Probabilistic Logics and Probabilistic Networks (eBook)

Rolf Haenni is professor at the Department of Engineering and Information Technology of the University of Applied Sciences of Berne (BFH-TI) in Biel, Switzerland. He holds a PhD degree in Computer Science from the University of Fribourg, for which he received the prize for the best thesis in 1996. Jan-Willem Romeijn is an assistant professor at the Philosophy Faculty of the University of Groningen. He obtained degrees cum laude in both physics and philosophy, worked as a financial mathematician and received his doctorate cum laude from the University of Groningen in 2005. Gregory Wheeler is Senior Research Scientist at the Centre for Artificial Intelligence at the New University of Lisbon. He received a joint PhD in Philosophy and Computer Science from the University of Rochester in 2002. Jon Williamson is Professor of Reasoning, Inference and Scientific Method at the University of Kent. He completed his PhD in Philosophy in 1998 and in 2007 was Times Higher Education UK Young Researcher of the Year.

Preface 7
Acknowledgements 8
Contents 9
Part I Probabilistic Logics 12
1 Introduction 13
1.1 The Fundamental Question of Probabilistic Logic 13
1.2 The Potential of Probabilistic Logic 14
1.3 Overview of the Book 15
1.4 Philosophical and Historical Background 17
1.5 Notation and Formal Setting 19
2 Standard Probabilistic Semantics 21
2.1 Background 21
2.1.1 Kolmogorov Probabilities 22
2.1.2 Interval-Valued Probabilities 23
2.1.3 Imprecise Probabilities 25
2.1.4 Convexity 26
2.2 Representation 28
2.3 Interpretation 29
3 Probabilistic Argumentation 31
3.1 Background 32
3.2 Representation 35
3.3 Interpretation 36
3.3.1 Generalizing the Standard Semantics 36
3.3.2 Premises from Unreliable Sources 38
4 Evidential Probability 42
4.1 Background 42
4.1.1 Calculating Evidential Probability 46
4.1.2 Extended Example: When Pigs Die 49
4.2 Representation 53
4.3 Interpretation 53
4.3.1 First-order Evidential Probability 54
4.3.2 Counterfactual Evidential Probability 55
4.3.3 Second-Order Evidential Probability 55
5 Statistical Inference 58
5.1 Background 58
5.1.1 Classical Statistics as Inference? 58
5.1.2 Fiducial Probability 61
5.1.3 Evidential Probability and Direct Inference 64
5.2 Representation 66
5.2.1 Fiducial Probability 66
5.2.2 Evidential Probability and the Fiducial Argument 67
5.3 Interpretation 68
5.3.1 Fiducial Probability 68
5.3.2 Evidential Probability 69
6 Bayesian Statistical Inference 71
6.1 Background 71
6.2 Representation 73
6.2.1 Infinitely Many Hypotheses 74
6.2.2 Interval-Valued Priors and Posteriors 76
6.3 Interpretation 77
6.3.1 Interpretation of Probabilities 77
6.3.2 Bayesian Confidence Intervals 78
7 Objective Bayesian Epistemology 80
7.1 Background 80
7.1.1 Determining Objective Bayesian Degrees of Belief 81
7.1.2 Constraints on Degrees of Belief 82
7.1.3 Propositional Languages 83
7.1.4 Predicate Languages 84
7.1.5 Objective Bayesianism in Perspective 86
7.2 Representation 87
7.3 Interpretation 87
Part II Probabilistic Networks 90
8 Credal and Bayesian Networks 91
8.1 Kinds of Probabilistic Network 92
8.1.1 Extensions 93
8.1.2 Extensions and Coordinates 94
8.1.3 Parameterised Credal Networks 96
8.2 Algorithms for Probabilistic Networks 97
8.2.1 Requirements of the Probabilistic Logic Framework 97
8.2.2 Compiling Probabilistic Networks 98
8.2.3 The Hill-Climbing Algorithm for Credal Networks 100
8.2.4 Complex Queries and Parameterised Credal Networks 102
9 Networks for the Standard Semantics 104
9.1 The Poverty of Standard Semantics 104
9.2 Constructing a Credal Net 105
9.3 Dilation and Independence 109
10 Networks for Probabilistic Argumentation 111
10.1 Probabilistic Argumentation with Credal Sets 111
10.2 Constructing and Applying the Credal Network 112
11 Networks for Evidential Probability 115
11.1 First-Order Evidential Probability 115
11.2 Second-Order Evidential Probability 117
11.3 Chaining Inferences 120
12 Networks for Statistical Inference 122
12.1 Functional Models and Networks 122
12.1.1 Capturing the Fiducial Argument in a Network 122
12.1.2 Aiding Fiducial Inference with Networks 123
12.1.3 Trouble with Step-by-Step Fiducial Probability 125
12.2 Evidential Probability and the Fiducial Argument 126
12.2.1 First-Order EP and the Fiducial Argument 126
12.2.2 Second-Order EP and the Fiducial Argument 127
13 Networks for Bayesian Statistical Inference 128
13.1 Credal Networks as Statistical Hypotheses 128
13.1.1 Construction of the Credal Network 129
13.1.2 Computational Advantages of Using the Credal Network 130
13.2 Extending Statistical Inference with Credal Networks 131
13.2.1 Interval-Valued Likelihoods 132
13.2.2 Logically Complex Statements with Statistical Hypotheses 134
14 Networks for Objective Bayesianism 135
14.1 Propositional Languages 135
14.2 Predicate Languages 137
15 Conclusion 140
References 141
Index 152