Bayesian Thinking

David A. Hoeflin, PhD (Mathematics from Iowa State University), over his nearly thirty-year career, first at AT&T Bell Labs then AT&T Labs, rose to be a Director in the Optimization, Reliability and Customer Analytics department. He led the development of advanced statistical methods for network performance analysis, reliability engineering, and cybersecurity applications. He is a named inventor on multiple patents and has authored numerous peer-reviewed papers.
Michael Tortorella, PhD, retired as a Distinguished Member of Technical Staff at Bell Laboratories, where he worked for 26 years. He subsequently worked as a Research Professor in the Rutgers Center for Operations Research (RUTCOR) at Rutgers University and?as an Adjunct Professor at Stevens Institute of Technology. He holds the Ph.D. in mathematics from Purdue University and is a recognized expert in reliability engineering and probabilistic methods for complex systems.
He is the author of the Wiley textbook Reliability, Maintainability, and Supportability: Best Practices for Systems Engineers and has published extensively in reliability theory, engineering, and management, numerical analysis, and operations research.

Foreword xi
Preface xv
About the Companion Website xix
1 Introduction to Bayesian Thinking 1
1.1 How Do Bayesians Think, Anyway? 1
1.2 Real-world Processes 2
1.3 Making Sense of "Making Sense" 3
1.4 Why Bayes? 5
1.5 Examples 6
1.5.1 At the Bank 6
1.5.2 Dr. Baize Will See You Now 8
1.5.3 The Queen of Hearts' Tarts 11
1.5.4 COVID-19 14
1.6 Making Sense: A Reprise 15
1.7 Exercises 16
2 Probability, Plausibility, and Life 19
2.1 Why Probability? 19
2.1.1 What Is Probability? 19
2.1.2 Probability and Plausibility: Jeffreys, Cox, Jaynes 20
2.2 Making Inferences About RWPs 22
3 Bayes' Theorem 23
3.1 Conditional Probability and Independence 23
3.2 Bayes' Theorem 24
3.3 Review of Previous Examples 25
3.3.1 Dr. Baize Will See You Now (Reprise) 25
3.3.2 The Queen of Hearts' Tarts (Reprise) 26
3.3.3 COVID-19 (Reprise) 26
3.4 Bayes' Theorem for Continuum Random Variables 26
3.5 How to Tell When Two Distributions Are Different 28
3.6 Quality of Knowledge 29
3.7 Exercises 29
4 Basic Bayesian Methodology 33
4.1 Overview of the Bayesian Method 34
4.2 Probability Models 35
4.3 Priors 37
4.3.1 Assigning a Prior 37
4.4 Data Collection 40
4.5 The Central Limit Theorem, or How We Used to Do This 41
4.6 Model Selection Criteria 47
4.6.1 Bayes Factors 47
4.6.2 Akaike's Information Criterion 49
4.6.3 Widely Accepted Information Criterion 50
4.6.4 Leave One Out 52
4.6.5 Deviance Information Criteria 52
4.6.6 Conclusions and Recommendations 52
4.7 Analysis of the Normal Distribution 53
4.7.1 Normal Distribution 53
4.7.1.2 Posterior Marginal Distribution for ;; 55
4.7.1.3 Posterior Marginal Distribution for ;; 55
4.7.1.4 Distribution of New Observations Given the Current Data 56
4.8 The Seven Pillars of Bayesian Wisdom 56
4.9 What We Have Learned 56
4.10 Exercises 57
5 Other Basic Bayesian Procedures 61
5.1 Contingency Tables 61
5.1.1 Types of Contingency Tables 62
5.1.2 Bayes Methods for Contingency Tables 62
5.1.3 Contingency Tables Using BF Analysis 63
5.2 Correlation 69Contents vii
5.2.1 Approximate Posterior Distribution of the Correlation 70
5.2.2 An Example 71
5.2.3 Small Steps 76
5.3 Bayesian Regression 77
5.3.1 Bayesian Approach for Estimating Regression 77
5.4 Design of Experiments 80
5.4.1 Introduction to Sampling in Statistical Inference 82
5.4.1.1 Frequentist Approaches to Sample Size Determination 82
5.4.2 Simulation-based Bayesian Methods for Complex Models 85
5.4.3 Additional Practical Considerations 86
5.5 What We Have Learned 87
5.6 Exercises 87
6 Using AI 91
6.1 Generating Prompts 91
6.1.1 General Rules for All Analyses 92
6.1.2 General Prompt Setup for EDA 92
6.2 Review and Revision 92
6.2.1 Special-case Suggestions 94
6.2.2 Final Thought 94
6.3 Examples 94
6.3.1 Example 1 95
6.3.2 Example 2 96
7 Building Better-Reliability Engineering Examples 103
7.1 Manufacturing Defects 105
7.1.1 Problem Description 105
7.1.2 Some Data Outcomes-Thinking About What You Might See 108
7.1.3 A New Prior 110
7.2 Service-affecting Incidents 111
7.2.1 Using AI 113
7.2.2 Using AI 115
7.3 An AI Solution for the Lifetime Estimation Case 116
7.3.1 Lifetime Estimation 116
7.3.2 2-Parameter (Shifted) Exponential Case 116
7.3.4 Choosing Between the Two Models 120
7.3.5 Using the AI 121
7.4 Helicopter Reliability 125
7.4.1 Problem Description 125
7.4.2 Parametric Models 126
7.5 Preventative Maintenance Scheduling 134
7.5.1 Using AI 136
7.6 Software Readiness 140
7.6.1 Problem Description: When to Stop Testing Software 140
7.6.2 New Project 144
7.6.3 Using AI 145
7.7 Repairable Systems 152
7.7.1 Problem Description: Repairable Systems Analysis 152
7.7.2 Using AI 160
7.8 What We Have Learned 163
7.9 Exercises 166
8 Examples from Sociology 171
8.1 The Queen of Hearts' Tarts 171
8.1.1 Description 171
8.1.2 Using AI 174
8.1.3 Afterword 176
8.2 Majors, Early Career Wages, and Underemployment 177
8.2.1 Description 177
8.2.2 Exploratory Data Analysis 177
8.2.3 Wage and Underemployment Distributions 178
8.2.4 Bayesian Cluster Analysis 182
8.2.5 Raw Data 185
8.2.6 Use of AI 185
8.3 Bankruptcies in 2020: A Poisson Study 193
8.3.1 Description 193
8.3.2 Bayesian Analysis Approach-One-way Layout 195
8.4 Bankruptcies in 2020: A Linear Model 196
8.4.1 Bayesian Analysis 196
8.5 Bankruptcies in 2020: A Two-way Layout 198
8.5.1 Description 198
8.5.2 Interpretation of Results 207
8.5.3 Using AI 208
8.6 Exercises 212
9 Examples from Medicine 217
9.1 Blood Type Modeling 217
9.1.1 Posterior Distributions 219
9.1.2 Bayes Factors 221
9.1.3 Using AI 221
9.2 Dementia 229
9.2.1 Analysis of Table 9.6: Using Increasingly Skeptical Priors 230
9.2.2 Results for Each Prior 232
9.2.3 Trend Analysis of Table 9.6 234
9.2.4 Using AI 236
9.3 Planning an Experiment 239
9.3.1 The Plan 239
9.3.2 Using AI 241
9.4 What We Have Learned 245
9.5 Exercises 246
10 Winners and Losers 249
10.1 Vive Le Bobsled! 249
10.1.1 Introduction to Olympic 2-person Bobsled Competition 249
10.1.2 Modeling Goals 250
10.1.3 Data 250
10.1.4 Physical Modeling (Use All That You Know) 252
10.1.5 Statistical Modeling 253
10.1.6 Combined Models 255
10.1.7 Men's Bobsled Results 263
10.1.8 Data 264
10.1.9 Using AI 265
10.2 Wine Tasting 270
10.2.1 Problem: Predicting the Quality of Portuguese Red Wine 270
10.2.2 EDA and Prior Elicitation 271
10.2.3 Modeling Effort 272
10.2.4 Using AI 278
10.3 What We Have Learned 284
10.4 Exercises 284
References 285
Index 287