Theories of Probability
An Examination of Foundations
- 1st Edition - January 28, 1973
- Imprint: Academic Press
- Author: Terrence L. Fine
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 2 - 3 8 7 9 - 1
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 6 3 8 9 - 2
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… Read more
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Request a sales quoteTheories 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.
Preface
I. Introduction
IA. Motivation
IB. Types of Probability Theories
1. Characteristics of Probability Theories
2. Domains of Application
3. Forms of Probability Statements
4. Relations Between Statements
5. Measurement
6. Goals and Their Achievement
7. Tentative Classification of Some of the Theories to Be Discussed
IC. Guide to the Discussion
1. Outline of Topics
2. References
3. Known Omissions
References
II. Axiomatic Comparative Probability
IIA. Introduction
IIB. Structure of Comparative Probability
1. Fundamental Axioms That Suffice for the Finite Case
2. Compatibility with Quantitative Probability
3. An Archimedean Axiom
4. An Axiom of Monotone Continuity
5. Compatibility of CP Relations
IIC. Compatibility with Finite Additivity
1. Introduction
2. Necessary and Sufficient Conditions for Compatibility
3. Should CP Be Compatible with Finite Additivity?
4. Sufficient Conditions for Compatibility with Finite Additivity
IID. Compatibility with Countable Additivity
IIE. Comparative Conditional Probability
1. Comparative Conditional Probability as a Ternary Relation
2. Comparative Conditional Probability as a Quaternary Relation
3. Relationship Between QCCP and CP
4. Bayes Theorem
5. Compatibility of CCP with Kolmogorov's Quantitative Probability
IIF. Independence
1. Independent Events
2. Mutually Independent Events
3. Independent Experiments
4. Concluding Remarks
IIG. Application to Decision-Making
1. Formulation of the Decision Problem
2. Axioms for Rational Decision-Making
3. Representations of ≳α and ≳
4. Extensions
IIH. Expectation in Comparative Probability
Appendix: Proofs of Results
References
III. Axiomatic Quantitative Probability
IIIA. Introduction
IIIB. Overspecification in the Kolmogorov Setup: Sample Space and Event Field
1. The Sample Space Ω
2. The σ-Field of Events ℱ
3. The Λ- and π-Fields of Events
4. The von Mises Field of Events
IIIC. Overspecification in the Probability Axioms: View from Comparative Probability
1. Unit Normalization and Nonnegativity Axioms
2. Finite Additivity Axiom
3. Continuity Axiom
IIID. Overspecification in the Probability Axioms: View from Measurement Theory
1. Fundamentals of Measurement Theory
2. Probability Measurement Scale
3. Necessity for an Additive Probability Scale
IIIE. Further Specification of the Event Field and Probability Measure
1. Preface
2. Selecting the Event Field
3. Selecting P
IIIF. Conditional Probability
1. Structure of Conditional Probability
2. Motivations for the Product Rule
IIIG. Independence
1. Role of Independence
2. Structure of Independence
IIIH. The Status of Axiomatic Probability
References
IV. Relative-Frequency and Probability
IVA. Introduction
IVB. Search for a Physical Interpretation of Probability Based on Finite Data
1. Reduction through Exchangeability
2. Maturity of Chances and Practical Certainty
IVC. Search for a Physical Interpretation of Probability Based on Infinite Data
1. Introduction
2. Definition of Apparent Convergence and Random Binary Sequence
3. Relations Between Apparent Convergence of Relative-Frequency and Randomness
4. Conclusion
IVD. Bernoulli/Borel Formalization of the Relation Between Probability and Relative-Frequency: Strong Laws of Large Numbers
IVE. Von Mises' Formalization of the Relation Between Probability and Relative-Frequency: The Collective
IVF. Role of Relative-Frequency in the Measurement of Probability
IVG. Predication of Outcomes from Probability Interpreted as Relative-Frequency
IVH. The Argument of the "Long Run"
IVI. Preliminary Conclusions and New Directions
IVJ. Axiomatic Approaches to the Measurement of Probability
1. Introduction
2. An Approach without Explicit Probability Models
3. The Approach of Statistical Decision Theory
4. Bayesian Approaches
5. What Has Been Accomplished
IVK. Measurement of Comparative Probability: Induction by Enumeration
1. Introduction
2. Defining Induction by Enumeration
3. Justifying Induction by Enumeration
IVL. Conclusion
References
V. Computational Complexity, Random Sequences, and Probability
VA. Introduction
VB. Definition of Random Finite Sequence Using Place-Selection Functions
VC. Definition of the Complexity of Finite Sequences
1. Background to Absolute Complexity
2. A Definition of Absolute Complexity
3. Properties of the Definition of Absolute Complexity
4. Other Definitions of Absolute Complexity
5. Conditional Complexity
6. Ineffectiveness of Complexity Calculations
VD. Complexity and Statistics
1. Statistical Tests for Goodness-of-Fit
2. Universal Statistical Tests
3. Role of Complexity in Defining Probabilistic Models
4. A Relation Between Complexity and Critical Level
VE. Definition of Random Finite Sequence Using Complexity
VF. Random Infinite Sequences
1. Complexity-Based Definitions
2. Statistical Definition
3. Relations Between the Complexity and Statistical Definitions
VG. Exchangeable and Bernoulli Finite Sequences
1. Exchangeable Sequences
2. Bernoulli Sequences
VH. Independence and Complexity
1. Introduction
2. Relative-Frequency, Complexity, and Stochastic Independence
3. Empirical Independence
4. Conclusions
VI. Complexity-Based Approaches to Prediction and Probability
1. Introduction
2. Comparative Probability
3. SolomonofFs Definitions of Probability
4. Critique of the Complexity Approach to Probability
VJ. Reflections on Complexity and Randomness: Determinism Versus Chance
VK. Potential Applications for the Complexity Approach
Appendix: Proofs of Results
References
VI. Classical Probability and Its Renaissance
VIA. Introduction
VIB. Illustrations of the Classical Argument and Assignments of Equiprobability
VIC. Axiomatic Formulations of the Classical Approach
1. The Principle of Invariance
2. Information-Theoretic Principles
VID. Justifying the Classical Approach and Its Axiomatic Reformulations
1. Classical Probability and Decision Under Uncertainty
2. Principle of Invariance
3. Information-Theoretic Principles
VIE. Conclusions
References
VII. Logical (Conditional) Probability
VIIA. Introduction
VIIB. Classificatory Probability and Modal Logic
VIIC. Koopman's Theory of Comparative Logical Probability
1. Structure of Comparative Probability
2. Relation to Conditional Quantitative Probability
3. Relation to Relative-Frequency
4. Conclusions
VIID. Carnap's Theory of Logical Probability
1. Introduction
2. Compatibility with Rational Decision-Making
3. Axioms of Invariance
4. Learning from Experience
5. Selection of a Unique Confirmation Function
VIIE. Logical Probability and Relative-Frequency
VIIF. Applications of C* and Cλ
VIIG. Critique of Logical Probability
1. Roles for Logical Probability
2. Formulation of Logical Probability
3. Justifying Logical Probability
Appendix: Proofs of Results
References
VIII. Probability as a Pragmatic Necessity: Subjective or Personal Probability
VIIIA. Introduction
VIIIB. Preferences and Utilities
VIIIC. An Approach to Subjective Probability through Reference to Preexisting Probability
1. Axioms of Anscombe and Aumann Type
2. The Associated Objective Distribution
VIIID. Approaches to Subjective Probability through Decision-Making
1. Formulation of Savage
2. Formulation of Krantz and Luce
VIIIE. Subjective Versus Arbitrary: Learning from Experience
VIIIF. Measurement of Subjective Probability
VIIIG. Roles for Subjective Probability
VIIIH. Critique of Subjective Probability
1. Role of Subjective Probability
2. Formulation of Subjective Probability
3. Measurement of Subjective Probability
4. Justification of Subjective Probability
References
IX. Conclusions
IXA. Where Do We Stand?
1. With Respect to Definitions of Probability
2. With Respect to Definitions of Associated Concepts
IXB. Probability in Physics
1. Introduction
2. Statistical Mechanics
3. Quantum Mechanics
4. Conclusions
IXC. What Can We Expect from a Theory of Probability?
IXD. Is Probability Needed?
References
Author Index
Subject Index
- Edition: 1
- Published: January 28, 1973
- Imprint: Academic Press
- No. of pages: 276
- Language: English
- Paperback ISBN: 9781483238791
- eBook ISBN: 9781483263892
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