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Notes on Byesian Confirmation Theory by Michael Strevens
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  Notes on Bayesian Confirmation Theory  Michael StrevensSeptember 2012 Contents 1 Introduction 52 Credence or Subjective Probability  73 Axioms of Probability  103.1 The Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Conditional Probability  . . . . . . . . . . . . . . . . . . . . 153.3 Probabilistic Independence . . . . . . . . . . . . . . . . . . 173.4 Justifying the Axioms . . . . . . . . . . . . . . . . . . . . . 184 Bayesian Conditionalization 224.1 Bayes’ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Background Knowledge . . . . . . . . . . . . . . . . . . . 254.4 Justifying Bayes’ Rule . . . . . . . . . . . . . . . . . . . . . 265 The Machinery of Modern Bayesianism 285.1 From Conditionalization to Confirmation . . . . . . . . . . 285.2 Constraining the Likelihood . . . . . . . . . . . . . . . . . 315.3 Constraining the Probability of the Evidence . . . . . . . . 365.4 Modern Bayesianism: ASummary  . . . . . . . . . . . . . . 396 Modern Bayesianism in Action 416.1 AWorked Example . . . . . . . . . . . . . . . . . . . . . . 416.2 General Properties of Bayesian Confirmation . . . . . . . . 446.3 Working with Infinitely Many Hypotheses . . . . . . . . . . 506.4 When Explicit Physical Probabilities Are Not Available . . . 581  7 Does Bayesianism Solve the Problem of Induction? 607.1 Subjective and Objective In Bayesian Confirmation Theory  . 607.2 The Uniformity of Nature . . . . . . . . . . . . . . . . . . 627.3 Goodman’s New Riddle . . . . . . . . . . . . . . . . . . . . 647.4 Simplicity  . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 668 Bayesian Confirmation Theory and the Problems of Confirmation 678.1 The Paradox of the Ravens . . . . . . . . . . . . . . . . . . 678.2 Variety of Evidence . . . . . . . . . . . . . . . . . . . . . . 738.3 The Problem of Irrelevant Conjuncts . . . . . . . . . . . . 789 The Subjectivity of Bayesian Confirmation Theory  819.1 The Problem of Subjectivity  . . . . . . . . . . . . . . . . . 819.2 Washing Out and Convergence . . . . . . . . . . . . . . . . 849.3 Radical Personalism . . . . . . . . . . . . . . . . . . . . . . 959.4 Constraining the Priors . . . . . . . . . . . . . . . . . . . . 9910 Bayesianism, Holism, and Auxiliary Hypotheses 10710.1 Auxiliary Hypotheses . . . . . . . . . . . . . . . . . . . . . 10710.2 The Bayesian’s Quine-Duhem Problem . . . . . . . . . . . 10810.3 The Problem of Ad Hoc Reasoning . . . . . . . . . . . . . . 11010.4 The Old Bayesian Approach to the Quine-Duhem Problem . 11410.5 ANew Bayesian Approach to the Quine-Duhem Problem . 11611 The Problem of Old Evidence 12311.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . 12311.2 Replaying History  . . . . . . . . . . . . . . . . . . . . . . . 12611.3 Learning about Entailment . . . . . . . . . . . . . . . . . . 12711.4 The Problem of Novel Theories . . . . . . . . . . . . . . . 12912 Further Reading 133Proofs 138Glossary  1432  List of Figures 1 Probability density  . . . . . . . . . . . . . . . . . . . . . . 522 Prior probability distributions . . . . . . . . . . . . . . . . 533 Effect of conditionalization I . . . . . . . . . . . . . . . . . 544 Effect of conditionalization II . . . . . . . . . . . . . . . . 555 Physical likelihoods . . . . . . . . . . . . . . . . . . . . . . 866 Washing out . . . . . . . . . . . . . . . . . . . . . . . . . . 887 Apportioning the blame . . . . . . . . . . . . . . . . . . . 118 List of Tech Boxes 2.1 Bayesian Theories of Acceptance . . . . . . . . . . . . . . . 82.2 What Do Credences Range Over? . . . . . . . . . . . . . . 93.1 Sigma Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 The Axiom of Countable Additivity  . . . . . . . . . . . . . 143.3 Conditional Probability Introduced Axiomatically  . . . . . 164.1 What the Apriorist Must Do . . . . . . . . . . . . . . . . . 275.1 Conditional Probability Characterized Dispositionally  . . . 305.2 Logical Probability  . . . . . . . . . . . . . . . . . . . . . . 325.3 The Probability Coordination Principle . . . . . . . . . . . 345.4 Subjectivism about Physical Probability  . . . . . . . . . . . 355.5 Inadmissible Information . . . . . . . . . . . . . . . . . . 375.6 Prior Probabilities . . . . . . . . . . . . . . . . . . . . . . 406.1 Weight of Evidence . . . . . . . . . . . . . . . . . . . . . . 456.2 The Law of Large Numbers . . . . . . . . . . . . . . . . . . 568.1 Hempel’s Ravens Paradox . . . . . . . . . . . . . . . . . . . 689.1 Why Radical? . . . . . . . . . . . . . . . . . . . . . . . . . 969.2 Origins of the Principle of Indifference . . . . . . . . . . . 10011.1 Prediction versus Accommodation . . . . . . . . . . . . . . 1303  There were three ravens sat on a tree,Downe a downe, hay downe, hay downeThere were three ravens sat on a tree,With a downeThere were three ravens sat on a tree,They were as blacke as they might be.With a downe derrie, derrie, derrie, downe, downe.Anonymous, 16th century 4
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