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Reliability and Risk: A Bayesian Perspective



Reliability and Risk: A Bayesian Perspective

Nozer D. Singpurwalla

ISBN: 978-0-470-06033-9 August 2006 396 Pages

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We all like to know how reliable and how risky certain situations are, and our increasing reliance on technology has led to the need for more precise assessments than ever before. Such precision has resulted in efforts both to sharpen the notions of risk and reliability, and to quantify them. Quantification is required for normative decision-making, especially decisions pertaining to our safety and wellbeing. Increasingly in recent years Bayesian methods have become key to such quantifications.

Reliability and Risk provides a comprehensive overview of the mathematical and statistical aspects of risk and reliability analysis, from a Bayesian perspective. This book sets out to change the way in which we think about reliability and survival analysis by casting them in the broader context of decision-making. This is achieved by:

  • Providing a broad coverage of the diverse aspects of reliability, including: multivariate failure models, dynamic reliability, event history analysis, non-parametric Bayes, competing risks, co-operative and competing systems, and signature analysis.
  • Covering the essentials of Bayesian statistics and exchangeability, enabling readers who are unfamiliar with Bayesian inference to benefit from the book.
  • Introducing the notion of “composite reliability”, or the collective reliability of a population of items.
  • Discussing the relationship between notions of reliability and survival analysis and econometrics and financial risk.

Reliability and Risk can most profitably be used by practitioners and research workers in reliability and survivability as a source of information, reference, and open problems. It can also form the basis of a graduate level course in reliability and risk analysis for students in statistics, biostatistics, engineering (industrial, nuclear, systems), operations research, and other mathematically oriented scientists, wherein the instructor could supplement the material with examples and problems.



1 Introduction and Overview.

1.1 Preamble: What do ‘Reliability’, ‘Risk’ and ‘Robustness’ Mean?

1.2 Objectives and Prospective Readership.

1.3 Reliability, Risk and Survival: State-of-the-Art.

1.4 Risk Management: A Motivation for Risk Analysis.

1.5 Books on Reliability, Risk and Survival Analysis.

1.6 Overview of the Book.

2 The Quantification of Uncertainty.

2.1 Uncertain Quantities and Uncertain Events: Their Definition and Codification.

2.2 Probability: A Satisfactory Way to Quantify Uncertainty.

2.3 Overview of the Different Interpretations of Probability.

2.4 Extending the Rules of Probability: Law of Total Probability and Bayes’ Law.

2.5 The Bayesian Paradigm: A Prescription for Reliability, Risk and Survival.


2.6 Probability Models, Parameters, Inference and Prediction.

2.7 Testing Hypotheses: Posterior Odds and Bayes Factors.

2.8 Utility as Probability and Maximization of Expected Utility.

2.9 Decision Trees and Influence Diagrams for Risk Analysis.

3 Exchangeability and Indifference.

3.1 Introduction to Exchangeability: de Finetti’s Theorem.

3.2 de Finetti-style Theorems for Infinite Sequences of Non-binary Random.

3.3 Error Bounds on de Finetti-style Results for Finite Sequences of Random.

4 Stochastic Models of Failure.

4.1 Introduction.

4.2 Preliminaries: Univariate, Multivariate and Multi-indexed Distribution Functions.

4.3 The Predictive Failure Rate Function of a Univariate Probability Distribution.

4.4 Interpretation and Uses of the Failure Rate Function – the Model Failure Rate.

4.5 Multivariate Analogues of the Failure Rate Function.

4.6 The Hazard Potential of Items and Individuals.

4.7 Probability Models for Interdependent Lifelengths.

4.8 Causality and Models for Cascading Failures.

4.9 Failure Distributions with Multiple Scales.

5 Parametric Failure Data Analysis.

5.1 Introduction and Perspective.

5.2 Assessing Predictive Distributions in the Absence of Data.

5.3 Prior Distributions in Chance Distributions.

5.4 Predictive Distributions Incorporating Failure Data.

5.5 Information from Life-tests: Learning from Data.

5.6 Optimal Testing: Design of Life-testing Experiments.

5.7 Adversarial Life-testing and Acceptance Sampling.

5.8 Accelerated Life-testing and Dose–response Experiments.

6 Composite Reliability: Signatures.

6.1 Introduction: Hierarchical Models.

6.2 ‘Composite Reliability’: Partial Exchangeability.

6.3 Signature Analysis and Signatures as Covariates.

7 Survival in Dynamic Environments.

7.1 Introduction: Why Stochastic Hazard Functions?

7.2 Hazard Rate Processes.

7.3 Cumulative Hazard Processes.

7.4 Competing Risks and Competing Risk Processes.

7.5 Degradation and Aging Processes.

8 Point Processes for Event Histories.

8.1 Introduction: What is Event History?

8.2 Other Point Processes in Reliability and Life-testing.

8.3 Multiplicative Intensity and Multivariate Point Processes.

8.4 Dynamic Processes and Statistical Models: Martingales.

8.5 Point Processes with Multiplicative Intensities.

9 Non-parametric Bayes Methods in Reliability.

9.1 The What and Why of Non-parametric Bayes.

9.2 The Dirichlet Distribution and its Variants.

9.3 A Non-parametric Bayes Approach to Bioassay.

9.4 Prior Distributions on the Hazard Function.

9.5 Prior Distributions for the Cumulative Hazard Function.

9.6 Priors for the Cumulative Distribution Function.

10 Survivability of Co-operative, Competing and Vague Systems.

10.1 Introduction: Notion of Systems and their Components.

10.2 Coherent Systems and their Qualitative Properties.

10.3 The Survivability of Coherent Systems.

10.4 Machine Learning Methods in Survivability Assessment.

10.5 Reliability Allocation: Optimal System Design.


10.6 The Utility of Reliability: Optimum System Selection.

10.7 Multi-state and Vague Stochastic Systems.

11 Reliability and Survival in Econometrics and Finance.

11.1 Introduction and Overview.

11.2 Relating Metrics of Reliability to those of Income Inequality.

11.3 Invoking Reliability Theory in Financial Risk Assessment.

11.4 Inferential Issues in Asset Pricing.

11.5 Concluding Comments.

Appendix A Markov Chain Monté Carlo Simulation.

A.1 The Gibbs Sampling Algorithm.

Appendix B Fourier Series Models and the Power Spectrum.

Appendix C Network Survivability and Borel’s Paradox.



"The book is written by an expert in reliability analysis and it is a very valuable source of information for mathematical models for reliability problems ... An extensive bibliography concludes the book." (Stat Papers, 2011)

"As the author mentions in his preface, the book can be read in several different ways, as a text for a graduate level course on reliability or as a source book for “information and open problems." This book has been a joy to read for this reviewer." (International Statistical Review, August 2008)

"Singpurwalla seems to be at his best in probabilistic modeling of reality. He has written what must be one of the first books reliability written from a subjective, Bayesian point of view." (International Statistical Review, August 2008)

"The material of this book will be most profitable for practitioners and researchers in reliability and survivability, who will greatly appreciate it as a source of information and open problems." (Mathematical Reviews, 2008h)

"This is a very interesting, provocative, and worthwhile book." (Biometrics, June 2008)

"What I liked most about this book, however, is the way it blends interesting technical material with foundational discussion about the nature of uncertainty." (Biometrics, June 2008)

"The investigation of the theoretical models under consideration in the book is first class…" (Law, Probability and Risk Advance Access, September 2007)

"I feel that I have learned an effective plotting technique from these plots…" (Technometrics, February 2008)

"…a cornucopia of probability models and inference methods for different problems…[that] serve as a rich taxonomy that statisticians can use to fit models…works as both an educational tool and as a reference." (MAA Reviews, March 6, 2007)