| Preface |
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vii | |
| About the Author |
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ix | |
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1 | (16) |
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1 | (1) |
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2 | (1) |
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1.3 Summary of the Main Claims |
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3 | (8) |
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3 | (1) |
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1.3.2 Critique of frequency and subjective philosophies of probability |
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4 | (1) |
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1.3.2.1 Positive philosophical ideas |
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5 | (1) |
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1.3.2.2 Negative philosophical ideas |
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6 | (2) |
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1.3.2.3 Innovative technical ideas |
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8 | (1) |
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1.3.3 Scientific laws of probability |
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8 | (1) |
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1.3.4 Statistics and philosophy |
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9 | (2) |
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1.4 Historical and Social Context |
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11 | (2) |
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13 | (4) |
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Philosophy of Probability |
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17 | (124) |
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2 Main Philosophies of Probability |
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19 | (22) |
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20 | (1) |
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20 | (2) |
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2.3 The Propensity Theory |
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22 | (1) |
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2.4 The Subjective Theory |
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22 | (9) |
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2.4.1 Interpreting subjectivity |
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23 | (1) |
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2.4.2 Verification of probabilistic statements |
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24 | (2) |
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2.4.3 Subjectivity as an escape from the shackles of verification |
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26 | (1) |
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2.4.4 The Dutch book argument |
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27 | (2) |
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2.4.5 The axiomatic system |
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29 | (1) |
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2.4.6 Identification of probabilities and decisions |
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30 | (1) |
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30 | (1) |
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31 | (2) |
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2.6 Summary of Philosophical Theories of Probability |
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33 | (1) |
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2.7 From Ideas to Theories |
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34 | (1) |
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35 | (3) |
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2.9 Is There Life Beyond Von Mises and De Finetti? |
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38 | (3) |
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41 | (18) |
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3.1 How Do You Prove that You are Not a Camel? |
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41 | (1) |
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42 | (5) |
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3.2.1 Frustration of philosophers |
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46 | (1) |
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47 | (1) |
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3.4 Von Mises' Brand of Skepticism |
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48 | (3) |
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49 | (1) |
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3.4.2 Inconsistencies in von Mises' theory |
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49 | (2) |
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3.5 De Finetti's Brand of Skepticism |
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51 | (4) |
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3.5.1 How to eat the cake and have it too |
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52 | (3) |
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3.6 On Approximate Theories |
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55 | (1) |
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3.7 Temperature, Beauty and Probability |
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56 | (1) |
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3.8 Latter Day Subjectivism |
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57 | (2) |
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4 The Frequency Philosophy of Probability |
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59 | (20) |
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4.1 Collective as an Elementary Concept |
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59 | (2) |
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4.2 Applications of Probability Do Not Rely on Collectives |
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61 | (4) |
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4.2.1 Stochastic processes |
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61 | (1) |
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62 | (1) |
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4.2.3 Graphical communication |
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63 | (2) |
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4.3 Collectives in Real Life |
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65 | (2) |
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4.4 Collectives and Symmetry |
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67 | (1) |
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4.5 Frequency Theory and the Law of Large Numbers |
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68 | (1) |
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4.6 Why is Mathematics Useful? |
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69 | (1) |
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4.7 Benefits of Imagination and Imaginary Benefits |
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70 | (1) |
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4.8 Imaginary Collectives |
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71 | (1) |
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72 | (1) |
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4.10 Frequency Theory and Individual Events |
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73 | (1) |
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4.11 Collectives and Populations |
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74 | (1) |
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4.12 Are All i.i.d. Sequences Collectives? |
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75 | (1) |
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4.13 Are Collectives i.i.d. Sequences? |
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76 | (1) |
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4.14 Martin-Lof Sequences |
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77 | (2) |
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5 The Subjective Philosophy of Probability |
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79 | (46) |
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5.1 "Subjective" --- A Word with a Subjective Meaning |
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80 | (1) |
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5.2 The Subjective Theory of Probability is Objective |
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81 | (2) |
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5.3 A Science without Empirical Content |
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83 | (1) |
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5.4 If Probability does not Exist, Everything is Permitted |
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84 | (4) |
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5.4.1 Creating something out of nothing |
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85 | (1) |
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5.4.2 The essence of probability |
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86 | (2) |
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5.5 De Finetti's Ultimate Failure |
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88 | (7) |
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5.5.1 Lazy decision maker |
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90 | (1) |
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5.5.2 Interpreting Dutch book |
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91 | (1) |
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5.5.3 Dutch book with a lapse of time |
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92 | (1) |
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5.5.3.1 The butterfly effect |
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93 | (1) |
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5.5.4 Rule of conditionalization |
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94 | (1) |
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5.6 All Sequential Decisions are Consistent |
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95 | (1) |
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96 | (2) |
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5.8 Cohabitation with an Evil Demiurge |
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98 | (2) |
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5.9 Why Bother to Use Probability? |
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100 | (1) |
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5.10 The Dutch Book Argument is Rejected by Bayesians |
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100 | (1) |
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5.11 Insurance Against Everything |
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101 | (1) |
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5.12 No Need to Collect Data |
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102 | (1) |
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103 | (1) |
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5.14 The Meaning of Consistency |
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104 | (1) |
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5.15 Interpreting Miracles |
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105 | (2) |
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5.16 Science, Probability and Subjectivism |
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107 | (1) |
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108 | (1) |
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109 | (1) |
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5.19 Subjective Theory and Atheism |
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110 | (1) |
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5.20 Imagination and Probability |
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111 | (2) |
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113 | (1) |
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5.22 Axiomatic System as a Magical Trick |
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114 | (1) |
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5.23 The Meaning of Subjectivity |
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114 | (5) |
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5.24 Probability and Chance |
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119 | (3) |
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122 | (3) |
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6 The Logical Philosophy of Probability |
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125 | (6) |
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125 | (2) |
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6.2 Why Do Scientists Ignore the Logical Philosophy of Probability? |
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127 | (1) |
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6.3 Probabilities of Propositions and Events |
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128 | (3) |
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131 | (10) |
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131 | (2) |
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7.2 The Value of Extremism |
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133 | (1) |
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7.3 Common Elements in Frequency and Subjective Theories |
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133 | (1) |
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7.4 Common Misconceptions |
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134 | (2) |
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7.5 Shattered Dreams of Perfection |
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136 | (1) |
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137 | (1) |
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137 | (4) |
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138 | (1) |
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138 | (1) |
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7.7.3 Subjective opinions |
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138 | (1) |
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7.7.4 Optimal Bayesian decisions |
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138 | (1) |
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7.7.5 Confidence intervals |
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138 | (1) |
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7.7.6 Significant difference |
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138 | (1) |
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139 | (1) |
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7.7.8 Objective Bayesian methods |
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139 | (1) |
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140 | (1) |
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7.7.10 Non-informative prior |
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140 | (1) |
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141 | (74) |
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143 | (34) |
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8.1 The Problem of Induction |
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143 | (6) |
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8.1.1 An ill posed problem |
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144 | (2) |
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8.1.1.1 On intuitively obvious propositions |
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146 | (1) |
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8.1.2 Induction is a law of nature |
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147 | (1) |
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8.1.3 Anthropic principle |
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148 | (1) |
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149 | (15) |
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8.2.1 Information and knowledge |
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149 | (1) |
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8.2.2 Resonance complexity |
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150 | (1) |
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151 | (2) |
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153 | (2) |
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8.2.5 Resonance level reduction |
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155 | (1) |
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8.2.6 Properties of resonance |
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156 | (1) |
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8.2.7 Physical basis of resonance |
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157 | (2) |
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8.2.8 Information and knowledge revisited |
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159 | (1) |
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8.2.9 Resonance and subjectivity |
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160 | (1) |
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161 | (1) |
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8.2.11 Resonance and philosophy of Hume |
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162 | (1) |
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8.2.12 Is resonance a new concept? |
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163 | (1) |
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164 | (2) |
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166 | (5) |
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8.4.1 Artificial Intelligence |
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169 | (1) |
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8.4.2 Social Context of Intelligence |
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170 | (1) |
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171 | (3) |
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8.6 From Philosophy to Science |
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174 | (3) |
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177 | (4) |
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181 | (34) |
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10.1 Science as a Communication System |
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181 | (3) |
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10.2 Some Attributes of Science |
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184 | (3) |
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10.2.1 Interpersonal character |
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184 | (1) |
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10.2.2 The role and limitations of resonance |
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185 | (1) |
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10.2.3 Convergence to the truth? |
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185 | (1) |
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186 | (1) |
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10.2.5 Beyond simple induction |
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186 | (1) |
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187 | (1) |
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10.3 Science for Scientists |
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187 | (1) |
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188 | (2) |
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189 | (1) |
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10.5 Sources and Perils of Loyalty |
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190 | (5) |
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10.5.1 Science as an antidote to manipulation |
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193 | (1) |
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10.5.2 Dependent information sources |
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194 | (1) |
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10.6 Falsificationism and Resonance |
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195 | (2) |
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10.7 Falsificationism as a Religion |
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197 | (3) |
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10.7.1 Sainthood in science |
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198 | (2) |
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200 | (1) |
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10.9 Multiple Personality Disorder |
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200 | (2) |
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10.10 Reality, Philosophy and Science |
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202 | (2) |
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204 | (1) |
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10.12 Major Trends in Philosophy of Science |
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205 | (8) |
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10.12.1 Real science --- the big picture |
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207 | (1) |
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10.12.2 Probabilism and Bayesianism in philosophy of science |
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208 | (2) |
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10.12.3 Levels of philosophical analysis |
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210 | (1) |
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10.12.4 Position of my theory in philosophy of science |
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211 | (2) |
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213 | (2) |
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215 | (116) |
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11 The Science of Probability |
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217 | (54) |
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11.1 Interpretation of (L1)--(L6) |
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218 | (8) |
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218 | (1) |
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219 | (2) |
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221 | (1) |
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11.1.4 Limits of applicability |
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221 | (1) |
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11.1.5 (L1)--(L6) as a starting point |
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222 | (1) |
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11.1.6 Approximate probabilities |
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223 | (1) |
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11.1.7 Statistical models |
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224 | (1) |
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225 | (1) |
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11.1.9 Probability of past events |
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225 | (1) |
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11.1.10 Purely mathematical independence |
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226 | (1) |
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11.1.11 Ruelle's view of probability |
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226 | (1) |
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11.2 Scientific Verification of (L1)--(L6) |
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226 | (3) |
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229 | (13) |
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11.3.1 Predictions at various reliability levels |
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230 | (2) |
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11.3.2 Predictions in existing scientific and philosophical theories |
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232 | (1) |
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11.3.3 Predictions, conditioning and hypothesis tests |
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233 | (1) |
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11.3.4 Prediction examples |
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233 | (1) |
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11.3.5 Histograms and image reconstruction |
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234 | (2) |
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11.3.6 Contradictory predictions |
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236 | (2) |
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11.3.7 Multiple predictions |
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238 | (4) |
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11.4 Symmetry, Independence and Resonance |
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242 | (1) |
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11.5 Symmetry is Relative |
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243 | (1) |
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11.6 Moderation is Golden |
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244 | (2) |
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11.7 Applications of (L1)--(L6): Some Examples |
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246 | (4) |
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246 | (1) |
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11.7.2 Laws (L1)--(L6) as a basis for statistics |
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247 | (1) |
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11.7.3 Long run frequencies and (L1)--(L6) |
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247 | (1) |
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248 | (2) |
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250 | (1) |
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11.9 Probability of a Single Event |
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251 | (1) |
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11.10 On Events that Belong to Two Sequences |
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252 | (1) |
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11.11 Events Are More Fundamental Than Random Variables |
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253 | (1) |
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254 | (2) |
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11.13 Are Coin Tosses i.i.d. or Exchangeable? |
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256 | (1) |
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11.14 Mathematical Foundations of Probability |
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257 | (1) |
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11.15 Axioms versus Laws of Science |
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258 | (1) |
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11.16 Objective and Subjective Probabilities |
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259 | (1) |
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11.17 Physical and Epistemic Probabilities |
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260 | (1) |
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11.18 Can Probability Be Explained? |
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261 | (2) |
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263 | (1) |
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11.20 Countable Additivity |
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264 | (1) |
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265 | (1) |
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11.22 Are Laws (L1)--(L6) Necessary? |
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266 | (1) |
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266 | (1) |
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11.24 The History of (L1)--(L6) in Philosophy of Probability |
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267 | (1) |
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11.25 Symmetry and Theories of Probability |
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268 | (3) |
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271 | (20) |
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271 | (1) |
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12.2 Decision Making in the Context of (L1)--(L6) |
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272 | (8) |
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12.2.1 Maximization of expected gain |
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272 | (3) |
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12.2.2 Maximization of expected gain as an axiom |
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275 | (1) |
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12.2.3 Stochastic ordering of decisions |
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275 | (3) |
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12.2.4 Generating predictions |
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278 | (1) |
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12.2.5 Intermediate decision problems |
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279 | (1) |
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12.3 Decision Making and Resonance |
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280 | (2) |
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12.4 Events with No Probabilities |
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282 | (2) |
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284 | (6) |
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12.5.1 Variability of utility in time |
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284 | (1) |
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12.5.2 Nonlinearity of utility |
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285 | (2) |
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12.5.3 Utility of non-monetary rewards |
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287 | (1) |
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12.5.4 Unobservable utilities |
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288 | (1) |
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12.5.5 Can utility be objective? |
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288 | (1) |
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12.5.6 What is the utility of gazillion dollars? |
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289 | (1) |
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12.6 Identification of Decisions and Probabilities |
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290 | (1) |
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291 | (16) |
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13.1 Confidence Intervals |
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291 | (3) |
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13.1.1 Practical challenges with statistical predictions |
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292 | (2) |
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13.1.2 Making predictions is necessary |
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294 | (1) |
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294 | (5) |
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13.2.1 Estimation and (L1)--(L6) |
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297 | (1) |
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13.2.2 Unbiasedness --- a concept with a single application |
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297 | (2) |
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299 | (4) |
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13.3.1 Hypothesis tests and collectives |
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300 | (1) |
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13.3.2 Hypothesis tests and the frequency interpretation of probability |
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301 | (1) |
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13.3.3 Hypothesis testing and (L1)--(L6) |
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301 | (1) |
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13.3.3.1 Sequences of hypothesis tests |
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301 | (1) |
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13.3.3.2 Single hypothesis test |
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302 | (1) |
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13.4 Hypothesis Testing and (L6) |
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303 | (1) |
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13.5 Hypothesis Testing and Falsificationism |
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304 | (1) |
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13.6 Does Frequency Statistics Need the Frequency Philosophy of Probability? |
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304 | (3) |
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307 | (24) |
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14.1 Two Faces of Subjectivity |
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307 | (2) |
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14.1.1 Non-existence versus informal assessment |
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307 | (1) |
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14.1.2 Are all probabilities subjective? |
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308 | (1) |
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14.1.3 Conditioning versus individuality |
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308 | (1) |
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14.1.4 Non-existent decisions |
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309 | (1) |
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14.2 Elements of Bayesian Analysis |
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309 | (1) |
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310 | (1) |
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14.3.1 Bayesian models are totally objective |
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310 | (1) |
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14.3.2 Bayesian models are totally subjective |
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311 | (1) |
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311 | (6) |
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312 | (1) |
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14.4.2 Bayesian statistics as an iterative method |
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313 | (2) |
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14.4.3 Truly subjective priors |
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315 | (2) |
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317 | (1) |
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318 | (1) |
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318 | (3) |
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14.7.1 Non-convergence of posterior distributions |
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319 | (2) |
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14.8 Bayesian Statistics and (L1)--(L6) |
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321 | (1) |
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14.9 Spurious Predictions |
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321 | (1) |
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14.10 Who Needs Subjectivism? |
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322 | (1) |
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14.11 Preaching to the Converted |
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323 | (3) |
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14.12 Constants and Random Variables |
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326 | (2) |
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328 | (3) |
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331 | (60) |
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333 | (20) |
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15.1 On Ideologies and Their Photo-Negatives |
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333 | (4) |
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15.2 Experimental Statistics --- A Missing Science |
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337 | (3) |
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15.3 Statistical Time Capsules |
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340 | (1) |
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15.4 Is Statistics a Science? |
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341 | (2) |
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15.5 Psychoanalytic Interpretation of Philosophy of Probability |
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343 | (1) |
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15.6 From Intuition to Science |
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344 | (2) |
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346 | (1) |
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15.8 The Three Aspects of Probability |
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347 | (1) |
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15.9 Is Probability a Science? |
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348 | (1) |
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15.10 Are Probability and Logic Experimental Sciences? |
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349 | (4) |
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16 Paradoxes, Wagers and Rules |
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353 | (22) |
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16.1 St. Petersburg Paradox |
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353 | (2) |
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355 | (6) |
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16.2.1 Scientific aspects of Pascal's wager |
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356 | (1) |
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16.2.1.1 Two kinds of infinity |
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356 | (1) |
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357 | (1) |
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16.2.1.3 On the utility of eternal life in hell |
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358 | (1) |
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16.2.1.4 Exponential discounting |
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358 | (1) |
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16.2.2 A sociological analysis of Pascal's wager |
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359 | (2) |
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361 | (5) |
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16.3.1 Cromwell's rule: practical implementation |
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361 | (3) |
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16.3.2 Cromwell's rule: philosophical problems |
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364 | (2) |
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366 | (1) |
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16.5 A New Prisoner Paradox |
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367 | (2) |
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16.5.1 Analysis of the new prisoner paradox |
|
|
368 | (1) |
|
|
|
369 | (3) |
|
16.7 The Probability of God |
|
|
372 | (3) |
|
|
|
375 | (8) |
|
17.1 Teaching Independence |
|
|
378 | (1) |
|
17.2 Probability and Frequency |
|
|
379 | (1) |
|
17.3 Undergraduate Textbooks |
|
|
380 | (3) |
|
18 Mathematical Methods of Probability and Statistics |
|
|
383 | (8) |
|
|
|
383 | (3) |
|
18.1.1 Law of large numbers, central limit theorem and large deviations principle |
|
|
385 | (1) |
|
18.1.2 Exchangeability and de Finetti's theorem |
|
|
385 | (1) |
|
18.2 Frequency Statistics |
|
|
386 | (1) |
|
|
|
387 | (1) |
|
18.4 Contradictory Predictions |
|
|
388 | (3) |
| Bibliography |
|
391 | (6) |
| Index |
|
397 | |
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