Theory of Probability : A Critical Introductory Treatment.

Intro -- Title Page -- Copyright Page -- Contents -- Foreword -- Preface by Adrian Smith -- Preface by Bruno de Finetti -- Translators' Preface -- Chapter 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 -- Chapter 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 -- Chapter 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 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 -- Chapter 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 -- 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 -- Chapter 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' -- Chapter 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 -- Chapter 7 A Preliminary Survey -- 7.1 Why a Survey at this Stage? -- 7.2 Heads and Tails: Preliminary Considerations -- 7.3 Heads and Tails: The Random Process -- 7.4 Some Particular Distributions -- 7.5 Laws of 'Large Numbers' -- 7.6 The 'Central Limit Theorem' -- The Normal Distribution -- 7.7 Proof of the Central Limit Theorem -- Chapter 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 -- Chapter 9 An Introduction to Other Types of Stochastic Process -- 9.1 Markov Processes -- 9.2 Stationary Processes -- Chapter 10 Problems in Higher Dimensions -- 10.1 Introduction -- 10.2 Second-Order Characteristics and the Normal Distribution -- 10.3 Some Particular Distributions: The Discrete Case -- 10.4 Some Particular Distributions: The Continuous Case -- 10.5 The Case of Spherical Symmetry -- Chapter 11 Inductive Reasoning -- Statistical Inference -- 11.1 Introduction -- 11.2 The Basic Formulation and Preliminary Clarifications -- 11.3 The Case of Independence and  the Case of Dependence -- 11.4 Exchangeability -- Chapter 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 -- EULA.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.