Preface xiii
Acknowledgments xvii
1 Introduction 1
1.1. Uncertainty and Its Significance / 1
1.2. Uncertainty-Based Information / 6
1.3. Generalized Information Theory / 7
1.4. Relevant Terminology and Notation / 10
1.5. An Outline of the Book / 20
Notes / 22
Exercises / 23
2 Classical Possibility-Based Uncertainty Theory 26
2.1. Possibility and Necessity Functions / 26
2.2. Hartley Measure of Uncertainty for Finite Sets / 27
2.2.1. Simple Derivation of the Hartley Measure / 28
2.2.2. Uniqueness of the Hartley Measure / 29
2.2.3. Basic Properties of the Hartley Measure / 31
2.2.4. Examples / 35
2.3. Hartley-Like Measure of Uncertainty for Infinite Sets / 45
2.3.1. Definition / 45
2.3.2. Required Properties / 46
2.3.3. Examples / 52
Notes / 56
Exercises / 57
3 Classical Probability-Based Uncertainty Theory 61
3.1. Probability Functions / 61
3.1.1. Functions on Finite Sets / 62
vii
3.1.2. Functions on Infinite Sets / 64
3.1.3. Bayes’ Theorem / 66
3.2. Shannon Measure of Uncertainty for Finite Sets / 67
3.2.1. Simple Derivation of the Shannon Entropy / 69
3.2.2. Uniqueness of the Shannon Entropy / 71
3.2.3. Basic Properties of the Shannon Entropy / 77
3.2.4. Examples / 83
3.3. Shannon-Like Measure of Uncertainty for Infinite Sets / 91
Notes / 95
Exercises / 97
4 Generalized Measures and Imprecise Probabilities 101
4.1. Monotone Measures / 101
4.2. Choquet Capacities / 106
4.2.1. Möbius Representation / 107
4.3. Imprecise Probabilities: General Principles / 110
4.3.1. Lower and Upper Probabilities / 112
4.3.2. Alternating Choquet Capacities / 115
4.3.3. Interaction Representation / 116
4.3.4. Möbius Representation / 119
4.3.5. Joint and Marginal Imprecise Probabilities / 121
4.3.6. Conditional Imprecise Probabilities / 122
4.3.7. Noninteraction of Imprecise Probabilities / 123
4.4. Arguments for Imprecise Probabilities / 129
4.5. Choquet Integral / 133
4.6. Unifying Features of Imprecise Probabilities / 135
Notes / 137
Exercises / 139
5 Special Theories of Imprecise Probabilities 143
5.1. An Overview / 143
5.2. Graded Possibilities / 144
5.2.1. Möbius Representation / 149
5.2.2. Ordering of Possibility Profiles / 151
5.2.3. Joint and Marginal Possibilities / 153
5.2.4. Conditional Possibilities / 155
5.2.5. Possibilities on Infinite Sets / 158
5.2.6. Some Interpretations of Graded Possibilities / 160
5.3. Sugeno l-Measures / 160
5.3.1. Möbius Representation / 165
5.4. Belief and Plausibility Measures / 166
5.4.1. Joint and Marginal Bodies of Evidence / 169
viii CONTENTS
5.4.2. Rules of Combination / 170
5.4.3. Special Classes of Bodies of Evidence / 174
5.5. Reachable Interval-Valued Probability Distributions / 178
5.5.1. Joint and Marginal Interval-Valued Probability
Distributions / 183
5.6. Other Types of Monotone Measures / 185
Notes / 186
Exercises / 190
6 Measures of Uncertainty and Information 196
6.1. General Discussion / 196
6.2. Generalized Hartley Measure for Graded Possibilities / 198
6.2.1. Joint and Marginal U-Uncertainties / 201
6.2.2. Conditional U-Uncertainty / 203
6.2.3. Axiomatic Requirements for the U-Uncertainty / 205
6.2.4. U-Uncertainty for Infinite Sets / 206
6.3. Generalized Hartley Measure in Dempster–Shafer
Theory / 209
6.3.1. Joint and Marginal Generalized Hartley Measures / 209
6.3.2. Monotonicity of the Generalized Hartley Measure / 211
6.3.3. Conditional Generalized Hartley Measures / 213
6.4. Generalized Hartley Measure for Convex Sets of Probability
Distributions / 214
6.5. Generalized Shannon Measure in Dempster-Shafer
Theory / 216
6.6. Aggregate Uncertainty in Dempster–Shafer Theory / 226
6.6.1. General Algorithm for Computing the Aggregate
Uncertainty / 230
6.6.2. Computing the Aggregated Uncertainty in Possibility
Theory / 232
6.7. Aggregate Uncertainty for Convex Sets of Probability
Distributions / 234
6.8. Disaggregated Total Uncertainty / 238
6.9. Generalized Shannon Entropy / 241
6.10. Alternative View of Disaggregated Total Uncertainty / 248
6.11. Unifying Features of Uncertainty Measures / 253
Notes / 253
Exercises / 255
7 Fuzzy Set Theory 260
7.1. An Overview / 260
7.2. Basic Concepts of Standard Fuzzy Sets / 262
7.3. Operations on Standard Fuzzy Sets / 266
CONTENTS ix
7.3.1. Complementation Operations / 266
7.3.2. Intersection and Union Operations / 267
7.3.3. Combinations of Basic Operations / 268
7.3.4. Other Operations / 269
7.4. Fuzzy Numbers and Intervals / 270
7.4.1. Standard Fuzzy Arithmetic / 273
7.4.2. Constrained Fuzzy Arithmetic / 274
7.5. Fuzzy Relations / 280
7.5.1. Projections and Cylindric Extensions / 281
7.5.2. Compositions, Joins, and Inverses / 284
7.6. Fuzzy Logic / 286
7.6.1. Fuzzy Propositions / 287
7.6.2. Approximate Reasoning / 293
7.7. Fuzzy Systems / 294
7.7.1. Granulation / 295
7.7.2. Types of Fuzzy Systems / 297
7.7.3. Defuzzification / 298
7.8. Nonstandard Fuzzy Sets / 299
7.9. Constructing Fuzzy Sets and Operations / 303
Notes / 305
Exercises / 308
8 Fuzzification of Uncertainty Theories 315
8.1. Aspects of Fuzzification / 315
8.2. Measures of Fuzziness / 321
8.3. Fuzzy-Set Interpretation of Possibility Theory / 326
8.4. Probabilities of Fuzzy Events / 334
8.5. Fuzzification of Reachable Interval-Valued Probability
Distributions / 338
8.6. Other Fuzzification Efforts / 348
Notes / 350
Exercises / 351
9 Methodological Issues 355
9.1. An Overview / 355
9.2. Principle of Minimum Uncertainty / 357
9.2.1. Simplification Problems / 358
9.2.2. Conflict-Resolution Problems / 364
9.3. Principle of Maximum Uncertainty / 369
9.3.1. Principle of Maximum Entropy / 369
x CONTENTS
9.3.2. Principle of Maximum Nonspecificity / 373
9.3.3. Principle of Maximum Uncertainty in GIT / 375
9.4. Principle of Requisite Generalization / 383
9.5. Principle of Uncertainty Invariance / 387
9.5.1. Computationally Simple Approximations / 388
9.5.2. Probability–Possibility Transformations / 390
9.5.3. Approximations of Belief Functions by Necessity
Functions / 399
9.5.4. Transformations Between l-Measures and Possibility
Measures / 402
9.5.5. Approximations of Graded Possibilities by Crisp
Possibilities / 403
Notes / 408
Exercises / 411
10 Conclusions 415
10.1. Summary and Assessment of Results in Generalized
Information Theory / 415
10.2. Main Issues of Current Interest / 417
10.3. Long-Term Research Areas / 418
10.4. Significance of GIT / 419
Notes / 421
Appendix A Uniqueness of the U-Uncertainty 425
Appendix B Uniqueness of Generalized Hartley Measure
in the Dempster–Shafer Theory 430
Appendix C Correctness of Algorithm 6.1 437
Appendix D Proper Range of Generalized
Shannon Entropy 442
Appendix E Maximum of GSa in Section 6.9 447
Appendix F Glossary of Key Concepts 449
Appendix G Glossary of Symbols 455
Bibliography 458
Subject Index 487
Name Index 494