# Practical Applications of Bayesian Reliability

Edited by

Yan Liu

Principal Reliability Engineer
Medtronic PLC, Minneapolis, USA

Athula I. Abeyratne

Senior Principal Statistician
Medtronic PLC, Minneapolis, USA

# Preface

Recently, groundbreaking work using Bayesian statistics for reliability analysis has emerged at various seminars and technical conferences which demonstrated great power in accurately predicting reliability and/or reducing sample size. Many engineers and scientists have expressed interest in learning Bayesian statistics. However, there is also much confusion in the learning process. This confusion comes mainly from three aspects:

1. What is Bayesian analysis exactly?
2. What are the benefits?
3. How can I apply the methods to solve my own problems?

This book is intended to provide basic knowledge and practical examples of Bayesian modeling in reliability and related science and engineering practices. We hope it will help engineers and scientists to find answers to the above common questions.

For scientists and engineers with no programming experience, coding is often considered too daunting. To help readers get started quickly, many Bayesian models using Just Another Gibbs Sampler (JAGS) are provided in this book (e.g. `3.4_Weibull.JAGS` in Section 3.4) containing fewer than ten lines of commands. Then all you need to do is to learn a few functions to run the Bayesian model and diagnose the results (discussed in Section 3.4). To help readers become familiar with R coding, this book also provides a number of short R scripts consisting of simple functions. There are some cases requiring longer R scripts. Those programs are divided into a few sections, and the function of each section is explained in detail separately.

Although some knowledge of Bayesian reliability is given to students in graduate courses of reliability engineering, the application of Bayesian reliability in industry is limited to special cases using conjugate prior distributions, due to mathematical tractability.

Thanks to the rapid development of computers in the past half century, the breakthroughs in computational algorithms and the increased computing power of personal computers have enabled complex Bayesian models to be built and solved, which has greatly promoted the progress and application of Bayesian modeling. However, most engineers and scientists may not know that these modeling and computational capabilities can help them solve more complex prediction problems, which may not have been feasible in the past using traditional statistical methods.

Bayesian models are expected to become increasingly popular among engineers and scientists. One advantage is that modern Bayesian statistics enables development of more complex reliability models for system level prediction. Some examples included in this book attempt to demonstrate this capability. These cases often require customized solutions. Most of the existing commercial statistical software provide traditional statistical methods, which are not suitable for solving complex reliability problems. In other cases, Bayesian modeling offers unique benefits to effectively utilize different sources of information to reduce sample size. This book is intended to provide readers with some examples of practical engineering applications. Hopefully readers can apply them to their own fields and get some inspiration for building new models.

The goal of this book is to help more engineers and scientists to understand Bayesian modeling capabilities, learn how to use Bayesian models to solve engineering prediction problems, and get inspiration for developing Bayesian models to solve complex problems. The main objectives of this book are

• to explain the differences and benefits of Bayesian methods compared to traditional frequentist methods
• to demonstrate how to develop models to propagate component‐level reliability to the final system level and quantify reliability uncertainty
• to demonstrate how to use different sources of information to reduce sample size
• to provide model examples for complex prediction problems
• to provide R and JAGS scripts for readers to understand and to use the models
• to design Bayesian reliability and substantiation test plans.

This book is intended for industry practitioners (reliability engineers, mechanical engineers, electrical engineers, product engineers, system engineers, materials scientists, Six Sigma Master Black Belts, Black Belts, Green Belts, etc.) whose work includes predicting design or manufacturing performance. Students in science and engineering, academic scholars, and researchers can also use this book as a reference.

Prerequisite knowledge includes basic knowledge of statistics and probability theory, and calculus. The goal is to enable engineers and scientists in different fields to acquire advanced Bayesian statistics skills and apply them to their work.

Throughout this book we extensively use the Markov chain Monte Carlo (MCMC) method to solve problems using JAGS software. We made an effort to reduce the use of complex Bayesian theory in this book and therefore it is not intended for people who want to learn the theory behind MCMC simulations.

Chapter 1 introduces basic concepts of reliability engineering, including random variables, discrete and continuous probability distributions, hazard function, and censored data. The Bayesian approach to reliability inference is briefly discussed. Non‐parametric estimation of survival function using the Kaplan–Meier method is introduced. The concepts of system reliability estimation, design capability prediction, and accelerated life testing are also discussed.

Basic concepts of Bayesian statistics and models are presented in Chapter 2. Basic ideas behind Bayesian reasoning, Bayesian probability theory, Bayes' theorem, selection of prior distributions, conjugate priors, Bayes' factor and its applications are discussed.

Bayesian computation, the Metropolis–Hastings algorithm, Gibbs sampling, BUGS/JAGS models for solving Bayesian problems, MCMC diagnostics and output analysis are introduced in Chapter 3. Discreate and continuous probability distributions that are frequently used in reliability analysis are discussed in detail in Chapter 4. Applications of these distributions in solving reliability problems using the Bayesian approach are also discussed in this chapter.

Chapter 5 introduces the concept of reliability testing and demonstration. The difference between substantiation and reliability testing is discussed. Classical and Bayesian methods for developing zero‐failure test plans for both substantiation and reliability testing are presented. Examples are given for developing these test plans assuming that the underlying time to failure model is Weibull.

In Chapter 1 we discuss the concepts of design capability and design for reliability. Monte Carlo simulation techniques are introduced from the Bayesian perspective for estimating design capability and reliability, with examples to demonstrate these techniques. Chapter 7 introduces Bayesian models for estimating system reliability. The theory of reliability block diagrams, fault trees, and Bayesian networks are introduced with practical examples.

Bayesian hierarchical models and their applications are discussed in Chapter 8. Chapter 9 introduces linear and logistic regression models in the Bayesian perspective. Examples and a case study are presented to show the reader how to apply Bayesian methods for solving different regression problems.

Please send comments, suggestions or any other feedback on this book to AbeyraLiu118@gmail.com.

Yan Liu

Athula I. Abeyratne

# Acknowledgments

The authors would like to sincerely thank Xingfu Chen, Donald Musgrove, Alicia Thode, Paul DeGroot, Pei Li, Vladimir Nikolski, and Norman Allie for their contributions in reviewing the manuscript. The authors would also like to sincerely thank Bradley P. Carlin and Harrison Quick for helping to answer questions related to Bayesian statistics. Many thanks to Greg Peterson and Shane Sondreal for their reviews and support to make this work presentable.

Thanks to our mentor Eric Maass, who has a great passion for teaching statistical methods to engineers. Tarek Haddad and Karen Hulting also provided valuable consulting on this topic. Some examples in this book are modified from actual engineering applications. The authors want to thank many Medtronic coworkers who contributed their case studies and/or provided valuable feedback, including Roger Berg, Mun‐Peng Tan, Scott Hareland, Paul Wisnewski, Patrick Zimmerman, Xiaobo Wang, Jim Haase, Anders Olmanson, and Craig Wiklund.