ring failure in the solid-rocket booster motors of the space shuttle.
Dana Kelly, Curtis Smith
Chapter 12. Bayesian Inference for Multilevel Fault Tree Models
Abstract
This chapter describes how information and data may be available at various levels in a fault tree model, and how these may be used in a Bayesian analysis framework to perform probabilistic inference on the model. For example, we might have information on the overall system performance, but we might also have subsystem and component level information. We demonstrate the analysis approach using a simple fault tree model containing a single top event (a “super-component”) and two sub-events (i.e., piece-parts). Also, we show how OpenBUGS can be used for the example models to estimate the probability of meeting a reliability goal at any level in the fault tree model.
Dana Kelly, Curtis Smith
Chapter 13. Additional Topics
Abstract
 This chapter introduces some aditional topics that are of a more specialized or advanced nature. This chapter begins by introducing Bayesian inference for extreme value processes, such as might be used to model high winds and flooding. It then gives an overview of the Bayesian treatment of expert opinion, and then proceeds to an example pointing out the pitfalls that can be encountered if ad hoc methods are employed. We next illustrate how to encode prior distributions into OpenBUGS that are not included as predefined distribution choices. We close this chapter with an example of Bayesian inference for a time-dependent Markov model of pipe rupture.
Dana Kelly, Curtis Smith
Backmatter
Metadaten
Titel
Bayesian Inference for Probabilistic Risk Assessment
verfasst von
Dana Kelly
Curtis Smith
Copyright-Jahr
2011
Verlag
Springer London
Electronic ISBN
978-1-84996-187-5
Print ISBN
978-1-84996-186-8
DOI
https://doi.org/10.1007/978-1-84996-187-5

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