Book description
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
 Cover
 Title Page
 Foreword
 Preface

1 Introduction
 1.1 Why a New Book on Probability?
 1.2 What are the Mathematical Differences?
 1.3 What are the Conceptual Differences?
 1.4 Preliminary Clarifications
 1.5 Some Implications to Note
 1.6 Implications for the Mathematical Formulation
 1.7 An Outline of the ‘Introductory Treatment’
 1.8 A Few Words about the ‘Critical’ Appendix
 1.9 Other Remarks
 1.10 Some Remarks on Terminology
 1.11 The Tyranny of Language
 1.12 References

2 Concerning Certainty and Uncertainty
 2.1 Certainty and Uncertainty
 2.2 Concerning Probability
 2.3 The Range of Possibility
 2.4 Critical Observations Concerning the ‘Space of Alternatives’
 2.5 Logical and Arithmetic Operations
 2.6 Assertion, Implication; Incompatibility
 2.7 Partitions; Constituents; Logical Dependence and Independence
 2.8 Representations in Linear form
 2.9 Means; Associative Means
 2.10 Examples and Clarifications
 2.11 Concerning Certain Conventions of Notation

3 Prevision and Probability
 3.1 From Uncertainty to Prevision
 3.2 Digressions on Decisions and Utilities
 3.3 Basic Definitions and Criteria
 3.4 A Geometric Interpretation: The Set 𝓟 of Coherent Previsions
 3.5 Extensions of Notation
 3.6 Remarks and Examples
 3.7 Prevision in the Case of Linear and Nonlinear Dependence
 3.8 Probabilities of Events
 3.9 Linear Dependence in General
 3.10 The Fundamental Theorem of Probability
 3.11 Zero Probabilities: Critical Questions
 3.12 Random Quantities with an Infinite Number of Possible Values
 3.13 The Continuity Property

4 Conditional Prevision and Probability
 4.1 Prevision and the State of Information
 4.2 Definition of Conditional Prevision (and Probability)
 4.3 Proof of the Theorem of Compound Probabilities
 4.4 Remarks
 4.5 Probability and Prevision Conditional on a Given Event H
 4.6 Likelihood
 4.7 Probability Conditional on a Partition H
 4.8 Comments
 4.9 Stochastic Dependence and Independence; Correlation
 4.10 Stochastic Independence Among (Finite) Partitions
 4.11 On the Meaning of Stochastic Independence
 4.12 Stochastic Dependence in the Direct Sense
 4.13 Stochastic Dependence in the Indirect Sense
 4.14 Stochastic Dependence through an Increase in Information
 4.15 Conditional Stochastic Independence
 4.16 Noncorrelation; Correlation (Positive or Negative)
 4.17 A Geometric Interpretation
 4.18 On the Comparability of Zero Probabilities
 4.19 On the Validity of the Conglomerative Property

5 The Evaluation of Probabilities
 5.1 How should Probabilities be Evaluated?
 5.2 Bets and Odds
 5.3 How to Think about Things
 5.4 The Approach Through Losses
 5.5 Applications of the Loss Approach
 5.6 Subsidiary Criteria for Evaluating Probabilities
 5.7 Partitions into Equally Probable Events
 5.8 The Prevision of a Frequency
 5.9 Frequency and ‘Wisdom after the Event’
 5.10 Some Warnings
 5.11 Determinism, Indeterminism, and other ‘Isms’

6 Distributions
 6.1 Introductory Remarks
 6.2 What we Mean by a ‘Distribution’
 6.3 The Parting of the Ways
 6.4 Distributions in Probability Theory
 6.5 An Equivalent Formulation
 6.6 The Practical Study of Distribution Functions
 6.7 Limits of Distributions
 6.8 Various Notions of Convergence for Random Quantities
 6.9 Distributions in Two (or More) Dimensions
 6.10 The Method of Characteristic Functions
 6.11 Some Examples of Characteristic Functions
 6.12 Some Remarks Concerning the Divisibility of Distributions
 7 A Preliminary Survey

8 Random Processes with Independent Increments
 8.1 Introduction
 8.2 The General Case: The Case of Asymptotic Normality
 8.3 The Wiener–Lévy Process
 8.4 Stable Distributions and Other Important Examples
 8.5 Behaviour and Asymptotic Behaviour
 8.6 Ruin Problems; the Probability of Ruin; the Prevision of the Duration of the Game
 8.7 Ballot Problems; Returns to Equilibrium; Strings
 8.8 The Clarification of Some So‐Called Paradoxes
 8.9 Properties of the Wiener–Lévy Process
 9 An Introduction to Other Types of Stochastic Process
 10 Problems in Higher Dimensions
 11 Inductive Reasoning; Statistical Inference

12 Mathematical Statistics
 12.1 The Scope and Limits of the Treatment
 12.2 Some Preliminary Remarks
 12.3 Examples Involving the Normal Distribution
 12.4 The Likelihood Principle and Sufficient Statistics
 12.5 A Bayesian Approach to ‘Estimation’ and ‘Hypothesis Testing’
 12.6 Other Approaches to ‘Estimation’ and ‘Hypothesis Testing’
 12.7 The Connections with Decision Theory

Appendix
 1 Concerning Various Aspects of the Different Approaches
 2 Events (true, false, and …)
 3 Events in an Unrestricted Field
 4 Questions Concerning ‘Possibility’
 5 Verifiability and the Time Factor
 6 Verifiability and the Operational Factor
 7 Verifiability and the Precision Factor
 8 Continuation: The Higher (or Infinite) Dimensional Case
 9 Verifiability and ‘Indeterminism’
 10 Verifiability and ‘Complementarity’
 11 Some Notions Required for a Study of the Quantum Theory Case
 12 The Relationship with ‘Three‐Valued Logic’
 13 Verifiability and Distorting Factors
 14 From ‘Possibility’ to ‘Probability’
 15 The First and Second Axioms
 16 The Third Axiom
 17 Connections with Aspects of the Interpretations
 18 Questions Concerning the Mathematical Aspects
 19 Questions Concerning Qualitative Formulations
 20 Conclusions
 Index
 End User License Agreement
Product information
 Title: Theory of Probability
 Author(s):
 Release date: April 2017
 Publisher(s): Wiley
 ISBN: 9781119286370
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