ISBN: 3790815276
TITLE: A Mathematical Theory of Arguments..
AUTHOR: Monney
TOC:

1. The Theory of Generalized Functional Models 1
1 1 The Theory of Hints on Finite Frames 1
1.1.1 Definition of a Hint and its Associated Functions 2
1.1.2 Combination of Hints 5
1.1.3 Mass functions and gem-functions 7
1.1.4 Dempster Specialization Matrices 10
1.1.5 A Representation of the Dempster Specialization Matrix 11
1.1.6 Combining Several Hints 12
1.2 Application 13
1.3 The Combination of Closed Hints 14
1.4 The Theories of Bayes and Fisher 18
1.4.1 Jessica's Pregnancy Test 18
1.4.2 The Solution of Bayes 18
1.4.3 The Solution of Fisher 19
1.5 The Definition and Analysis of a Generalized Functional Model 21
1 6 Generalized Functional Models and Hints 24
1.7 Examples of Generalized Functional Models 26
1.7.1 Jessica's Pregnancy Test 26
1.7.2 Policy Identification (I) 28
1.7.3 Policy Identification (II) 31
1 8 Prior Information 33
2. The Plausibility and Likelihood Functions 39
2.1 The Likelihood Ratio as a Weight of Evidence 39
2.2 The Weight of Evidence for Composite Hypotheses 40
2.3 Functional and Distribution Models 43
2.3.1 The Distribution Model of the Problem 43
2.3.2 A GFM Obtained by Conditional Embedding 44
2.3.3 A GFM Inspired by Dempster's Structures of the First Kind 46
2.4 Evidence About a Survival Rate 49
2.5 Degrees of Support as Weights of Evidence 55
3. Hints on Continuous Frames and Gaussian Linear Systems 59
3.1 Continuous Hints 59
3.2 Gaussian Hints 62
3.3 Precise Gaussian Hints 64
3.4 Gaussian Linear Systems 68
4. Assumption-Based Reasoning with Classical Regression Models 71
4.1 Classical Linear Regression Models as Special Gaussian Linear Systems 71
4.2 The Principle of the Inference 72
4.3 The Result of the Inference 76
4.4 Changing Basis 79
4.5 Permissible Bases and Admissible Matrices 82
4.6 Different Representations of the Result of the Inference 86
4.6.1 A Representation Derived from Linear Dependencies 86
4.6.2 A Representation Derived from a Special Class of Admissible Matrices 88
4.7 Full Rank Matrices T_2 such that T_2A = 0 90
4.7.1 A Matrix Based on Linear Dependencies 90
4.7.2 A Matrix Based on the Householder Method 91
4.7.3 A Matrix Based on Classical Variable Elimination 91
4.7.4 A Matrix Based on a Generalized Inverse of A 93
5. Assumption-Based Reasoning with General Gaussian Linear Systems 97
5.1 The Principle of the Inference 97
5.2 The Gaussian Hint Inferred from a Gaussian Linear System of the First Kind 98
5.3 The Gaussian Hint Inferred from a Gaussian Linear System of the Second Kind 99
5.4 The Gaussian Hint Inferred from a Gaussian Linear System of the Third Kind 104
5.5 The Gaussian Hint Inferred from a Gaussian Linear System of the Fourth Kind. 105
5.6 Canonical Proper Potentials 106
6. Gaussian Hints as a Valuation System 109
6.1 Shenoy-Shafer's Axiomatic Valuation Systems 109
6.2 Marginalization of Gaussian Hints 110
6.2.1 Marginalization of a Precise Gaussian Hint 111
6.2.2 Marginalization of a Non-Precise Gaussian Hint 112
6.3 Vacuous Extension of Gaussian Hints 117
6.4 Transport of Gaussian Hints 117
6.5 Projection and Composition of Potentials 118
6.6 Combination of Gaussian Hints Defined on the Same Domain 121
6.7 Computation of Combined Gaussian Hints 123
6.7.1 Combination when the Composed Potential is of the Third Kind 123
6.7.2 Combination with a Precise Gaussian Hint 124
6.7.3 Combination of Two Precise Gaussian Hints 125
6.7.4 Combination when the Composed Potential is of the First Kind 126
6.8 Combination of Hints Defined on Arbitrary Domains 127
7. Local Propagation of Gaussian Hints 129
7 1 The Axioms of Shenoy and Shafer 130
7.2 The Local Propagation Algorithm 133
8. Application to the Kalman Filter 137
8 1 Definition of the Random System 137
8.2 Derivation of the Kalman Filter 138
References 149
Index 153
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