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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 | |