Haenni / Romeijn / Wheeler Probabilistic Logics and Probabilistic Networks

1. Auflage 2010
ISBN: 978-94-007-0008-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 155 Seiten

While probabilistic logics in principle might be applied to solve a range of problems, in practice they are rarely applied - perhaps because they seem disparate, complicated, and computationally intractable. This programmatic book argues that several approaches to probabilistic logic fit into a simple unifying framework in which logically complex evidence is used to associate probability intervals or probabilities with sentences. Specifically, Part I shows that there is a natural way to present a question posed in probabilistic logic, and that various inferential procedures provide semantics for that question, while Part II shows that there is the potential to develop computationally feasible methods to mesh with this framework. The book is intended for researchers in philosophy, logic, computer science and statistics. A familiarity with mathematical concepts and notation is presumed, but no advanced knowledge of logic or probability theory is required.

Autoren/Hrsg.

Haenni, Rolf

Romeijn, Jan-Willem

Wheeler, Gregory

Williamson, Jon

Inhaltsverzeichnis

1;Preface;7
2;Acknowledgements;8
3;Contents;9
4;Part I Probabilistic Logics;12
4.1;1 Introduction;13
4.1.1;1.1 The Fundamental Question of Probabilistic Logic;13
4.1.2;1.2 The Potential of Probabilistic Logic;14
4.1.3;1.3 Overview of the Book;15
4.1.4;1.4 Philosophical and Historical Background;17
4.1.5;1.5 Notation and Formal Setting;19
4.2;2 Standard Probabilistic Semantics;21
4.2.1;2.1 Background;21
4.2.1.1;2.1.1 Kolmogorov Probabilities;22
4.2.1.2;2.1.2 Interval-Valued Probabilities;23
4.2.1.3;2.1.3 Imprecise Probabilities;25
4.2.1.4;2.1.4 Convexity;26
4.2.2;2.2 Representation;28
4.2.3;2.3 Interpretation;29
4.3;3 Probabilistic Argumentation;31
4.3.1;3.1 Background;32
4.3.2;3.2 Representation;35
4.3.3;3.3 Interpretation;36
4.3.3.1;3.3.1 Generalizing the Standard Semantics;36
4.3.3.2;3.3.2 Premises from Unreliable Sources;38
4.4;4 Evidential Probability;42
4.4.1;4.1 Background;42
4.4.1.1;4.1.1 Calculating Evidential Probability;46
4.4.1.2;4.1.2 Extended Example: When Pigs Die;49
4.4.2;4.2 Representation;53
4.4.3;4.3 Interpretation;53
4.4.3.1;4.3.1 First-order Evidential Probability;54
4.4.3.2;4.3.2 Counterfactual Evidential Probability;55
4.4.3.3;4.3.3 Second-Order Evidential Probability;55
4.5;5 Statistical Inference;58
4.5.1;5.1 Background;58
4.5.1.1;5.1.1 Classical Statistics as Inference?;58
4.5.1.2;5.1.2 Fiducial Probability;61
4.5.1.3;5.1.3 Evidential Probability and Direct Inference;64
4.5.2;5.2 Representation;66
4.5.2.1;5.2.1 Fiducial Probability;66
4.5.2.2;5.2.2 Evidential Probability and the Fiducial Argument;67
4.5.3;5.3 Interpretation;68
4.5.3.1;5.3.1 Fiducial Probability;68
4.5.3.2;5.3.2 Evidential Probability;69
4.6;6 Bayesian Statistical Inference;71
4.6.1;6.1 Background;71
4.6.2;6.2 Representation;73
4.6.2.1;6.2.1 Infinitely Many Hypotheses;74
4.6.2.2;6.2.2 Interval-Valued Priors and Posteriors;76
4.6.3;6.3 Interpretation;77
4.6.3.1;6.3.1 Interpretation of Probabilities;77
4.6.3.2;6.3.2 Bayesian Confidence Intervals;78
4.7;7 Objective Bayesian Epistemology;80
4.7.1;7.1 Background;80
4.7.1.1;7.1.1 Determining Objective Bayesian Degrees of Belief;81
4.7.1.2;7.1.2 Constraints on Degrees of Belief;82
4.7.1.3;7.1.3 Propositional Languages;83
4.7.1.4;7.1.4 Predicate Languages;84
4.7.1.5;7.1.5 Objective Bayesianism in Perspective;86
4.7.2;7.2 Representation;87
4.7.3;7.3 Interpretation;87
5;Part II Probabilistic Networks;90
5.1;8 Credal and Bayesian Networks;91
5.1.1;8.1 Kinds of Probabilistic Network;92
5.1.1.1;8.1.1 Extensions;93
5.1.1.2;8.1.2 Extensions and Coordinates;94
5.1.1.3;8.1.3 Parameterised Credal Networks;96
5.1.2;8.2 Algorithms for Probabilistic Networks;97
5.1.2.1;8.2.1 Requirements of the Probabilistic Logic Framework;97
5.1.2.2;8.2.2 Compiling Probabilistic Networks;98
5.1.2.3;8.2.3 The Hill-Climbing Algorithm for Credal Networks;100
5.1.2.4;8.2.4 Complex Queries and Parameterised Credal Networks;102
5.2;9 Networks for the Standard Semantics;104
5.2.1;9.1 The Poverty of Standard Semantics;104
5.2.2;9.2 Constructing a Credal Net;105
5.2.3;9.3 Dilation and Independence;109
5.3;10 Networks for Probabilistic Argumentation ;111
5.3.1;10.1 Probabilistic Argumentation with Credal Sets;111
5.3.2;10.2 Constructing and Applying the Credal Network;112
5.4;11 Networks for Evidential Probability;115
5.4.1;11.1 First-Order Evidential Probability;115
5.4.2;11.2 Second-Order Evidential Probability;117
5.4.3;11.3 Chaining Inferences;120
5.5;12 Networks for Statistical Inference;122
5.5.1;12.1 Functional Models and Networks;122
5.5.1.1;12.1.1 Capturing the Fiducial Argument in a Network;122
5.5.1.2;12.1.2 Aiding Fiducial Inference with Networks;123
5.5.1.3;12.1.3 Trouble with Step-by-Step Fiducial Probability;125
5.5.2;12.2 Evidential Probability and the Fiducial Argument;126
5.5.2.1;12.2.1 First-Order EP and the Fiducial Argument;126
5.5.2.2;12.2.2 Second-Order EP and the Fiducial Argument;127
5.6;13 Networks for Bayesian Statistical Inference;128
5.6.1;13.1 Credal Networks as Statistical Hypotheses;128
5.6.1.1;13.1.1 Construction of the Credal Network;129
5.6.1.2;13.1.2 Computational Advantages of Using the Credal Network;130
5.6.2;13.2 Extending Statistical Inference with Credal Networks;131
5.6.2.1;13.2.1 Interval-Valued Likelihoods;132
5.6.2.2;13.2.2 Logically Complex Statements with Statistical Hypotheses;134
5.7;14 Networks for Objective Bayesianism ;135
5.7.1;14.1 Propositional Languages;135
5.7.2;14.2 Predicate Languages;137
5.8;15 Conclusion;140
6;References;141
7;Index;152