Subjective Logic (eBook)

Foreword 6
Preface 8
Acknowledgements 10
Contents 13
Chapter 1 Introduction 20
Chapter 2 Elements of Subjective Opinions 26
2.1 Motivation for the Opinion Representation 26
2.2 Flexibility of Representation 27
2.3 Domains and Hyperdomains 27
2.4 Random Variables and Hypervariables 31
2.5 Belief Mass Distribution and Uncertainty Mass 32
2.6 Base Rate Distributions 33
2.7 Probability Distributions 36
Chapter 3 Opinion Representations 38
3.1 Belief and Trust Relationships 38
3.2 Opinion Classes 39
3.3 Aleatory and Epistemic Opinions 41
3.4 Binomial Opinions 43
3.4.1 Binomial Opinion Representation 43
3.4.2 The Beta Binomial Model 45
3.4.3 Mapping Between a Binomial Opinion and a Beta PDF 47
3.5 Multinomial Opinions 49
3.5.1 The Multinomial Opinion Representation 49
3.5.2 The Dirichlet Multinomial Model 50
3.5.3 Visualising Dirichlet Probability Density Functions 53
3.5.4 Coarsening Example: From Ternary to Binary 53
3.5.5 Mapping Between Multinomial Opinion and Dirichlet PDF 55
3.5.6 Uncertainty-Maximisation 56
3.6 Hyper-opinions 58
3.6.1 The Hyper-opinion Representation 58
3.6.2 Projecting Hyper-opinions to Multinomial Opinions 59
3.6.3 The Dirichlet Model Applied to Hyperdomains 60
3.6.4 Mapping Between a Hyper-opinion and a Dirichlet HPDF 61
3.6.5 Hyper-Dirichlet PDF 62
3.7 Alternative Opinion Representations 65
3.7.1 Probabilistic Notation of Opinions 65
3.7.2 Qualitative Opinion Representation 67
Chapter 4 Decision Making Under Vagueness and Uncertainty 70
4.1 Aspects of Belief and Uncertainty in Opinions 70
4.1.1 Sharp Belief Mass 70
4.1.2 Vague Belief Mass 71
4.1.3 Dirichlet Visualisation of Opinion Vagueness 73
4.1.4 Focal Uncertainty Mass 74
4.2 Mass-Sum 75
4.2.1 Mass-Sum of a Value 75
4.2.2 Total Mass-Sum 77
4.3 Utility and Normalisation 78
4.4 Decision Criteria 82
4.5 The Ellsberg Paradox 84
4.6 Examples of Decision Making 88
4.6.1 Decisions with Difference in Projected Probability 88
4.6.2 Decisions with Difference in Sharpness 90
4.6.3 Decisions with Difference in Vagueness and Uncertainty 92
4.7 Entropy in the Opinion Model 94
4.7.1 Outcome Surprisal 95
4.7.2 Opinion Entropy 96
4.8 Conflict Between Opinions 98
4.9 Ambiguity 101
Chapter 5Principles of Subjective Logic 102
5.1 Related Frameworks for Uncertain Reasoning 102
5.1.1 Comparison with Dempster-Shafer Belief Theory 102
5.1.2 Comparison with Imprecise Probabilities 104
5.1.3 Comparison with Fuzzy Logic 105
5.1.4 Comparison with Kleene’s Three-Valued Logic 106
5.2 Subjective Logic as a Generalisation of Probabilistic Logic 107
5.3 Overview of Subjective-Logic Operators 111
Chapter 6 Addition, Subtraction and Complement 114
6.1 Addition 114
6.2 Subtraction 116
6.3 Complement 118
Chapter 7 Binomial Multiplication and Division 120
7.1 Binomial Multiplication and Comultiplication 120
7.1.1 Binomial Multiplication 121
7.1.2 Binomial Comultiplication 122
7.1.3 Approximations of Product and Coproduct 123
7.2 Reliability Analysis 126
7.2.1 Simple Reliability Networks 126
7.2.2 Reliability Analysis of Complex Systems 128
7.3 Binomial Division and Codivision 129
7.3.1 Binomial Division 129
7.3.2 Binomial Codivision 131
7.4 Correspondence with Probabilistic Logic 133
Chapter 8 Multinomial Multiplication and Division 134
8.1 Multinomial Multiplication 134
8.1.1 Elements of Multinomial Multiplication 134
8.1.2 Normal Multiplication 137
8.1.3 Justification for Normal Multinomial Multiplication 139
8.1.4 Proportional Multiplication 139
8.1.5 Projected Multiplication 140
8.1.6 Hypernomial Product 141
8.1.7 Product of Dirichlet Probability Density Functions 142
8.2 Examples of Multinomial Product Computation 144
8.2.1 Comparing Normal, Proportional and Projected Products 145
8.2.2 Hypernomial Product Computation 146
8.3 Multinomial Division 147
8.3.1 Elements of Multinomial Division 147
8.3.2 Averaging Proportional Division 148
8.3.3 Selective Division 150
Chapter 9 Conditional Reasoning and Subjective Deduction 152
9.1 Introduction to Conditional Reasoning 152
9.2 Probabilistic Conditional Inference 155
9.2.1 Bayes’ Theorem 155
9.2.2 Binomial Probabilistic Deduction and Abduction 158
9.2.3 Multinomial Probabilistic Deduction and Abduction 159
9.3 Notation for Subjective Conditional Inference 161
9.3.1 Notation for Binomial Deduction and Abduction 162
9.3.2 Notation for Multinomial Deduction and Abduction 163
9.4 Binomial Deduction 166
9.4.1 Marginal Base Rate for Binomial Opinions 166
9.4.2 Free Base-Rate Interval 167
9.4.3 Method for Binomial Deduction 169
9.4.4 Justification for the Binomial Deduction Operator 171
9.5 Multinomial Deduction 173
9.5.1 Marginal Base Rate Distribution 174
9.5.2 Free Base-Rate Distribution Intervals 174
9.5.3 Constraints for Multinomial Deduction 176
9.5.4 Method for Multinomial Deduction 178
9.6 Example: Match-Fixing 181
9.7 Interpretation of Material Implication in Subjective Logic 183
9.7.1 Truth-Functional Material Implication 183
9.7.2 Material Probabilistic Implication 184
9.7.3 Relevance in Implication 186
9.7.4 Subjective Interpretation of Material Implication 187
9.7.5 Comparison with Subjective Logic Deduction 188
9.7.6 How to Interpret Material Implication 189
Chapter 10 Subjective Abduction 190
10.1 Introduction to Abductive Reasoning 190
10.2 Relevance and Dependence 192
10.2.1 Relevance and Irrelevance 193
10.2.2 Dependence and Independence 194
10.3 Binomial Subjective Bayes’ Theorem 194
10.3.1 Principles for Inverting Binomial Conditional Opinions 194
10.3.2 Uncertainty Mass of Inverted Binomial Conditionals 196
10.3.3 Deriving Binomial Inverted Conditionals 199
10.3.4 Convergence of Repeated Inversions 200
10.4 Binomial Abduction 202
10.5 Illustrating the Base-Rate Fallacy 203
10.6 The Multinomial Subjective Bayes’ Theorem 206
10.6.1 Principles for Inverting Multinomial Conditional Opinions 206
10.6.2 Uncertainty Mass of Inverted Multinomial Conditionals 208
10.6.3 Deriving Multinomial Inverted Conditionals 211
10.7 Multinomial Abduction 212
10.8 Example: Military Intelligence Analysis 213
10.8.1 Example: Intelligence Analysis with Probability Calculus 213
10.8.2 Example: Intelligence Analysis with Subjective Logic 215
Chapter 11 Joint and Marginal Opinions 218
11.1 Joint Probability Distributions 218
11.2 Joint Opinion Computation 220
11.2.1 Joint Base Rate Distribution 220
11.2.2 Joint Uncertainty Mass 221
11.2.3 Assembling the Joint Opinion 222
11.3 Opinion Marginalisation 222
11.3.1 Opinion Marginalisation Method 223
11.4 Example: Match-Fixing Revisited 224
11.4.1 Computing the Join Opinion 224
11.4.2 Computing Marginal Opinions 225
Chapter 12 Belief Fusion 226
12.1 Interpretation of Belief Fusion 226
12.1.1 Correctness and Consistency Criteria for Fusion Models 228
12.1.2 Classes of Fusion Situations 230
12.1.3 Criteria for Fusion Operator Selection 232
12.2 Belief Constraint Fusion 234
12.2.1 Method of Constraint Fusion 235
12.2.2 Frequentist Interpretation of Constraint Fusion 236
12.2.3 Expressing Preferences with Subjective Opinions 240
12.2.4 Example: Going to the Cinema, First Attempt 242
12.2.5 Example: Going to the Cinema, Second Attempt 243
12.2.6 Example: Not Going to the Cinema 244
12.3 Cumulative Fusion 244
12.3.1 Aleatory Cumulative Fusion 244
12.3.2 Epistemic Cumulative Fusion 247
12.4 Averaging Belief Fusion 248
12.5 Weighted Belief Fusion 250
12.6 Consensus & Compromise Fusion
12.7 Example Comparison of Fusion Operators 254
Chapter 13 Unfusion and Fission of Subjective Opinions 256
13.1 Unfusion of Opinions 256
13.1.1 Cumulative Unfusion 257
13.1.2 Averaging Unfusion 258
13.1.3 Example: Cumulative Unfusion of Binomial Opinions 259
13.2 Fission of Opinions 259
13.2.1 Cumulative Fission 259
13.2.2 Example Fission of Opinion 261
13.2.3 Averaging Fission 261
Chapter 14 Computational Trust 262
14.1 The Notion of Trust 262
14.1.1 Reliability Trust 263
14.1.2 Decision Trust 265
14.1.3 Reputation and Trust 267
14.2 Trust Transitivity 268
14.2.1 Motivating Example for Transitive Trust 268
14.2.2 Referral Trust and Functional Trust 270
14.2.3 Notation for Transitive Trust 271
14.2.4 Compact Notation for Transitive Trust Paths 272
14.2.5 Semantic Requirements for Trust Transitivity 272
14.3 The Trust-Discounting Operator 273
14.3.1 Principle of Trust Discounting 273
14.3.2 Trust Discounting with Two-Edge Paths 274
14.3.3 Example: Trust Discounting of Restaurant Advice 276
14.3.4 Trust Discounting for Multi-edge Path 278
14.4 Trust Fusion 281
14.5 Trust Revision 284
14.5.1 Motivation for Trust Revision 284
14.5.2 Trust Revision Method 285
14.5.3 Example: Conflicting Restaurant Recommendations 287
Chapter 15 Subjective Trust Networks 290
15.1 Graphs for Trust Networks 290
15.1.1 Directed Series-Parallel Graphs 290
15.2 Outbound-Inbound Set 291
15.2.1 Parallel-Path Subnetworks 292
15.2.2 Nesting Level 293
15.3 Analysis of DSPG Trust Networks 294
15.3.1 Algorithm for Analysis of DSPG 295
15.3.2 Soundness Requirements for Receiving Advice Opinions 296
15.4 Analysing Complex Non-DSPG Trust Networks 298
15.4.1 Synthesis of DSPG Trust Network 301
15.4.2 Criteria for DSPG Synthesis 303
Chapter 16 Bayesian Reputation Systems 308
16.1 Computing Reputation Scores 310
16.1.1 Binomial Reputation Score 310
16.1.2 Multinomial Reputation Scores 310
16.2 Collecting and Aggregating Ratings 311
16.2.1 Collecting Ratings 311
16.2.2 Aggregating Ratings with Ageing 312
16.2.3 Reputation Score Convergence with Time Decay 312
16.3 Base Rates for Ratings 313
16.3.1 Individual Base Rates 313
16.3.2 Total History Base Rate 314
16.3.3 Sliding Time Window Base Rate 314
16.3.4 High Longevity Factor Base Rate 314
16.3.5 Dynamic Community Base Rate 315
16.4 Reputation Representation 316
16.4.1 Multinomial Probability Representation 316
16.4.2 Point Estimate Representation 317
16.4.3 Continuous Ratings 318
16.5 Simple Scenario Simulation 318
16.6 Combining Trust and Reputation 320
Chapter 17 Subjective Networks 322
17.1 Bayesian Networks 323
17.1.1 Example: Lung Cancer Situation 325
17.1.2 Variable Structures 327
17.1.3 The Chain Rule of Conditional Probability 328
17.1.4 Na¨ive Bayes Classifier 329
17.1.5 Independence and Separation 329
17.2 Chain Rules for Subjective Bayesian Networks 331
17.2.1 Chained Conditional Opinions 331
17.2.2 Chained Inverted Opinions 332
17.2.3 Validation of the Subjective Bayes’ Theorem 334
17.2.4 Chained Joint Opinions 335
17.3 Subjective Bayesian Networks 335
17.3.1 Subjective Predictive Reasoning 336
17.3.2 Subjective Diagnostic Reasoning 337
17.3.3 Subjective Intercausal Reasoning 338
17.3.4 Subjective Combined Reasoning 339
17.4 Independence Properties in Subjective Bayesian Networks 340
17.5 Subjective Network Modelling 342
17.5.1 Subjective Network with Source Opinions 343
17.5.2 Subjective Network with Trust Fusion 343
17.6 Perspectives on Subjective Networks 344
References 346
Acronyms 351
Index 353