Probability Applications

Contents
xi
10.2.6 How can we solve the corresponding optimization
problem? 222
10.3 How to determine probabilities from observed values of excess energy:
Optimal way (use of simulated annealing) 222
10.3.1 An expression for probabilities 222
10.3.2 Best in what sense? 223
10.3.3 An optimality criterion can be nonnumeric 223
10.3.4 The optimality criterion must be final
224
10.3.5 The criterion must not change if we change the measuring
unit for energy 224
10.3.6 Definitions and the main result 225
10.4 How to determine the probability of detection: Optimal way (use of
neural networks) 226
10.4.1 The POD function must be smooth and monotonic .... 226
10.4.2 We must choose a family of functions, not a
single function 227
10.4.3 Definition and the main result 227
10.5 How to combine probabilities (use of fuzzy techniques) 228
10.5.1 Traditional probabilistic approach: Maximum entropy . . 228
10.5.2 Traditional approach is not always sufficient 228
10.5.3 Main idea: Describe general combination operations . . . 229
10.5.4 The notions of t-norms and ^-conorms 230
10.6 Preliminary results 231
10.7 Alternative approach to fusing probabilities: Fuzzy rules 231
10.7.1 Main problems with the above approach 231
10.7.2 The use of fuzzy rules 231
10.7.3 Expert rules for fault detection 232
10.7.4 The problem with this rule base and how we solve it . . . 233
10.7.5 Experimental results 234
10.8 Applications to aircraft reliability 235
10.8.1 Reliability: General problem 235
10.8.2 Traditional approach to reliability 235
10.8.3 Traditional approach is not always sufficient: A problem . 235
10.8.4 Proposed approach to fusing probabilities: Main idea . . 236
10.8.5 Resulting solution 237
10.9 Closing thoughts 237
Appendix: Proofs 237
Proof of Theorem 10.1 237
Proof of Theorem 10.2 238
References 240
11 Auto Reliability Project 243
Jane M. Booker and Thomas R. Bement
11.1 Description of the reliability problem 243
11.2 Implementing the probability approach 246
11.2.1 Logic and reliability models 246
11.2.2 Expert elicitation 247
11.2.3 Updating methods 249
11.2.4 Calculating reliabilities 250