Cover......Page 1
Springer Series in Reliability Engineering......Page 2
Bayesian Inference for Probabilistic Risk Assessment......Page 4
ISBN 9781849961868......Page 5
Preface......Page 6
Contents......Page 10
1.1…Introduction......Page 14
1.2…Background for Bayesian Inference......Page 15
1.3…An Overview of the Bayesian Inference Process......Page 16
References......Page 19
2.2…Bayes’ Theorem......Page 20
2.3.1 The Discrete Case......Page 22
2.3.2 The Continuous Case......Page 24
3.2…The Binomial Distribution......Page 28
3.2.1 Binomial Inference with Conjugate Prior......Page 29
3.2.2 Binomial Inference with Noninformative Prior......Page 33
3.2.3 Binomial Inference with Nonconjugate Prior......Page 34
3.3…The Poisson Model......Page 37
3.3.1 Poisson Inference with Conjugate Prior......Page 38
3.3.2 Poisson Inference with Noninformative Prior......Page 39
3.3.3 Poisson Inference with Nonconjugate Prior......Page 40
3.4…The Exponential Model......Page 41
3.4.1 Exponential Inference with Conjugate Prior......Page 42
3.4.3 Exponential Inference with Nonconjugate Prior......Page 43
3.5.1 Developing a Conjugate Prior......Page 44
3.5.3 Cautions in Developing an Informative Prior......Page 47
3.5.4 Ensure Prior is Consistent with Expected Data: Preposterior Analysis......Page 48
3.6…Exercises......Page 50
Reference......Page 51
4.1…Direct Inference Using the Posterior Distribution......Page 52
4.2…Posterior Predictive Distribution......Page 53
4.2.1 Graphical Checks Based on the Posterior Predictive Distribution......Page 56
4.3…Model Checking with Summary Statistics from the Posterior Predictive Distribution......Page 57
4.3.1 Bayesian Chi-Square Statistic......Page 58
4.3.2 Cramer-von Mises Statistic......Page 60
4.4…Exercises......Page 61
References......Page 63
5.1…Time Trend in p......Page 64
5.2…Time Trend in lambda......Page 68
References......Page 73
6.1…Qualitative Convergence Checks......Page 74
6.2…Quantitative Convergence Checks......Page 75
6.3…Ensuring Adequate Coverage of Posterior Distribution......Page 76
6.4…Determining Adequate Sample Size......Page 77
Reference......Page 78
7.1…Variability in Trend Models......Page 80
7.2…Source-to-Source Variability......Page 84
7.3…Dealing with Convergence Problems in Hierarchical Bayes Models of Variability......Page 90
7.4…Choice of First-Stage Prior......Page 94
7.5…Trend Models Revisited......Page 98
7.6…Summary......Page 99
7.7…Exercises......Page 100
References......Page 101
8 More Complex Models for Random Durations......Page 102
8.2.1 Frequentist Analysis......Page 103
8.2.2 Bayesian Analysis......Page 104
8.3…Analysis with Weibull Model......Page 105
8.4…Analysis with Lognormal Model......Page 106
8.5…Analysis with Gamma Model......Page 108
8.6…Estimating Nonrecovery Probability......Page 109
8.6.1 Propagating Uncertainty in Convolution Calculations......Page 111
8.7…Model Checking and Selection......Page 113
8.8…Exercises......Page 118
References......Page 122
9 Modeling Failure with Repair......Page 124
9.2…Repair Same as Old: Nonhomogeneous Poisson Process......Page 125
9.2.1 Graphical Check for Trend in Rate of Occurrence of Failure When Repair is same as Old......Page 126
9.2.2 Bayesian Inference Under Same-as-Old Repair Assumption......Page 128
9.3…Incorporating Results into PRA......Page 134
References......Page 135
10.1…Censored Data for Random Durations......Page 136
10.2…Uncertainty in Binomial Demands or Poisson Exposure Time......Page 139
10.3…Uncertainty in Binomial or Poisson Failure Counts......Page 141
10.4…Alternative Approaches for Including Uncertainty in Poisson or Binomial Event Counts......Page 143
10.5…Uncertainty in Common-Cause Event Counts......Page 148
10.6…Exercises......Page 150
References......Page 152
11.1…Aleatory Model for O-ring Distress......Page 154
11.2…Model-Checking......Page 158
11.3…Probability of Shuttle Failure......Page 159
11.5…Regression Models for Component Lifetime......Page 163
11.6…Battery Example......Page 167
11.8…Exercises......Page 174
References......Page 176
12.2…Example of a Super-Component with Two Piece-Parts......Page 178
12.3…Examples of a Super-Component with Multiple Piece-Parts......Page 180
12.4…The ‘‘Bayesian Anomaly’’......Page 182
12.5…A Super-Component with Piece-Parts and a Sub-System......Page 183
12.6…Emergency Diesel Generator Example......Page 185
12.7…Meeting Reliability Goals at Multiple Levels in a Fault Tree......Page 187
References......Page 189
13.1…Extreme Value Processes......Page 190
13.1.1 Bayesian Inference for the GEV Parameters......Page 192
13.1.2 Thresholds and the Generalized Pareto Distribution......Page 193
13.2…Treatment of Expert Opinion......Page 196
13.2.1 Information from a Single Expert......Page 197
13.2.2 Using Information from Multiple Experts......Page 198
13.3…Pitfalls of ad hoc Methods......Page 200
13.3.2 Using a Logistic-Normal First-Stage Prior......Page 201
13.3.3 Update with New Data......Page 203
13.4…Specifying a New Prior Distribution in OpenBUGS......Page 204
13.5.1 Aleatory Models for Failure......Page 205
13.5.2 Other Markov Model Parameters......Page 206
13.5.3 Markov System Equations......Page 207
13.5.4 Implementation in OpenBUGS......Page 208
References......Page 212
Appendix A......Page 214
Appendix B......Page 230