Understanding Uncertainty

Preface xi Prologue xiii 1. Uncertainty 1 1.1. Introduction 1 1.2. Examples 2 1.3. Suppression of Uncertainty 7 1.4. The Removal of Uncertainty 8 1.5. The Uses of Uncertainty 9 1.6. The Calculus of Uncertainty 11 1.7. Beliefs 12 1.8. Decision Analysis 13 2. Stylistic Questions 15 2.1. Reason 15 2.2. Unreason 17 Literature 17 Advertising 17 Politics 18 Law 18 Television 18 2.3. Facts 19 2.4. Emotion 19 2.5. Prescriptive and Descriptive Approaches 20 2.6. Simplicity 22 2.7. Mathematics 23 2.8. Writing 25 2.9. Mathematics Tutorial 26 3. Probability 30 3.1. Measurement 30 3.2. Randomness 32 3.3. A Standard for Probability 34 3.4. Probability 35 3.5. Coherence 36 3.6. Belief 37 3.7. Complementary Event 39 3.8. Odds 40 3.9. Knowledge Base 43 3.10. Examples 44 3.11. Retrospect 46 4. Two Events 47 4.1. Two Events 47 4.2. Conditional Probability 49 4.3. Independence 51 4.4. Association 53 4.5. Examples 54 4.6. Supposition and Fact 56 4.7. Seeing and Doing 57 5. The Rules of Probability 59 5.1. Combinations of Events 59 5.2. Addition Rule 61 5.3. Multiplication Rule 62 5.4. The Basic Rules 64 5.5. Examples 66 5.6. Extension of the Conversation 68 5.7. Dutch Books 70 5.8. Scoring Rules 72 5.9. Logic Again 73 5.10. Decision Analysis 74 5.11. The Prisoners' Dilemma 75 5.12. The Calculus and Reality 76 6. Bayes Rule 79 6.1. Transposed Conditionals 79 6.2. Learning 81 6.3. Bayes Rule 82 6.4. Medical Diagnosis 83 6.5. Odds Form of Bayes Rule 86 6.6. Forensic Evidence 88 6.7. Likelihood Ratio 89 6.8. Cromwell's Rule 90 6.9. A Tale of Two Urns 92 6.10. Ravens 94 6.11. Diagnosis and Related Matters 97 6.12. Information 98 7. Measuring Uncertainty 101 7.1. Classical Form 101 7.2. Frequency Data 103 7.3. Exchangeability 104 7.4. Bernoulli Series 106 7.5. De Finetti's Result 107 7.6. Large Numbers 109 7.7. Belief and Frequency 111 7.8. Chance 114 8. Three Events 117 8.1. The Rules of Probability 117 8.2. Simpson's Paradox 119 8.3. Source of the Paradox 121 8.4. Experimentation 122 8.5. Randomization 123 8.6. Exchangeability 125 8.7. Spurious Association 128 8.8. Independence 130 8.9. Conclusions 132 9. Variation 134 9.1. Variation and Uncertainty 134 9.2. Binomial Distribution 135 9.3. Expectation 137 9.4. Poisson Distribution 139 9.5. Spread 142 9.6. Variability as an Experimental Tool 144 9.7. Probability and Chance 145 9.8. Pictorial Representation 147 9.9. The Normal Distribution 150 9.10. Variation as a Natural Phenomenon 152 9.11. Ellsberg's Paradox 154 10. Decision Analysis 158 10.1. Beliefs and Actions 158 10.2. Comparison of Consequences 160 10.3. Medical Example 162 10.4. Maximization of Expected Utility 164 10.5. More on Utility 165 10.6. Some Complications 167 10.7. Reason and Emotion 168 10.8. Numeracy 170 10.9. Expected Utility 171 10.10. Decision Trees 172 10.11. The Art and Science of Decision Analysis 175 10.12. Further Complications 177 10.13. Combination of Features 179 10.14. Legal Applications 182 11. Science 186 11.1. Scientific Method 186 11.2. Science and Education 187 11.3. Data Uncertainty 188 11.4. Theories 190 11.5. Uncertainty of a Theory 193 11.6. The Bayesian Development 195 11.7. Modification of Theories 197 11.8. Models 199 11.9. Hypothesis Testing 202 11.10. Significance Tests 204 11.11. Repetition 206 11.12. Summary 208 12. Examples 211 12.1. Introduction 211 12.2. Cards 212 12.3. The Three Doors 213 12.4. The Newcomers to Your Street 215 12.5. The Two Envelopes 217 12.6. Y2K 220 12.7. UFOs 221 12.8. Conglomerability 224 13. Probability Assessment 226 13.1. Nonrepeatable Events 226 13.2. Two Events 227 13.3. Coherence 230 13.4. Probabilistic Reasoning 233 13.5. Trickle Down 234 13.6. Summary 236 Epilogue 238 Subject Index 243 Index of Examples 248 Index of Notations 250