Theories of Probability

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