The search for certainty : on the clash of science and philosophy of probability

This volume represents a radical departure from the current philosophical duopoly in the area of foundations of probability, that is, the frequency and subjective theories. One of the main new ideas is a set of scientific laws of probability. The new laws are simple, intuitive and, last but not least, they agree well with the contents of current textbooks on probability. Another major new claim is that the "frequency statistics" has nothing in common with the "frequency philosophy of probability," contrary to popular belief. Similarly, contrary to the general perception, the "Bayesian statistics" shares nothing in common with the "subjective philosophy of probability." The book is non-partisan on the scientific side -- it is supportive of both frequency statistics and Bayesian statistics. On the other hand, it contains well-documented and thoroughly-explained criticisms of the frequency and subjective philosophies of probability. Short reviews of other philosophical theories of probability and basic mathematical methods of probability and statistics are incorporated. The book includes substantial chapters on decision theory and teaching probability, and it is easily accessible to the general audience. Contents: * Main Philosophies of Probability * The Science of Probability * Decision Making * The Frequency Philosophy of Probability * Classical Statistics * The Subjective Philosophy of Probability * Bayesian Statistics * Teaching Probability * Abuse of Language * What is Science? * What is Philosophy? * Concluding Remarks * Mathematical Methods of Probability and Statistics * Literature Review 1. Introduction. 1.1. Reality and philosophy. 1.2. Summary of the main claims. 1.3. Historical and social context. 1.4. Disclaimers -- 2. Main Philosophies of probability. 2.1. The classical theory. 2.2. The logical theory. 2.3. The propensity theory. 2.4. The subjective theory. 2.5. The frequency theory. 2.6. Summary of philosophical theories of probability. 2.7. Incompleteness - the universal malady. 2.8. Popular philosophy -- 3. The science of probability. 3.1. Interpretation of (L1)-(L5). 3.2. A philosophy of probability and scientific verification of (L1)-(L5). 3.3. Predictions. 3.4. Is symmetry objective? 3.5. Symmetry is relative. 3.6. Moderation is golden. 3.7. Circularity in science and philosophy. 3.8. Applications of (L1)-(L5) : some examples. 3.9. Symmetry and data. 3.10. Probability of a single event. 3.11. On events that belong to two sequences. 3.12. Deformed coins. 3.13. Symmetry and theories of probability. 3.14. Are coin tosses i.i.d. or exchangeable? 3.15. Physical and epistemic probabilities. 3.16. Countable additivity. 3.17. Quantum mechanics -- 4. Decision making. 4.1. Decision making in the context of (L1)-(L5). 4.2. Events with no probabilities. 4.3. Law enforcement. 4.4. Utility in complex decision problems. 4.5. Identification of decisions and probabilities -- 5. The frequency philosophy of probability. 5.1. The smoking gun. 5.2. Inconsistencies in von Mises' theory. 5.3. Collective as an elementary concept. 5.4. Applications of probability do not rely on collectives. 5.5. Collectives in real life. 5.6. Collectives and symmetry. 5.7. Frequency theory and the law of large numbers. 5.8. Benefits of imagination and imaginary benefits. 5.9. Imaginary collectives. 5.10. Computer simulations. 5.11. Frequency theory and individual events. 5.12. Collectives and populations. 5.13. Are all i.i.d. sequences collectives? 5.14. Are collectives i.i.d. sequences? -- 6. Classical statistics. 6.1. Confidence intervals. 6.2. Estimation. 6.3. Hypothesis testing. 6.4. Experimental statistics - a missing science. 6.5. Hypothesis testing and (L5) 6.6. Does classical statistics need the frequency theory? -- 7. The subjective philosophy of probability. 7.1. The smoking gun. 7.2. How to eat the cake and have it too. 7.3. The subjective theory of probability is objective. 7.4. A science without empirical content. 7.5. The weakest scientific theory ever. 7.6. The subjective theory does not imply the Bayes theorem. 7.7. The Dutch book argument is rejected by Bayesians. 7.8. No need to collect data. 7.9. Empty promises. 7.10. The meaning of consistency. 7.11. Interpreting miracles. 7.12. Science, probability and subjectivism. 7.13. A word with a thousand meanings. 7.14. Apples and oranges. 7.15. Arbitrage. 7.16. Subjective theory and atheism. 7.17. Imagination and probability. 7.18. A misleading slogan. 7.19. Axiomatic system as a magical trick -- 8. Bayesian statistics. 8.1. Two faces of subjectivity. 8.2. Elements of Bayesian analysis. 8.3. Models. 8.4. Priors. 8.5. Data. 8.6. Posteriors. 8.7. Bayesian statistics and (L1)-(L5). 8.8. Spurious predictions. 8.9. Who needs subjectivism? 8.10. Preaching to the converted. 8.11. Constants and random variables. 8.12. Criminal trials -- 9. Teaching probability. 9.1. Teaching independence. 9.2. Probability and frequency. 9.3. Undergraduate textbooks -- 10. Abuse of language -- 11. What is science? 11.1. From intuition to science. 11.2. Science as service. 11.3. Decision making. 11.4. Mathematical foundations of probability. 11.5. Axioms versus laws of science -- 12. What is philosophy? 12.1. What is philosophy of probability? 12.2. Is probability a science? 12.3. Objective and subjective probabilities. 12.4. Yin and yang. 12.5. What exists? 12.6. Who needs philosophy? -- 13. Concluding remarks. 13.1. Does science have to be rational? 13.2. Common elements in frequency and subjective theories. 13.3. On peaceful coexistence. 13.4. Common misconceptions -- 14. Mathematical methods of probability and statistics. 14.1. Probability. 14.2. Classical statistics. 14.3. Bayesian statistics. 14.4. Contradictory predictions -- 15. Literature review. 15.1. Classics. 15.2. Philosophy. 15.3. Philosophy and mathematics This volume represents a radical departure from the current philosophical duopoly in the area of foundations of probability, that is, the frequency and subjective theories. One of the main new ideas is a set of scientific laws of probability. The new laws are simple, intuitive and, last but not least, they agree well with the contents of current textbooks on probability. Another major new claim is that the frequency statistics has nothing in common with the frequency philosophy of probability, contrary to popular belief. Similarly, contrary to the general perception, the Bayesian statistics shares nothing in common with the subjective philosophy of probability. This volume is non-partisan on the scientific side that it is supportive of both frequency statistics and Bayesian statistics. On the other hand, it contains well-documented and thoroughly-explained criticisms of the frequency and subjective philosophies of probability. Short reviews of other philosophical theories of probability and basic mathematical methods of probability and statistics are incorporated. This volume includes essential chapters on decision theory and teaching probability, and it is easily accessible to the general audience.