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Preface to the Second Edition | |

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Bayesian Principles | |

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Introduction | |

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The Problem of Induction | |

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Popper's Attempt to Solve the Problem of Induction | |

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Scientific Method in Practice | |

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Probabilistic Induction: The Bayesian Approach | |

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The Objectivity Ideal | |

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The Plan of the Book | |

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Exercises | |

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The Probability Calculus | |

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Introduction | |

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Some Logical Preliminaries | |

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The Probability Calculus | |

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The Axioms | |

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Two Different Interpretations of the Axioms | |

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Useful Theorems of the Calculus | |

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Random Variables | |

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Kolmogorov's Axioms | |

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Propositions | |

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Infinitary Operations | |

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Countable Additivity | |

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Exercises | |

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Distributions and Densities | |

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Distributions | |

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Probability Densities | |

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Expected Values | |

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The Mean and Standard Deviation | |

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Probabilistic Independence | |

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Conditional Distributions | |

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The Bivariate Normal | |

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The Binomial Distribution | |

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The Weak Law of Large Numbers | |

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Exercises | |

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The Classical and Logical Theories | |

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Introduction | |

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The Classical Theory | |

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The Principle of Indifference | |

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The Rule of Succession | |

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The Principle of Indifference and the Paradoxes | |

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Carnap's Logical Probability Measures | |

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Carnap's c[dagger] and c* | |

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The Dependence on A Priori Assumptions | |

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Exercises | |

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Subjective Probability | |

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Degrees of Belief and the Probability Calculus | |

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Betting Quotients and Degrees of Belief | |

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Why Should Degrees of Belief Obey the Probability Calculus? | |

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The Ramsey--de Finetti Theorem | |

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Conditional Betting-Quotients | |

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Fair Odds and Zero Expectations | |

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Fairness and Consistency | |

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Upper and Lower Probabilities | |

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Other Arguments for the Probability Calculus | |

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The Standard Dutch Book Argument | |

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Scoring Rules | |

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Using a Standard | |

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The Cox-Good-Lucas Argument | |

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Introducing Utilities | |

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Conclusion | |

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Exercises | |

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Updating Belief | |

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Bayesian Conditionalisation | |

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Jeffrey Conditionalisation | |

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Generalising Jeffrey's Rule to Partitions | |

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Dutch Books Again | |

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The Principle of Minimum Information | |

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Conclusion | |

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Exercises | |

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Bayesian Induction: Deterministic Theories | |

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Bayesian Versus Non-Bayesian Approaches | |

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The Bayesian Notion of Confirmation | |

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The Application of Bayes's Theorem | |

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Falsifying Hypotheses | |

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Checking a Consequence | |

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The Probability of the Evidence | |

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The Ravens Paradox | |

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The Design of Experiments | |

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The Duhem Problem | |

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The Problem | |

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Lakatos and Kuhn on the Duhem Problem | |

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The Duhem Problem Solved by Bayesian Means | |

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Good Data, Bad Data, and Data Too Good to Be True | |

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Ad Hoc Hypotheses | |

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Some Examples of Ad Hoc Hypotheses | |

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A Standard Account of Adhocness | |

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Popper's Defence of the Adhocness Criterion | |

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Why the Standard Account Must Be Wrong | |

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The Bayesian View of Ad Hoc Theories | |

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The Notion of Independent Evidence | |

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Infinitely Many Theories Compatible with the Data | |

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The Problem | |

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The Bayesian Approach to the Problem | |

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Conclusion | |

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Exercises | |

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Classical Inference in Statistics | |

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Fisher's Theory | |

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Falsificationism in Statistics | |

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Fisher's Theory | |

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Has Fisher's Theory a Rational Foundation? | |

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Which Test-Statistic? | |

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The Chi-Square Test | |

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Sufficient Statistics | |

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Conclusion | |

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The Neyman-Pearson Theory of Significance Tests | |

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An Outline of the Theory | |

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How the Neyman-Pearson Theory Improves on Fisher's | |

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The Choice of Critical Region | |

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The Choice of Test-Statistic and the Use of Sufficient Statistics | |

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Some Problems for the Neyman-Pearson Theory | |

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What Does It Mean to Accept and Reject a Hypothesis? | |

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The Neyman-Pearson Theory as an Account of Inductive Support | |

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A Well-Supported Hypothesis Rejected in a Significance Test | |

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A Subjective Element in Neyman-Pearson Testing: The Choice of Null Hypothesis | |

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A Further Subjective Element: Determining the Outcome Space | |

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Justifying the Stopping Rule | |

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Testing Composite Hypotheses | |

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Conclusion | |

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Exercises | |

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The Classical Theory of Estimation | |

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Introduction | |

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Point Estimation | |

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Sufficient Estimators | |

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Unbiased Estimators | |

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Consistent Estimators | |

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Efficient Estimators | |

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Interval Estimation | |

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Confidence Intervals | |

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The Categorical-Assertion Interpretation of Confidence Intervals | |

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The Subjective-Confidence Interpretation of Confidence Intervals | |

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The Stopping Rule Problem, Again | |

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Prior Knowledge | |

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The Multiplicity of Competing Intervals | |

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Principles of Sampling | |

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Random Sampling | |

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Judgment Sampling | |

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Objections to Judgment Sampling | |

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Some Advantages of Judgment Sampling | |

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Conclusion | |

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Exercises | |

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Statistical Inference in Practice | |

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Causal Hypotheses: Clinical and Agricultural Trials | |

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Introduction: The Problem | |

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Control and Randomization | |

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Significance-Test Justifications for Randomization | |

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The Problem of the Reference Population | |

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Fisher's Argument | |

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Some Difficulties with Fisher's Argument | |

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A Plausible Defence | |

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Why the Plausible Defence Doesn't Work | |

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The Eliminative-Induction Defence of Randomization | |

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Sequential Clinical Trials | |

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Practical and Ethical Considerations | |

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Conclusion | |

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Regression Analysis | |

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Introduction | |

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Simple Linear Regression | |

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The Method of Least Squares | |

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Why Least Squares? | |

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Intuition as a Justification | |

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The Gauss-Markov Justification | |

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The Maximum-Likelihood Justification | |

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Summary | |

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Prediction | |

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Prediction Intervals | |

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Prediction by Confidence Intervals | |

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Making a Further Prediction | |

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Examining the Form of a Regression | |

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Prior Knowledge | |

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Data Analysis | |

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Inspecting Scatter Plots | |

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Outliers | |

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Influential Points | |

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Conclusion | |

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Exercises | |

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The Bayesian Approach to Statistical Inference | |

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Objective Probability | |

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Introduction | |

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Von Mises's Frequency Theory | |

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Relative Frequencies in Collectives | |

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Probabilities in Collectives | |

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Independence in Derived Collectives | |

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Summary of the Main Features of Von Mises's Theory | |

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The Empirical Adequacy of Von Mises's Theory | |

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The Fast-Convergence Argument | |

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The Laws of Large Numbers Argument | |

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The Limits-Occur-Elsewhere-in-Science Argument | |

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Preliminary Conclusion | |

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Popper's Propensity Theory, and Single-Case Probabilities | |

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Popper's Propensity Theory | |

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Jacta Alea Est | |

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The Theory of Objective Chance | |

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A Bayesian Reconstruction of Von Mises's Theory | |

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Are Objective Probabilities Redundant? | |

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Exchangeability and the Existence of Objective Probability | |

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Conclusion | |

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Exercises | |

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Bayesian Induction: Statistical Hypotheses | |

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The Prior Distribution and the Question of Subjectivity | |

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Estimating the Mean of a Normal Population | |

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Estimating a Binomial Proportion | |

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Credible Intervals and Confidence Intervals | |

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The Principle of Stable Estimation | |

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Describing the Evidence | |

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Sufficient Statistics | |

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Methods of Sampling | |

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Testing Causal Hypotheses | |

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A Bayesian Analysis of Clinical Trials | |

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Clinical Trials without Randomization | |

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Conclusion | |

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Exercises | |

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Finale | |

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The Objections to the Subjective Bayesian Theory | |

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Introduction | |

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The Bayesian Theory Is Prejudiced in Favour of Weak Hypotheses | |

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The Prior Probability of Universal Hypotheses Must Be Zero | |

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Probabilistic Induction Is Impossible | |

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The Principal Principle Is Inconsistent (Miller's Paradox) | |

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The Paradox of Ideal Evidence | |

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Hypotheses Cannot Be Supported by Evidence Already Known | |

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P(h | |

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Evidence Doesn't Confirm Theories Constructed to Explain It | |

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The Principle of Explanatory Surplus | |

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Prediction Scores Higher Than Accommodation | |

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The Problem of Subjectivism | |

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Entropy, Symmetry, and Objectivity | |

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Simplicity | |

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People Are Not Bayesians | |

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The Dempster-Shafer Theory | |

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Belief Functions | |

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What Are Belief Functions? | |

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Representing Ignorance | |

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Evaluating Probabilities with Imprecise Information | |

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Are We Calibrated? | |

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Reliable Inductive Methods | |

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Finale | |

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Exercises | |

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Bibliography | |

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Index | |