The Search for Certainty: On the Clash of Science and Philosophy of Probability

Contents......Page 10
Preface......Page 8
1.1 Reality and Philosophy......Page 16
1.2 Summary of the Main Claims......Page 17
1.2.1 Critique of the frequency and subjective theories......Page 18
1.2.2 Scienti c laws of probability......Page 21
1.2.3 Statistics and philosophy......Page 23
1.3 Historical and Social Context......Page 25
1.4 Disclaimers......Page 27
2. Main Philosophies of Probability......Page 30
2.2 The Logical Theory......Page 31
2.4 The Subjective Theory......Page 33
2.4.1 Interpreting subjectivity......Page 34
2.4.2 Verification of probabilistic statements......Page 35
2.4.3 Subjectivity as an escape from the shackles of verification......Page 37
2.4.4 The Dutch book argument......Page 38
2.4.5 The axiomatic system......Page 40
2.4.6 Identification of probabilities and decisions......Page 41
2.5 The Frequency Theory......Page 42
2.6 Summary of Philosophical Theories of Probability......Page 44
2.7 Incompleteness—The Universal Malady......Page 45
2.8 Popular Philosophy......Page 46
3. The Science of Probability......Page 50
3.1 Interpretation of (L1)-(L5)......Page 51
3.2 A Philosophy of Probability and Scienti c Veri cation of (L1)-(L5)......Page 55
3.3 Predictions......Page 58
3.4 Is Symmetry Objective?......Page 68
3.5 Symmetry is Relative......Page 69
3.6 Moderation is Golden......Page 70
3.6.1 A sixth law?......Page 72
3.7 Circularity in Science and Philosophy......Page 73
3.8 Applications of (L1)-(L5): Some Examples......Page 74
3.8.2 Laws (L1)-(L5) as a basis for statistics......Page 75
3.8.3 Long run frequencies and (L1)-(L5)......Page 76
3.8.4 Life on Mars......Page 77
3.9 Symmetry and Data......Page 79
3.10 Probability of a Single Event......Page 80
3.11 On Events that Belong to Two Sequences......Page 81
3.12 Deformed Coins......Page 82
3.13 Symmetry and Theories of Probability......Page 83
3.14 Are Coin Tosses i.i.d. or Exchangeable?......Page 85
3.15 Physical and Epistemic Probabilities......Page 86
3.16 Countable Additivity......Page 87
3.17 Quantum Mechanics......Page 89
4.1 Decision Making in the Context of (L1)-(L5)......Page 92
4.1.1 Maximization of expected gain......Page 93
4.1.2 Maximization of expected gain as an axiom......Page 95
4.1.3 Stochastic ordering of decisions......Page 96
4.1.4 Generating predictions......Page 98
4.1.5 A new prisoner paradox......Page 99
4.2 Events with No Probabilities......Page 101
4.3 Law Enforcement......Page 103
4.4 Utility in Complex Decision Problems......Page 105
4.4.2 Nonlinearity of utility......Page 106
4.4.3 Utility of non-monetary rewards......Page 108
4.4.4 Unobservable utilities......Page 109
4.5 Identification of Decisions and Probabilities......Page 110
5. The Frequency Philosophy of Probability......Page 112
5.2 Inconsistencies in von Mises\' Theory......Page 113
5.3 Collective as an Elementary Concept......Page 115
5.4 Applications of Probability Do Not Rely on Collectives......Page 116
5.5 Collectives in Real Life......Page 118
5.6 Collectives and Symmetry......Page 120
5.7 Frequency Theory and the Law of Large Numbers......Page 121
5.8 Benefits of Imagination and Imaginary Bene ts......Page 122
5.9 Imaginary Collectives......Page 123
5.10 Computer Simulations......Page 124
5.11 Frequency Theory and Individual Events......Page 125
5.12 Collectives and Populations......Page 126
5.13 Are All i.i.d. Sequences Collectives?......Page 127
5.14 Are Collectives i.i.d. Sequences?......Page 128
6.1 Confidence Intervals......Page 130
6.2 Estimation......Page 133
6.3 Hypothesis Testing......Page 136
6.3.1 Hypothesis tests and collectives......Page 137
6.3.2 Hypothesis tests and the frequency interpretation of probability......Page 138
6.3.3 Hypothesis testing and (L1)-(L5)......Page 139
6.4 Experimental Statistics—A Missing Science......Page 141
6.6 Does Classical Statistics Need the Frequency Theory?......Page 144
7.1 The Smoking Gun......Page 146
7.2 How to Eat the Cake and Have It Too......Page 148
7.3 The Subjective Theory of Probability is Objective......Page 151
7.4 A Science without Empirical Content......Page 152
7.5.1 Creating something out of nothing......Page 154
7.5.2 The essence of probability......Page 155
7.6 The Subjective Theory Does Not Imply the Bayes Theorem......Page 158
7.6.1 Sequential decisions in statistics......Page 159
7.6.2 Honest mistakes......Page 160
7.6.3 The past and the future are decoupled......Page 162
7.6.4 The Dutch book argument is static......Page 165
7.6.5 Cohabitation with an evil demiurge......Page 168
7.6.6 The Bayes theorem is unobservable......Page 170
7.6.7 All statistical strategies are Bayesian......Page 171
7.7 The Dutch Book Argument is Rejected by Bayesians......Page 173
7.8 No Need to Collect Data......Page 174
7.9 Empty Promises......Page 175
7.10 The Meaning of Consistency......Page 176
7.11 Interpreting Miracles......Page 177
7.12 Science, Probability and Subjectivism......Page 178
7.13 A Word with a Thousand Meanings......Page 180
7.14 Apples and Oranges......Page 184
7.15 Arbitrage......Page 186
7.16 Subjective Theory and Atheism......Page 187
7.17 Imagination and Probability......Page 188
7.18 A Misleading Slogan......Page 190
7.19 Axiomatic System as a Magical Trick......Page 191
8.1.1 Non-existence vs. informal assessment......Page 192
8.1.3 Conditioning vs. individuality......Page 193
8.2 Elements of Bayesian Analysis......Page 194
8.3 Models......Page 195
8.4 Priors......Page 196
8.4.1 Objective priors......Page 197
8.4.2 Bayesian statistics as an iterative method......Page 198
8.4.3 Truly subjective priors......Page 199
8.6 Posteriors......Page 202
8.6.1 Non-convergence of posterior distributions......Page 203
8.8 Spurious Predictions......Page 205
8.9 Who Needs Subjectivism?......Page 206
8.10 Preaching to the Converted......Page 207
8.11 Constants and Random Variables......Page 210
8.12 Criminal Trials......Page 211
9. Teaching Probability......Page 214
9.1 Teaching Independence......Page 217
9.2 Probability and Frequency......Page 218
9.3 Undergraduate Textbooks......Page 219
10. Abuse of Language......Page 222
11. What is Science?......Page 226
11.1 From Intuition to Science......Page 229
11.2 Science as Service......Page 231
11.3 Decision Making......Page 232
11.4 Mathematical Foundations of Probability......Page 233
11.5 Axioms versus Laws of Science......Page 235
12. What is Philosophy?......Page 236
12.1 What is Philosophy of Probability?......Page 238
12.2 Is Probability a Science?......Page 241
12.3 Objective and Subjective Probabilities......Page 242
12.4 Yin and Yang......Page 243
12.5 What Exists?......Page 244
12.6 Who Needs Philosophy?......Page 245
13.1 Does Science Have to be Rational?......Page 246
13.2 Common Elements in Frequency and Subjective Theories......Page 247
13.4 Common Misconceptions......Page 248
14.1 Probability......Page 252
14.1.1 Law of Large Numbers, Central Limit Theorem and Large Deviations Principle......Page 253
14.1.2 Exchangeability and de Finetti\'s theorem......Page 254
14.2 Classical Statistics......Page 255
14.3 Bayesian Statistics......Page 256
14.4 Contradictory Predictions......Page 257
15.1 Classics......Page 260
15.3 Philosophy and Mathematics......Page 261
Bibliography......Page 264
Index......Page 268