Theories of Probability

Theories of Probability

An Examination of Foundations

1st Edition - January 28, 1973

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  • Author: Terrence L. Fine
  • eBook ISBN: 9781483263892

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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.

Table of Contents

  • 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


    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


    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


    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


    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


    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


    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


    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


    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?


    Author Index

    Subject Index

Product details

  • No. of pages: 276
  • Language: English
  • Copyright: © Academic Press 1973
  • Published: January 28, 1973
  • Imprint: Academic Press
  • eBook ISBN: 9781483263892

About the Author

Terrence L. Fine

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