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Theory of Probability

Book Description

First issued in translation as a two-volume work in 1975, this classic book provides the first complete development of the theory of probability from a subjectivist viewpoint. It proceeds from a detailed discussion of the philosophical mathematical aspects to a detailed mathematical treatment of probability and statistics.

De Finetti’s theory of probability is one of the foundations of Bayesian theory. De Finetti stated that probability is nothing but a subjective analysis of the likelihood that something will happen and that that probability does not exist outside the mind.  It is the rate at which a person is willing to bet on something happening.  This view is directly opposed to the classicist/ frequentist view of the likelihood of a particular outcome of an event, which assumes that the same event could be identically repeated many times over, and the 'probability' of a particular outcome has to do with the fraction of the time that outcome results from the repeated trials.

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