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Reliability and Risk: A Bayesian Perspective

Nozer D. Singpurwalla
John Wiley
Publication Date: 
Number of Pages: 
Wiley Series in Probability and Statistics
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Tom Schulte
, on

Using stochastic models and a general Bayesian approach to quantifying uncertainty, this text provides a rigorous mathematical treatment that can put bounds of certainty on such well-worn phrases as “credibility”, “reliability”, “risk”, “robustness”, and “survivability” that, as the author points out, have now become very much a part of our daily vocabulary.

Bayesian approaches have proven successful in combating spam by applying an intelligent algorithm to pre-screening our e-mail for us. This is an example of our increasing reliance on technology for precise assessments. In assessing failure rates, drug efficacy, election outcomes, hurricane paths and more, we are seeing the results of continued efforts to quantify the ideas of risk and reliability. Quantification for decision-making as regards our safety and well-being has increased in recent years through Bayesian methods. This text is a comprehensive overview of the mathematical and statistical aspects of risk and reliability analysis from a Bayesian perspective. The reader should be comfortable, in a high-level general and abstract sense, with Weibull and other distributions, Markov processes, and the like. The author introduces each concept when it is appropriate, but only in the general, theoretical sense, so these concept introductions need to be a refresher for the reader. This is a graduate-level text.

Let me first unburden myself of two complaints. First, the material in this text is well-suited and indeed designed for real-world application, but application examples in the text are few. When present, they are detailed and illustrative. For instance, signature analysis is explored through a thorough one-page example of vibration in traction motors. Secondly, completing some proof steps is left to the reader, but exercises are not provided.

Now let me sing the book’s praises. The first three chapters are good introductory material and some skipping ahead here may be possible. Chapter 1 introduces the book’s structure and the definition of and motivation for tackling reliability and risk. A reader comfortable with the ideas of utility or exchangeability within Bayesian statistics can bypass Chapters 2 and 3, although the second chapter on "The Quantification of Uncertainty" is an excellent historical and intellectual overview of why we do and should take a Bayesian approach to quantifying uncertainty.

Thereafter, Prof. Singpurwalla takes the reader on a journey, exploring the topics with a keen interest in the motivation and theory behind each one. For instance, the finer points between choosing exponential versus Weibull distribution as a model for reliability are explored here at a level few textbooks that only partly treat Bayesian statistical methods can afford.

Much of the rest of book is a cornucopia of probability models and inference methods for different problems in risk and reliability. Taken from the author's decades of work in reliability, these sections serve as a rich taxonomy that statisticians can use to fit models. Thus, the book works as both as an educational tool and as a reference. Sections on stochastic models of failure include discussion on failure rates, common lifetime models, accelerated tests, and dose-response experiments. Later, the book presents advanced chapters on survival in dynamic environments, point processes for event histories, and more. The final two chapters explore competing and vague systems as well as the use of reliability in econometrics and finance. Supporting appendices cover Markov chain Monté Carlo simulation, Fourier Series models, and Borel’s Paradox.

Tom Schulte ( is a graduate student at Oakland University and with an appropriate relative frequency can be found working as a software engineer specializing in designing and implementing statistical process control (SPC) systems.



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.

2.2.1 The Rules of Probability.

2.2.2 Justifying the Rules of Probability.

2.3 Overview of the Different Interpretations of Probability.

2.3.1 A Brief History of Probability.

2.3.2 The Different Kinds of Probability.

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

2.4.1 Marginalization.

2.4.2 The Law of Total Probability.

2.4.3 Bayes’ Law: The Incorporation of Evidence and the Likelihood.

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


2.6 Probability Models, Parameters, Inference and Prediction.

2.6.1 The Genesis of Probability Models and Their Parameters.

2.6.2 Statistical Inference and Probabilistic Prediction.

2.7 Testing Hypotheses: Posterior Odds and Bayes Factors.

2.7.1 Bayes Factors: Weight of Evidence and Change in Odds.

2.7.2 Uses of the Bayes Factor.

2.7.3 Alternatives to Bayes Factors.

2.8 Utility as Probability and Maximization of Expected Utility.

2.8.1 Utility as a Probability.

2.8.2 Maximization of Expected Utility.

2.8.3 Attitudes to Risk: The Utility of Money.

2.9 Decision Trees and Influence Diagrams for Risk Analysis.

2.9.1 The Decision Tree.

2.9.2 The Influence Diagram.

3 Exchangeability and Indifference.

3.1 Introduction to Exchangeability: de Finetti’s Theorem.

3.1.1 Motivation for the Judgment of Exchangeability.

3.1.2 Relationship Between Independence and Exchangeability.

3.1.3 de Finetti’s Representation Theorem for Zero-one Exchangeable.


3.1.4 Exchangeable Sequences and the Law of Large Numbers.

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


3.2.1 Sufficiency and Indifference in Zero-one Exchangeable Sequences.

3.2.2 Invariance Conditions Leading to Mixtures of Other Distributions.

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


3.3.1 Bounds for Finitely Extendable Zero-one Random Quantities.

3.3.2 Bounds for Finitely Extendable Non-binary Random Quantities.

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.

4.3.1 The Case of Discontinuity.

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

4.4.1 The True Failure Rate: Does it Exist?

4.4.2 Decreasing Failure Rates, Reliability Growth, Burn-in and the Bathtub.


4.4.3 The Retrospective (or Reversed) Failure Rate.

4.5 Multivariate Analogues of the Failure Rate Function.

4.5.1 The Hazard Gradient.

4.5.2 The Multivariate Failure Rate Function.

4.5.3 The Conditional Failure Rate Functions.

4.6 The Hazard Potential of Items and Individuals.

4.6.1 Hazard Potentials and Dependent Lifelengths.

4.6.2 The Hazard Gradient and Conditional Hazard Potentials.

4.7 Probability Models for Interdependent Lifelengths.

4.7.1 Preliminaries: Bivariate Distributions.

4.7.2 The Bivariate Exponential Distributions of Gumbel.

4.7.3 Freund’s Bivariate Exponential Distribution.

4.7.4 The Bivariate Exponential of Marshall and Olkin.

4.7.5 The Bivariate Pareto as a Failure Model.

4.7.6 A Bivariate Exponential Induced by a Shot-noise Process.

4.7.7 A Bivariate Exponential Induced by a Bivariate Pareto’s Copula.

4.7.8 Other Specialized Bivariate Distributions.

4.8 Causality and Models for Cascading Failures.

4.8.1 Probabilistic Causality and Causal Failures.

4.8.2 Cascading and Models of Cascading Failures.

4.9 Failure Distributions with Multiple Scales.

4.9.1 Model Development.

4.9.2 A Failure Model Indexed by Two Scales.

5 Parametric Failure Data Analysis.

5.1 Introduction and Perspective.

5.2 Assessing Predictive Distributions in the Absence of Data.

5.2.1 The Exponential as a Chance Distribution.

5.2.2 The Weibull (and Gamma) as a Chance Distribution.

5.2.3 The Bernoulli as a Chance Distribution.

5.2.4 The Poisson as a Chance Distribution.

5.2.5 The Generalized Gamma as a Chance Distribution.

5.2.6 The Inverse Gaussian as a Chance Distribution.

5.3 Prior Distributions in Chance Distributions.

5.3.1 Eliciting Prior Distributions via Expert Testimonies.

5.3.2 Using Objective (or Default) Priors.

5.4 Predictive Distributions Incorporating Failure Data.

5.4.1 Design Strategies for Industrial Life-testing.

5.4.2 Stopping Rules: Non-informative and Informative.

5.4.3 The Total Time on Test.

5.4.4 Exponential Life-testing Procedures.

5.4.5 Weibull Life-testing Procedures.

5.4.6 Life-testing Under the Generalized Gamma and the Inverse.


5.4.7 Bernoulli Life-testing Procedures.

5.4.8 Life-testing and Inference Under the BVE.

5.5 Information from Life-tests: Learning from Data.

5.5.1 Preliminaries: Entropy and Information.

5.5.2 Learning for Inference from Life-test Data: Testing for Confidence.

5.5.3 Life-testing for Decision Making: Acceptance Sampling.

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.

5.8.1 Formulating Accelerated Life-testing Problems.

5.8.2 The Kalman Filter Model for Prediction and Smoothing.

5.8.3 Inference from Accelerated Tests Using the Kalman Filter.

5.8.4 Designing Accelerated Life-testing Experiments.

6 Composite Reliability: Signatures.

6.1 Introduction: Hierarchical Models.

6.2 ‘Composite Reliability’: Partial Exchangeability.

6.2.1 Simulating Exchangeable and Partially Exchangeable Sequences.

6.2.2 The Composite Reliability of Ultra-reliable Units.

6.2.3 Assessing Reliability and Composite Reliability.

6.3 Signature Analysis and Signatures as Covariates.

6.3.1 Assessing the Power Spectrum via a Regression Model.

6.3.2 Bayesian Assessment of the Power Spectrum.

6.3.3 A Hierarchical Bayes Assessment of the Power Spectrum.

6.3.4 The Spectrum as a Covariate Using an Accelerated Life Model.

6.3.5 Closing Remarks on Signatures and Covariates.

7 Survival in Dynamic Environments.

7.1 Introduction: Why Stochastic Hazard Functions?

7.2 Hazard Rate Processes.

7.2.1 Hazard Rates as Shot-noise Processes.

7.2.2 Hazard Rates as Lévy Processes.

7.2.3 Hazard Rates as Functions of Diffusion Processes.

7.3 Cumulative Hazard Processes.

7.3.1 The Cumulative Hazard as a Compound Poisson Process.

7.3.2 The Cumulative Hazard as an Increasing Lévy Process.

7.3.3 Cumulative Hazard as Geometric Brownian Motion.

7.3.4 The Cumulative Hazard as a Markov Additive Process.

7.4 Competing Risks and Competing Risk Processes.

7.4.1 Deterministic Competing Risks.

7.4.2 Stochastic Competing Risks and Competing Risk Processes.

7.5 Degradation and Aging Processes.

7.5.1 A Probabilistic Framework for Degradation Modeling.

7.5.2 Specifying Degradation Processes.

8 Point Processes for Event Histories.

8.1 Introduction: What is Event History?

8.1.1 Parameterizing the Intensity Function.

8.2 Other Point Processes in Reliability and Life-testing.

8.2.1 Multiple Failure Modes and Competing Risks.

8.2.2 Items Experiencing Degradation and Deterioration.

8.2.3 Units Experiencing Maintenance and Repair.

8.2.4 Life-testing Under Censorship and Withdrawals.

8.3 Multiplicative Intensity and Multivariate Point Processes.

8.3.1 Multivariate Counting and Intensity Processes.

8.4 Dynamic Processes and Statistical Models: Martingales.

8.4.1 Decomposition of Continuous Time Processes.

8.4.2 Stochastic Integrals and a Martingale Central Limit Theorem.

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.

9.2.1 The Ordered Dirichlet Distribution.

9.2.2 The Generalized Dirichlet – Concept of Neutrality.

9.3 A Non-parametric Bayes Approach to Bioassay.

9.3.1 A Prior for Potency.

9.3.2 The Posterior Potency.

9.4 Prior Distributions on the Hazard Function.

9.4.1 Independent Beta Priors on Piecewise Constant Hazards.

9.4.2 The Extended Gamma Process as a Prior.

9.5 Prior Distributions for the Cumulative Hazard Function.

9.5.1 Neutral to the Right Probabilities and Gamma Process Priors.

9.5.2 Beta Process Priors for the Cumulative Hazard.

9.6 Priors for the Cumulative Distribution Function.

9.6.1 The Dirichlet Process Prior.

9.6.2 Neutral to the Right-prior Processes.

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

10.1 Introduction: Notion of Systems and their Components.

10.1.1 Overview of the Chapter.

10.2 Coherent Systems and their Qualitative Properties.

10.2.1 The Reliability of Coherent Systems.

10.3 The Survivability of Coherent Systems.

10.3.1 Performance Processes and their Driving Processes.

10.3.2 System Survivability Under Hierarchical Independence.

10.3.3 System Survivability Under Interdependence.

10.3.4 Prior Distributions on the Unit Hypercube.

10.4 Machine Learning Methods in Survivability Assessment.

10.4.1 An Overview of the Neural Net Methodology.

10.4.2 A Two-phased Neural Net for System Survivability.

10.5 Reliability Allocation: Optimal System Design.

10.5.1 The Decision Theoretic Formulation.

10.5.2 Reliability Apportionment for Series Systems.

10.5.3 Reliability Apportionment for Parallel Redundant.


10.5.4 Apportioning Node Reliabilities in Networks.

10.5.5 Apportioning Reliability Under Interdependence.

10.6 The Utility of Reliability: Optimum System Selection.

10.6.1 Decision-making for System Selection.

10.6.2 The Utility of Reliability.

10.7 Multi-state and Vague Stochastic Systems.

10.7.1 Vagueness or Imprecision.

10.7.2 Many-valued Logic: A Synopsis.

10.7.3 Consistency Profiles and Probabilities of Vague Sets.

10.7.4 Reliability of Components in Vague Binary States.

10.7.5 Reliability of Systems in Vague Binary States.

10.7.6 Concluding Comments on 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.

11.2.1 Some Metrics of Reliability and Survival.

11.2.2 Metrics of Income Inequality.

11.2.3 Relating the Metrics.

11.2.4 The Entropy of Income Shares.

11.2.5 Lorenz Curve Analysis of Failure Data.

11.3 Invoking Reliability Theory in Financial Risk Assessment.

11.3.1 Asset Pricing of Risk-free Bonds: An Overview.

11.3.2 Re-interpreting the Exponentiation Formula.

11.3.3 A Characterization of Present Value Functions.

11.3.4 Present Value Functions Under Stochastic Interest Rates.

11.4 Inferential Issues in Asset Pricing.

11.4.1 Formulating the Inferential Problem.

11.4.2 A Strategy for Pooling Present Value Functions.

11.4.3 Illustrative Example: Pooling Present Value Functions.

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.

B.1 Preliminaries: Trigonometric Functions.

B.2 Orthogonality of Trigonometric Functions.

B.3 The Fourier Representation of a Finite Sequence of Numbers.

B.4 Fourier Series Models for Time Series Data.

Appendix C Network Survivability and Borel’s Paradox.

C.1 Preamble.

C.2 Re-assessing Testimonies of Experts Who have Vanished.

C.3 The Paradox in Two Dimensions.

C.4 The Paradox in Network Survivability Assessment.