STUDIES IN BAYESIAN theory

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Contents
Abstract ii
Acknowledgements iii
1 Introduction 1
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background on Bayesian Confirmation . . . . . . . . . . . . . . . . 1
1.2.1 Probability Theory I: Kolmogorov’s Axioms . . . . . . . . . 1
1.2.2 Probability Theory II: Interpretation(s) . . . . . . . . . . . . 3
1.2.3 Qualitative BayesianConfirmation . . . . . . . . . . . . . . 4
1.2.4 Quantitative Confirmation I: The Basic Concepts . . . . . . 5
1.2.5 Quantitative Confirmation II: The Many Measures . . . . . 7
2 The Plurality of BayesianMeasures of Confirmation and the Problem
of Measure Sensitivity 9
2.1 A General Overviewof the Problem . . . . . . . . . . . . . . . . . . 9
2.2 Contemporary Examples of the Problem . . . . . . . . . . . . . . . 10
2.2.1 Gillies’s Rendition of the Popper-Miller Argument . . . . . . 11
2.2.2 Rosenkrantz & Earman on “Irrelevant Conjunction” . . . . 12
2.2.2.1 A New Analysis of “Irrelevant Conjunction” . . . . 13
2.2.3 Eells & Sober on the Grue Paradox . . . . . . . . . . . . . . 17
2.2.4 Horwich et al. on Ravens and Variety . . . . . . . . . . . . . 19
2.2.5 An Important Theme in Our Examples . . . . . . . . . . . . 20
2.3 Two Arguments Against r . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 The “Deductive Insensitivity” Argument Against r . . . . . 21
vii
2.3.2 The “Unintuitive Confirmation” Argument Against r . . . . 22
2.4 Summary of Results So Far . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Where DoWe Go From Here? . . . . . . . . . . . . . . . . . . . . . 25
3 Independent Evidence, Measures of Confirmation, and The Value
of Evidential Diversity 27
3.1 Three Existing Attempts to Solve the Problem of Measure Sensitivity 27
3.1.1 Milne’s Reductionistic Argument for r . . . . . . . . . . . . 28
3.1.2 Carnap’s Symmetry Argument for r . . . . . . . . . . . . . . 33
3.1.3 Good’s “Best Explicatum” Argument for l . . . . . . . . . . 35
3.2 A BayesianAccount of Independent Evidence . . . . . . . . . . . . 37
3.2.1 The Fundamental Peirceian Desiderata . . . . . . . . . . . . 38
3.2.2 A Negation SymmetryDesideratum . . . . . . . . . . . . . . 40
3.2.3 Conclusive Evidence and Measures of Confirmation . . . . . 41
3.2.4 Screening-Off and Confirmational Independence . . . . . . . 43
3.2.4.1 Wittgenstein’s Example and Sober’s Analysis . . . 43
3.2.4.2 A FormalModel . . . . . . . . . . . . . . . . . . . 45
3.3 An Application to Evidential Diversity . . . . . . . . . . . . . . . . 48
3.3.1 Comparison with the ‘Correlation’ Approach . . . . . . . . . 50
3.3.2 Comparison with Wayne and Horwich on Diversity . . . . . 53
3.3.2.1 Wayne’s Reconstruction of Horwich’s Account . . . 53
3.3.2.2 Wayne’s Counterexample to H3 . . . . . . . . . . . 54
3.3.2.3 Why Wayne’s Counterexample is Not Salient . . . 55
3.3.2.4 Charitably Reconstructing Horwich’s Account . . . 56
3.3.2.5 A Remaining Worry About Horwich’s Account . . 58
3.3.2.6 The Robustness of Our Reconstruction H3∗ . . . . 62
4 Future Directions 64
viii
4.1 Some Remaining Open Questions . . . . . . . . . . . . . . . . . . . 64
4.2 An Analogous Philosophical Problem . . . . . . . . . . . . . . . . . 65
A Technical Details 66
A.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.3 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
A.4 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A.5 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.6 Proof of Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.7 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
A.8 Proof of Theorem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.9 Proof of Theorem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A.10 Proofs Concerning Milne’s Desiderata (7)–(11) . . . . . . . . . . . . 83
A.11Proofs ofWayne’s (20), (21), and (22) . . . . . . . . . . . . . . . . 88
A.11.1 Proof of (20) . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A.11.2 Proof of (21) . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A.11.3 Proof of (22) . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.12 Proof of the Robustness of H3∗ . . . . . . . . . . . . . . . . . . . . 90
A.13 Counterexample to CP1 =⇒H3 . . . . . . . . . . . . . . . . . . . . 91
A.14 Using˛ˇto Reason About the Probability Calculus . . . 93

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