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Janus-faced probability PDF

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Paolo Rocchi Janus-Faced Probability Janus-Faced Probability Paolo Rocchi Janus-Faced Probability 123 PaoloRocchi IBM Italia LUISS GuidoCarliUniversity Roma Italy ISBN 978-3-319-04860-4 ISBN 978-3-319-04861-1 (eBook) DOI 10.1007/978-3-319-04861-1 Springer ChamHeidelberg New YorkDordrecht London LibraryofCongressControlNumber:2014933942 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) ‘‘When you travel often, you will be addicted to it forever. Our destination is not the place, but new way to see things.’’ Henry Miller Preface PaoloRocchi’sbookarguesthatthediversityoftheinterpretationsofprobability, in fact, is not a problem crying for immediate resolution, but a very natural situation for a scientist working on concrete applications of probability and sta- tistics.Oneneednotkeeptojustoneconcreteinterpretation,saytothefrequency interpretation. In the debate on interpretations of probability, which has been continuing for centuries, the idea that interpretations can vary depending on the problems under study can be considered as opportunistic. The probabilistic com- munity is sharply divided into camps struggling to justify one or other concrete interpretation. For example, Kolmogorov and Gnedenko actively supported the objective interpretation, and they strongly criticized the subjective approach (however, this did not affect the friendly relations between Kolmogorov and De Finetti). Although both Kolmogorov and von Mises kept to the objective inter- pretation, they fiercely debated the primacy of frequency or measure-theoretic probability. It is interesting that although Kolmogorov debated von Mises’ fun- damental frequentism, till his death he remained not completely satisfied by his (Kolmogorov’s) measure-theoretic axiomatics. He worked hard to find a mathe- maticallyrigorousdefinitionofarandomsequenceand,inspiteofaneraofafew brilliant new ideas, such as Kolmogorov’s algorithmic complexity, the afore- mentioned problem is still open. This book presents an in-depth analysis of the basic interpretations of proba- bilityanditcanbeconsideredasanattempttocombinethemharmoniouslyonthe basis of the pluralist approach: ‘‘interpretations can be chosen depending on applications.’’Thisisacomplexresearchprojectanditisfarfrombeingcomplete. Nevertheless, the book can be considered as one of the most important contribu- tionstotheanalysisoftheinterpretationalproblemsofprobabilitytheory,atleast in the last 10–15 years. Although personally I do not support the author’s pluralism (I consider the diversity of interpretations a sign of the deep crises in the foundations of proba- bility as the result of the absence of rigorous mathematical theory of individual random sequence), I have to agree with his observation that if one of the basic interpretations, e.g., frequency or subjective, were wrong, we would have long seen problems with applications. vii viii Preface I also point to the similarity between the present interpretational statuses of probability theory and quantum mechanics. It might be that the interpretational problemsofthelatteraregeneratedbytheinterpretationalproblemsoftheformer. A collateral message of this book is that developing the pluralist viewpoint on probabilitywillcontributetomultidisciplinaryapplicationsofprobability,butitis not the final word on the problem of interpretations of probability. I hope it will pave the way for many new books to come. Andrei Khrennikov Professor of Applied Mathematics, Intl. Center for Math. Modeling Linnaeus University Växjö, Sweden Contents Part I On the Meaning of Probability 1 Interpretations of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Variety of Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Pluralists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 A Mathematical Approach to the Interpretation Problem . . . . . . 9 2.1 Why I am a Pluralist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Guidelines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 An Analytical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Theorem of Large Numbers (Strong) . . . . . . . . . . . . 11 2.3.2 Theorem of a Single Number. . . . . . . . . . . . . . . . . . 12 2.4 Two Remarks on Mathematics and Science. . . . . . . . . . . . . . 12 2.4.1 Physical Mapping. . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.2 Division of Labor. . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Different Arguments, Different Probabilities . . . . . . . . . . . . . 17 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Probability Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 A Parameter that Cannot be Validated Does Not Exist . . . . . . 23 3.2 Theoretical Proof and Empirical Validation. . . . . . . . . . . . . . 24 3.3 From Controllability to Realism. . . . . . . . . . . . . . . . . . . . . . 25 3.3.1 Statistics of Long-Term Events . . . . . . . . . . . . . . . . 26 3.4 From Uncontrollability to Unrealism. . . . . . . . . . . . . . . . . . . 27 3.4.1 Statistics of Single Events. . . . . . . . . . . . . . . . . . . . 30 3.5 Sorry Science. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5.1 Severe Constraint. . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5.2 Opponents to Unrealism . . . . . . . . . . . . . . . . . . . . . 32 3.5.3 Things are No Longer the Same. . . . . . . . . . . . . . . . 34 3.6 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ix x Contents 4 About the Compatibility of the Diverging Interpretations . . . . . . 39 4.1 Not Contradictory Approaches. . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Method of Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.1 Lemma of Obligatory Subjectivism. . . . . . . . . . . . . . 40 4.2.2 Mutually Exclusive Tactics . . . . . . . . . . . . . . . . . . . 41 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Criticism on the Philosophical Pollution . . . . . . . . . . . . . . . . . . . 45 5.1 Analytical Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Separation of Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3 Inclusiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.4 Terms of Comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.5 Moderated Debates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.5.1 Death of Dogmatism. . . . . . . . . . . . . . . . . . . . . . . . 50 5.5.2 Empirical Problem Solving . . . . . . . . . . . . . . . . . . . 52 5.6 Open Doors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Part II Considerations Around the Probability Axiomatization 6 Some Remarks on the Argument of Probability. . . . . . . . . . . . . . 59 6.1 Abstract Axioms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2.1 DFDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2.2 ERD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.3 A Negligible Topic?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.3.1 Famous Correspondence . . . . . . . . . . . . . . . . . . . . . 63 6.3.2 Pascal’s Conjecture. . . . . . . . . . . . . . . . . . . . . . . . . 63 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7 Classical Modeling of the Probability Argument . . . . . . . . . . . . . 69 7.1 Subset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.2 Sentence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.3 Precise and Generic Arguments . . . . . . . . . . . . . . . . . . . . . . 73 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8 Structural Modeling of the Probability Argument . . . . . . . . . . . . 77 8.1 Structural Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.1.1 Graph Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.1.2 Algebraic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.1.3 Use of the Structural Model. . . . . . . . . . . . . . . . . . . 80 8.2 Compound Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.2.1 Meta-Relationships. . . . . . . . . . . . . . . . . . . . . . . . . 81 8.2.2 Basic Structural Forms . . . . . . . . . . . . . . . . . . . . . . 83 Contents xi 8.3 Structure of Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.3.1 A Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9 Some Topics on Probability and Structural Models. . . . . . . . . . . 91 9.1 Demanding Structural Analysis . . . . . . . . . . . . . . . . . . . . . . 91 9.2 Illustration of Events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 9.2.1 Complete Structure. . . . . . . . . . . . . . . . . . . . . . . . . 93 9.2.2 Incomplete Structure. . . . . . . . . . . . . . . . . . . . . . . . 94 9.2.3 Ignorance, Uncertainty and Perfect Knowledge . . . . . 95 9.2.4 Structures and Probability . . . . . . . . . . . . . . . . . . . . 96 9.3 In Search of a Precise Argument . . . . . . . . . . . . . . . . . . . . . 97 9.3.1 Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.3.2 Classical Definition . . . . . . . . . . . . . . . . . . . . . . . . 98 9.3.3 Definitional Arguments of Probability. . . . . . . . . . . . 99 9.3.4 Useful Arguments of Probability . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 10 Exploring into the Essence of Events. . . . . . . . . . . . . . . . . . . . . . 103 10.1 The Core. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 10.1.1 Equivocal Description. . . . . . . . . . . . . . . . . . . . . . . 104 10.1.2 More or Less Steady. . . . . . . . . . . . . . . . . . . . . . . . 106 10.1.3 Multiplication Law. . . . . . . . . . . . . . . . . . . . . . . . . 108 10.1.4 Addition Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.2 About Event’s Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10.3 Final Remark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Appendix A: Interpretations of Probability—An Outline . . . . . . . . . . 113 Appendix B: Pluralist Works—A Partial Bibliography. . . . . . . . . . . . 135 Appendix C: Law of Large Numbers—A Proof. . . . . . . . . . . . . . . . . 139 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

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