Janus-faced probability

150 Pages·2014·1.488 MB·English

by  Rocchi, Paolo

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
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‘‘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