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El. knyga: Subjective Logic: A Formalism for Reasoning Under Uncertainty

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This is the first comprehensive treatment of subjective logic and all its operations. The author developed the approach, and in this book he first explains subjective opinions, opinion representation, and decision-making under vagueness and uncertainty, and he then offers a full definition of subjective logic, harmonising the key notations and formalisms, concluding with chapters on trust networks and subjective Bayesian networks, which when combined form general subjective networks. The author shows how real-world situations can be realistically modelled with regard to how situations are perceived, with conclusions that more correctly reflect the ignorance and uncertainties that result from partially uncertain input arguments.The book will help researchers and practitioners to advance, improve and apply subjective logic to build powerful artificial reasoning models and tools for solving real-world problems. A good grounding in discrete mathematics is a prerequisite.

Introduction.- Elements of Subjective Opinions.- Opinion Representations.- Decision-Making Under Vagueness and Uncertainty.- Principles of Subjective Logic.- Addition, Subtraction and Complement.- Binomial Multiplication and Division.- Multinomial Multiplication and Division.- Conditional Deduction.- Conditional Abduction.- Joint and Marginal Opinions.- Fusion of Subjective Opinions.- Unfusion and Fission of Subjective Opinions.- Computational Trust.- Trust Networks.- Bayesian Reputation Systems.- Subjective Networks.
1 Introduction
1(6)
2 Elements of Subjective Opinions
7(12)
2.1 Motivation for the Opinion Representation
7(1)
2.2 Flexibility of Representation
8(1)
2.3 Domains and Hyperdomains
8(4)
2.4 Random Variables and Hypervariables
12(1)
2.5 Belief Mass Distribution and Uncertainty Mass
13(1)
2.6 Base Rate Distributions
14(3)
2.7 Probability Distributions
17(2)
3 Opinion Representations
19(32)
3.1 Belief and Trust Relationships
19(1)
3.2 Opinion Classes
20(2)
3.3 Aleatory and Epistemic Opinions
22(2)
3.4 Binomial Opinions
24(6)
3.4.1 Binomial Opinion Representation
24(2)
3.4.2 The Beta Binomial Model
26(2)
3.4.3 Mapping Between a Binomial Opinion and a Beta PDF
28(2)
3.5 Multinomial Opinions
30(9)
3.5.1 The Multinomial Opinion Representation
30(1)
3.5.2 The Dirichlet Multinomial Model
31(3)
3.5.3 Visualising Dirichlet Probability Density Functions
34(1)
3.5.4 Coarsening Example: From Ternary to Binary
34(2)
3.5.5 Mapping Between Multinomial Opinion and Dirichlet PDF
36(1)
3.5.6 Uncertainty-Maximisation
37(2)
3.6 Hyper-opinions
39(7)
3.6.1 The Hyper-opinion Representation
39(1)
3.6.2 Projecting Hyper-opinions to Multinomial Opinions
40(1)
3.6.3 The Dirichlet Model Applied to Hyperdomains
41(1)
3.6.4 Mapping Between a Hyper-opinion and a Dirichlet HPDF
42(1)
3.6.5 Hyper-Dirichlet PDF
43(3)
3.7 Alternative Opinion Representations
46(5)
3.7.1 Probabilistic Notation of Opinions
46(2)
3.7.2 Qualitative Opinion Representation
48(3)
4 Decision Making Under Vagueness and Uncertainty
51(32)
4.1 Aspects of Belief and Uncertainty in Opinions
51(5)
4.1.1 Sharp Belief Mass
51(1)
4.1.2 Vague Belief Mass
52(2)
4.1.3 Dirichlet Visualisation of Opinion Vagueness
54(1)
4.1.4 Focal Uncertainty Mass
55(1)
4.2 Mass-Sum
56(3)
4.2.1 Mass-Sum of a Value
56(2)
4.2.2 Total Mass-Sum
58(1)
4.3 Utility and Normalisation
59(4)
4.4 Decision Criteria
63(2)
4.5 The Ellsberg Paradox
65(4)
4.6 Examples of Decision Making
69(6)
4.6.1 Decisions with Difference in Projected Probability
69(2)
4.6.2 Decisions with Difference in Sharpness
71(2)
4.6.3 Decisions with Difference in Vagueness and Uncertainty
73(2)
4.7 Entropy in the Opinion Model
75(4)
4.7.1 Outcome Surprisal
76(1)
4.7.2 Opinion Entropy
77(2)
4.8 Conflict Between Opinions
79(3)
4.9 Ambiguity
82(1)
5 Principles of Subjective Logic
83(12)
5.1 Related Frameworks for Uncertain Reasoning
83(5)
5.1.1 Comparison with Dempster-Shafer Belief Theory
83(2)
5.1.2 Comparison with Imprecise Probabilities
85(1)
5.1.3 Comparison with Fuzzy Logic
86(1)
5.1.4 Comparison with Kleene's Three-Valued Logic
87(1)
5.2 Subjective Logic as a Generalisation of Probabilistic Logic
88(4)
5.3 Overview of Subjective-Logic Operators
92(3)
6 Addition, Subtraction and Complement
95(6)
6.1 Addition
95(2)
6.2 Subtraction
97(2)
6.3 Complement
99(2)
7 Binomial Multiplication and Division
101(14)
7.1 Binomial Multiplication and Comultiplication
101(6)
7.1.1 Binomial Multiplication
102(1)
7.1.2 Binomial Comultiplication
103(1)
7.1.3 Approximations of Product and Coproduct
104(3)
7.2 Reliability Analysis
107(3)
7.2.1 Simple Reliability Networks
107(2)
7.2.2 Reliability Analysis of Complex Systems
109(1)
7.3 Binomial Division and Codivision
110(4)
7.3.1 Binomial Division
110(2)
7.3.2 Binomial Codivision
112(2)
7.4 Correspondence with Probabilistic Logic
114(1)
8 Multinomial Multiplication and Division
115(18)
8.1 Multinomial Multiplication
115(10)
8.1.1 Elements of Multinomial Multiplication
115(3)
8.1.2 Normal Multiplication
118(2)
8.1.3 Justification for Normal Multinomial Multiplication
120(1)
8.1.4 Proportional Multiplication
120(1)
8.1.5 Projected Multiplication
121(1)
8.1.6 Hypernomial Product
122(1)
8.1.7 Product of Dirichlet Probability Density Functions
123(2)
8.2 Examples of Multinomial Product Computation
125(3)
8.2.1 Comparing Normal, Proportional and Projected Products
126(1)
8.2.2 Hypernomial Product Computation
127(1)
8.3 Multinomial Division
128(5)
8.3.1 Elements of Multinomial Division
128(1)
8.3.2 Averaging Proportional Division
129(2)
8.3.3 Selective Division
131(2)
9 Conditional Reasoning and Subjective Deduction
133(38)
9.1 Introduction to Conditional Reasoning
133(3)
9.2 Probabilistic Conditional Inference
136(6)
9.2.1 Bayes' Theorem
136(3)
9.2.2 Binomial Probabilistic Deduction and Abduction
139(1)
9.2.3 Multinomial Probabilistic Deduction and Abduction
140(2)
9.3 Notation for Subjective Conditional Inference
142(5)
9.3.1 Notation for Binomial Deduction and Abduction
143(1)
9.3.2 Notation for Multinomial Deduction and Abduction
144(3)
9.4 Binomial Deduction
147(7)
9.4.1 Marginal Base Rate for Binomial Opinions
147(1)
9.4.2 Free Base-Rate Interval
148(2)
9.4.3 Method for Binomial Deduction
150(2)
9.4.4 Justification for the Binomial Deduction Operator
152(2)
9.5 Multinomial Deduction
154(8)
9.5.1 Marginal Base Rate Distribution
155(1)
9.5.2 Free Base-Rate Distribution Intervals
155(2)
9.5.3 Constraints for Multinomial Deduction
157(2)
9.5.4 Method for Multinomial Deduction
159(3)
9.6 Example: Match-Fixing
162(2)
9.7 Interpretation of Material Implication in Subjective Logic
164(7)
9.7.1 Truth-Functional Material Implication
164(1)
9.7.2 Material Probabilistic Implication
165(2)
9.7.3 Relevance in Implication
167(1)
9.7.4 Subjective Interpretation of Material Implication
168(1)
9.7.5 Comparison with Subjective Logic Deduction
169(1)
9.7.6 How to Interpret Material Implication
170(1)
10 Subjective Abduction
171(28)
10.1 Introduction to Abductive Reasoning
171(2)
10.2 Relevance and Dependence
173(2)
10.2.1 Relevance and Irrelevance
174(1)
10.2.2 Dependence and Independence
175(1)
10.3 Binomial Subjective Bayes' Theorem
175(8)
10.3.1 Principles for Inverting Binomial Conditional Opinions
175(2)
10.3.2 Uncertainty Mass of Inverted Binomial Conditionals
177(3)
10.3.3 Deriving Binomial Inverted Conditionals
180(1)
10.3.4 Convergence of Repeated Inversions
181(2)
10.4 Binomial Abduction
183(1)
10.5 Illustrating the Base-Rate Fallacy
184(3)
10.6 The Multinomial Subjective Bayes' Theorem
187(6)
10.6.1 Principles for Inverting Multinomial Conditional Opinions
187(2)
10.6.2 Uncertainty Mass of Inverted Multinomial Conditionals
189(3)
10.6.3 Deriving Multinomial Inverted Conditionals
192(1)
10.7 Multinomial Abduction
193(1)
10.8 Example: Military Intelligence Analysis
194(5)
10.8.1 Example: Intelligence Analysis with Probability Calculus
194(2)
10.8.2 Example: Intelligence Analysis with Subjective Logic
196(3)
11 Joint and Marginal Opinions
199(8)
11.1 Joint Probability Distributions
199(2)
11.2 Joint Opinion Computation
201(2)
11.2.1 Joint Base Rate Distribution
201(1)
11.2.2 Joint Uncertainty Mass
202(1)
11.2.3 Assembling the Joint Opinion
203(1)
11.3 Opinion Marginalisation
203(2)
11.3.1 Opinion Marginalisation Method
204(1)
11.4 Example: Match-Fixing Revisited
205(2)
11.4.1 Computing the Join Opinion
205(1)
11.4.2 Computing Marginal Opinions
206(1)
12 Belief Fusion
207(30)
12.1 Interpretation of Belief Fusion
207(8)
12.1.1 Correctness and Consistency Criteria for Fusion Models
209(2)
12.1.2 Classes of Fusion Situations
211(2)
12.1.3 Criteria for Fusion Operator Selection
213(2)
12.2 Belief Constraint Fusion
215(10)
12.2.1 Method of Constraint Fusion
216(1)
12.2.2 Frequentist Interpretation of Constraint Fusion
217(4)
12.2.3 Expressing Preferences with Subjective Opinions
221(2)
12.2.4 Example: Going to the Cinema, First Attempt
223(1)
12.2.5 Example: Going to the Cinema, Second Attempt
224(1)
12.2.6 Example: Not Going to the Cinema
225(1)
12.3 Cumulative Fusion
225(4)
12.3.1 Aleatory Cumulative Fusion
225(3)
12.3.2 Epistemic Cumulative Fusion
228(1)
12.4 Averaging Belief Fusion
229(2)
12.5 Weighted Belief Fusion
231(2)
12.6 Consensus & Compromise Fusion
233(2)
12.7 Example Comparison of Fusion Operators
235(2)
13 Unfusion and Fission of Subjective Opinions
237(6)
13.1 Unfusion of Opinions
237(3)
13.1.1 Cumulative Unfusion
238(1)
13.1.2 Averaging Unfusion
239(1)
13.1.3 Example: Cumulative Unfusion of Binomial Opinions
240(1)
13.2 Fission of Opinions
240(3)
13.2.1 Cumulative Fission
240(2)
13.2.2 Example Fission of Opinion
242(1)
13.2.3 Averaging Fission
242(1)
14 Computational Trust
243(28)
14.1 The Notion of Trust
243(6)
14.1.1 Reliability Trust
244(2)
14.1.2 Decision Trust
246(2)
14.1.3 Reputation and Trust
248(1)
14.2 Trust Transitivity
249(5)
14.2.1 Motivating Example for Transitive Trust
249(2)
14.2.2 Referral Trust and Functional Trust
251(1)
14.2.3 Notation for Transitive Trust
252(1)
14.2.4 Compact Notation for Transitive Trust Paths
253(1)
14.2.5 Semantic Requirements for Trust Transitivity
253(1)
14.3 The Trust-Discounting Operator
254(8)
14.3.1 Principle of Trust Discounting
254(1)
14.3.2 Trust Discounting with Two-Edge Paths
255(2)
14.3.3 Example: Trust Discounting of Restaurant Advice
257(2)
14.3.4 Trust Discounting for Multi-edge Path
259(3)
14.4 Trust Fusion
262(3)
14.5 Trust Revision
265(6)
14.5.1 Motivation for Trust Revision
265(1)
14.5.2 Trust Revision Method
266(2)
14.5.3 Example: Conflicting Restaurant Recommendations
268(3)
15 Subjective Trust Networks
271(18)
15.1 Graphs for Trust Networks
271(1)
15.1.1 Directed Series-Parallel Graphs
271(1)
15.2 Outbound-Inbound Set
272(3)
15.2.1 Parallel-Path Subnetworks
273(1)
15.2.2 Nesting Level
274(1)
15.3 Analysis of DSPG Trust Networks
275(4)
15.3.1 Algorithm for Analysis of DSPG
276(1)
15.3.2 Soundness Requirements for Receiving Advice Opinions
277(2)
15.4 Analysing Complex Non-DSPG Trust Networks
279(10)
15.4.1 Synthesis of DSPG Trust Network
282(2)
15.4.2 Criteria for DSPG Synthesis
284(5)
16 Bayesian Reputation Systems
289(14)
16.1 Computing Reputation Scores
291(1)
16.1.1 Binomial Reputation Score
291(1)
16.1.2 Multinomial Reputation Scores
291(1)
16.2 Collecting and Aggregating Ratings
292(2)
16.2.1 Collecting Ratings
292(1)
16.2.2 Aggregating Ratings with Ageing
293(1)
16.2.3 Reputation Score Convergence with Time Decay
293(1)
16.3 Base Rates for Ratings
294(3)
16.3.1 Individual Base Rates
294(1)
16.3.2 Total History Base Rate
295(1)
16.3.3 Sliding Time Window Base Rate
295(1)
16.3.4 High Longevity Factor Base Rate
295(1)
16.3.5 Dynamic Community Base Rate
296(1)
16.4 Reputation Representation
297(2)
16.4.1 Multinomial Probability Representation
297(1)
16.4.2 Point Estimate Representation
298(1)
16.4.3 Continuous Ratings
299(1)
16.5 Simple Scenario Simulation
299(2)
16.6 Combining Trust and Reputation
301(2)
17 Subjective Networks
303(24)
17.1 Bayesian Networks
304(8)
17.1.1 Example: Lung Cancer Situation
306(2)
17.1.2 Variable Structures
308(1)
17.1.3 The Chain Rule of Conditional Probability
309(1)
17.1.4 Naive Bayes Classifier
310(1)
17.1.5 Independence and Separation
310(2)
17.2 Chain Rules for Subjective Bayesian Networks
312(4)
17.2.1 Chained Conditional Opinions
312(1)
17.2.2 Chained Inverted Opinions
313(2)
17.2.3 Validation of the Subjective Bayes' Theorem
315(1)
17.2.4 Chained Joint Opinions
316(1)
17.3 Subjective Bayesian Networks
316(5)
17.3.1 Subjective Predictive Reasoning
317(1)
17.3.2 Subjective Diagnostic Reasoning
318(1)
17.3.3 Subjective Intercausal Reasoning
319(1)
17.3.4 Subjective Combined Reasoning
320(1)
17.4 Independence Properties in Subjective Bayesian Networks
321(2)
17.5 Subjective Network Modelling
323(2)
17.5.1 Subjective Network with Source Opinions
324(1)
17.5.2 Subjective Network with Trust Fusion
324(1)
17.6 Perspectives on Subjective Networks
325(2)
References 327(6)
Acronyms 333(2)
Index 335
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