Theory of Probability

Table of contents

1. Cover
2. Title Page
3. Foreword
4. Preface
5. 1 Introduction
   1. 1.1 Why a New Book on Probability?
   2. 1.2 What are the Mathematical Differences?
   3. 1.3 What are the Conceptual Differences?
   4. 1.4 Preliminary Clarifications
   5. 1.5 Some Implications to Note
   6. 1.6 Implications for the Mathematical Formulation
   7. 1.7 An Outline of the ‘Introductory Treatment’
   8. 1.8 A Few Words about the ‘Critical’ Appendix
   9. 1.9 Other Remarks
   10. 1.10 Some Remarks on Terminology
   11. 1.11 The Tyranny of Language
   12. 1.12 References
6. 2 Concerning Certainty and Uncertainty
   1. 2.1 Certainty and Uncertainty
   2. 2.2 Concerning Probability
   3. 2.3 The Range of Possibility
   4. 2.4 Critical Observations Concerning the ‘Space of Alternatives’
   5. 2.5 Logical and Arithmetic Operations
   6. 2.6 Assertion, Implication; Incompatibility
   7. 2.7 Partitions; Constituents; Logical Dependence and Independence
   8. 2.8 Representations in Linear form
   9. 2.9 Means; Associative Means
   10. 2.10 Examples and Clarifications
   11. 2.11 Concerning Certain Conventions of Notation
7. 3 Prevision and Probability
   1. 3.1 From Uncertainty to Prevision
   2. 3.2 Digressions on Decisions and Utilities
   3. 3.3 Basic Definitions and Criteria
   4. 3.4 A Geometric Interpretation: The Set 𝓟 of Coherent Previsions
   5. 3.5 Extensions of Notation
   6. 3.6 Remarks and Examples
   7. 3.7 Prevision in the Case of Linear and Nonlinear Dependence
   8. 3.8 Probabilities of Events
   9. 3.9 Linear Dependence in General
   10. 3.10 The Fundamental Theorem of Probability
   11. 3.11 Zero Probabilities: Critical Questions
   12. 3.12 Random Quantities with an Infinite Number of Possible Values
   13. 3.13 The Continuity Property
8. 4 Conditional Prevision and Probability
   1. 4.1 Prevision and the State of Information
   2. 4.2 Definition of Conditional Prevision (and Probability)
   3. 4.3 Proof of the Theorem of Compound Probabilities
   4. 4.4 Remarks
   5. 4.5 Probability and Prevision Conditional on a Given Event H
   6. 4.6 Likelihood
   7. 4.7 Probability Conditional on a Partition H
   8. 4.8 Comments
   9. 4.9 Stochastic Dependence and Independence; Correlation
   10. 4.10 Stochastic Independence Among (Finite) Partitions
   11. 4.11 On the Meaning of Stochastic Independence
   12. 4.12 Stochastic Dependence in the Direct Sense
   13. 4.13 Stochastic Dependence in the Indirect Sense
   14. 4.14 Stochastic Dependence through an Increase in Information
   15. 4.15 Conditional Stochastic Independence
   16. 4.16 Noncorrelation; Correlation (Positive or Negative)
   17. 4.17 A Geometric Interpretation
   18. 4.18 On the Comparability of Zero Probabilities
   19. 4.19 On the Validity of the Conglomerative Property
9. 5 The Evaluation of Probabilities
   1. 5.1 How should Probabilities be Evaluated?
   2. 5.2 Bets and Odds
   3. 5.3 How to Think about Things
   4. 5.4 The Approach Through Losses
   5. 5.5 Applications of the Loss Approach
   6. 5.6 Subsidiary Criteria for Evaluating Probabilities
   7. 5.7 Partitions into Equally Probable Events
   8. 5.8 The Prevision of a Frequency
   9. 5.9 Frequency and ‘Wisdom after the Event’
   10. 5.10 Some Warnings
   11. 5.11 Determinism, Indeterminism, and other ‘Isms’
10. 6 Distributions
    1. 6.1 Introductory Remarks
    2. 6.2 What we Mean by a ‘Distribution’
    3. 6.3 The Parting of the Ways
    4. 6.4 Distributions in Probability Theory
    5. 6.5 An Equivalent Formulation
    6. 6.6 The Practical Study of Distribution Functions
    7. 6.7 Limits of Distributions
    8. 6.8 Various Notions of Convergence for Random Quantities
    9. 6.9 Distributions in Two (or More) Dimensions
    10. 6.10 The Method of Characteristic Functions
    11. 6.11 Some Examples of Characteristic Functions
    12. 6.12 Some Remarks Concerning the Divisibility of Distributions
11. 7 A Preliminary Survey
    1. 7.1 Why a Survey at this Stage?
    2. 7.2 Heads and Tails: Preliminary Considerations
    3. 7.3 Heads and Tails: The Random Process
    4. 7.4 Some Particular Distributions
    5. 7.5 Laws of ‘Large Numbers’
    6. 7.6 The ‘Central Limit Theorem’; The Normal Distribution
    7. 7.7 Proof of the Central Limit Theorem
12. 8 Random Processes with Independent Increments
    1. 8.1 Introduction
    2. 8.2 The General Case: The Case of Asymptotic Normality
    3. 8.3 The Wiener–Lévy Process
    4. 8.4 Stable Distributions and Other Important Examples
    5. 8.5 Behaviour and Asymptotic Behaviour
    6. 8.6 Ruin Problems; the Probability of Ruin; the Prevision of the Duration of the Game
    7. 8.7 Ballot Problems; Returns to Equilibrium; Strings
    8. 8.8 The Clarification of Some So‐Called Paradoxes
    9. 8.9 Properties of the Wiener–Lévy Process
13. 9 An Introduction to Other Types of Stochastic Process
    1. 9.1 Markov Processes
    2. 9.2 Stationary Processes
14. 10 Problems in Higher Dimensions
    1. 10.1 Introduction
    2. 10.2 Second‐Order Characteristics and the Normal Distribution
    3. 10.3 Some Particular Distributions: The Discrete Case
    4. 10.4 Some Particular Distributions: The Continuous Case
    5. 10.5 The Case of Spherical Symmetry
15. 11 Inductive Reasoning; Statistical Inference
    1. 11.1 Introduction
    2. 11.2 The Basic Formulation and Preliminary Clarifications
    3. 11.3 The Case of Independence and the Case of Dependence
    4. 11.4 Exchangeability
16. 12 Mathematical Statistics
    1. 12.1 The Scope and Limits of the Treatment
    2. 12.2 Some Preliminary Remarks
    3. 12.3 Examples Involving the Normal Distribution
    4. 12.4 The Likelihood Principle and Sufficient Statistics
    5. 12.5 A Bayesian Approach to ‘Estimation’ and ‘Hypothesis Testing’
    6. 12.6 Other Approaches to ‘Estimation’ and ‘Hypothesis Testing’
    7. 12.7 The Connections with Decision Theory
17. Appendix
    1. 1 Concerning Various Aspects of the Different Approaches
    2. 2 Events (true, false, and …)
    3. 3 Events in an Unrestricted Field
    4. 4 Questions Concerning ‘Possibility’
    5. 5 Verifiability and the Time Factor
    6. 6 Verifiability and the Operational Factor
    7. 7 Verifiability and the Precision Factor
    8. 8 Continuation: The Higher (or Infinite) Dimensional Case
    9. 9 Verifiability and ‘Indeterminism’
    10. 10 Verifiability and ‘Complementarity’
    11. 11 Some Notions Required for a Study of the Quantum Theory Case
    12. 12 The Relationship with ‘Three‐Valued Logic’
    13. 13 Verifiability and Distorting Factors
    14. 14 From ‘Possibility’ to ‘Probability’
    15. 15 The First and Second Axioms
    16. 16 The Third Axiom
    17. 17 Connections with Aspects of the Interpretations
    18. 18 Questions Concerning the Mathematical Aspects
    19. 19 Questions Concerning Qualitative Formulations
    20. 20 Conclusions
18. Index
19. End User License Agreement