Fine Theories of Probability

An Examination of Foundations
1. Auflage 2014
ISBN: 978-1-4832-6389-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 276 Seiten, Web PDF

Theories of Probability: An Examination of Foundations reviews the theoretical foundations of probability, with emphasis on concepts that are important for the modeling of random phenomena and the design of information processing systems. Topics covered range from axiomatic comparative and quantitative probability to the role of relative frequency in the measurement of probability. Computational complexity and random sequences are also discussed. Comprised of nine chapters, this book begins with an introduction to different types of probability theories, followed by a detailed account of axiomatic formalizations of comparative and quantitative probability and the relations between them. Subsequent chapters focus on the Kolmogorov formalization of quantitative probability; the common interpretation of probability as a limit of the relative frequency of the number of occurrences of an event in repeated, unlinked trials of a random experiment; an improved theory for repeated random experiments; and the classical theory of probability. The book also examines the origin of subjective probability as a by-product of the development of individual judgments into decisions. Finally, it suggests that none of the known theories of probability covers the whole domain of engineering and scientific practice. This monograph will appeal to students and practitioners in the fields of mathematics and statistics as well as engineering and the physical and social sciences.

Autoren/Hrsg.

Fine, Terrence L.

Inhaltsverzeichnis

1;Front Cover;1
2;Theories of Probability: An Examination of Foundations;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;12
6;Chapter I. Introduction;14
6.1;... Motivation;14
6.2;IB. Types of Probability Theories;16
6.3;IC. Guide to the Discussion;23
6.4;References;27
7;Chapter Il. Axiomatic Comparative Probability;28
7.1;IIA. Introduction;28
7.2;IIB. Structure of Comparative Probability;29
7.3;IIC. Compatibility with Finite Additivity;35
7.4;IID. Compatibility with Countable Additivity;40
7.5;IIE. Comparative Conditional Probability;41
7.6;IIF. Independence;45
7.7;IIG. Application to Decision-Making;50
7.8;IlH. Expectation in Comparative Probability;55
7.9;II. Appendix: Proofs of Results;56
7.10;References;69
8;Chapter Ill. Axiomatic Quantitative Probability;71
8.1;IIIA. Introduction;71
8.2;IIIB. Overspecification in the Kolmogorov Setup: Sample Space and Event Field;73
8.3;IIIC. Overspecification in the Probability Axioms: View from Comparative Probability;78
8.4;IIID. Overspecìficatìon in the Probability Axioms: View from Measurement Theory;81
8.5;IIIE. Further Specification of the Event Field and Probability Measure;87
8.6;IIIF. Conditional Probability;89
8.7;IIIG. Independence;93
8.8;IIIH. The Status of Axiomatic Probability;96
8.9;References;97
9;Chapter IV. Relative-Frequency and Probability;98
9.1;IVA. Introduction;98
9.2;IVB. Search for a Physical Interpretation of Probability Based on Finite Data;99
9.3;IVC. Search for a Physical Interpretation of Probability Based on Infinite Data;102
9.4;IVD. Bernoulli/Borel Formalization of the Relation between Probability and Relative-Frequency: Strong Laws of Large Numbers;108
9.5;IVE. Von Mises' Formalization of the Relation between Probability and Relative-Frequency: The Collective;110
9.6;IVF. Role of Relative-Frequency in the Measurement of Probability;113
9.7;IVG. Prediction of Outcomes from Probability Interpreted as Relative-Frequency;115
9.8;IVH. The Argument of the "Long Run";116
9.9;IVI. Preliminary Conclusions and New Directions;116
9.10;IVJ. Axiomatic Approaches to the Measurement of Probability;118
9.11;IVK. Measurement of Comparative Probability: Induction by Enumeration;124
9.12;IVL. Conclusion;128
9.13;References;129
10;Chapter V. Computational Complexity, Random Sequences, and Probability;131
10.1;VA. Introduction;131
10.2;VB. Definition of Random Finite Sequence Using Place-Selection Functions;132
10.3;VC. Definition of the Complexity of Finite Sequences;134
10.4;VD. Complexity and Statistics;143
10.5;VE. Definition of Random Finite Sequence Using Complexity;147
10.6;VF. Random Infinite Sequences;149
10.7;VG. Exchangeable and Bernoulli Finite Sequences;151
10.8;VH. Independence and Complexity;154
10.9;VI. Complexity-Based Approaches to Prediction and Probability;159
10.10;VJ. Reflections on Complexity and Randomness: Determinism versus Chance;165
10.11;VK. Potential Applications for the Complexity Approach;166
10.12;V. Appendix: Proofs of Results;167
10.13;References;177
11;Chapter VI. Classical Probability and its Renaissance;179
11.1;VIA. Introduction;179
11.2;VIB. Illustrations of the Classical Argument and Assignments of Equiprobability;180
11.3;VIC. Axiomatic Formulations of the Classical Approach;183
11.4;VID. Justifying the Classical Approach and Its Axiomatic Reformulations;187
11.5;VIE. Conclusions;189
11.6;References;190
12;Chapter VII. Logical (Conditional) Probability;192
12.1;VIIA. Introduction;192
12.2;VIIB. Classificatory Probability and Modal Logic;194
12.3;VIIC. Koopman's Theory of Comparative Logical Probability;196
12.4;VIID. Carnap's Theory of Logical Probability;200
12.5;VIIE. Logical Probability and Relative-Frequency;210
12.6;VIIF. Applications of C* and C.;211
12.7;VIIG. Critique of Logical Probability;214
12.8;VII. Appendix: Proofs of Results;217
12.9;References;223
13;Chapter VIlI. Probability as a Pragmatic Necessity: Subjective or Personal Probability;225
13.1;VIIIA. Introduction;225
13.2;VIIIB. Preferences and Utilities;227
13.3;VIIIC. An Approach to Subjective Probability through Reference to Preexisting Probability;229
13.4;VIIID. Approaches to Subjective Probability through Decision-Making;232
13.5;VIIIE. Subjective versus Arbitrary: Learning from Experience;239
13.6;VIIIF. Measurement of Subjective Probability;241
13.7;VIIIG. Roles for Subjective Probability;242
13.8;VIIIH. Critique of Subjective Probability;243
13.9;References;249
14;Chapter IX. Conclusions;251
14.1;IXA. Where Do We Stand?;251
14.2;IXB. Probability in Physics;255
14.3;IXC. What Can We Expect from a Theory of Probability?;259
14.4;IXD. Is Probability Needed?;262
14.5;References;264
15;Author Index;266
16;Subject index;269