Janus-Faced Probability
1st Edition
Published on 25. April 2014
XIII, 147 pages
978-3-319-04861-1 (ISBN)
Description
The problem of probability interpretation was long overlooked before exploding in the 20th century, when the frequentist and subjectivist schools formalized two conflicting conceptions of probability. Beyond the radical followers of the two schools, a circle of pluralist thinkers tends to reconcile the opposing concepts. The author uses two theorems in order to prove that the various interpretations of probability do come into opposition and can be used in different contexts. The goal here is to clarify the multi fold nature of probability by means of a purely mathematical approach and to show how philosophical arguments can only serve to deepen actual intellectual contrasts. The book can be considered as one of the most important contributions in the analysis of probability interpretation in the last 10-15 years.
Reviews / Votes
"I highly recommend this book for those readers interested in an accessible introduction to the problems related to the foundations of probability theory and statistics, with a very interesting compilation of references on the subject. Beside this, the ideas presented in the book open the door to interesting research inquiries in the field, especially those related to the foundations of quantum probability theory, quantum information theory and quantum statistical techniques." (Federico Holik, Mathematical Reviews, December, 2015)
"Rocchi (IBM and LUISS Univ., Italy) begins by discussing two of these, the frequentist and the subjectivist approaches. . Each of the book's ten chapters ends with a list of recent papers on the topic, and the book ends with an interesting series of three appendixes. . This specialized work will mainly appeal to philosophers of probability. Summing Up: Recommended. Only comprehensive mathematics collections serving researchers/faculty." (M. Bona, Choice, Vol. 52 (9), May, 2015)
More details
Series
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
Professional and scholarly
Product notice
Reflowable
Illustrations
XIII, 147 p. 26 illus.
File size
2,49 MB
ISBN-13
978-3-319-04861-1 (9783319048611)
DOI
10.1007/978-3-319-04861-1
Schweitzer Classification
Other editions
Additional editions
Person
Prof. Rocchi has already produced original proposals in various research fields such as computing, reliability theory, didactics and software programming and this book also suggests a creative solution.
Content
1 - Preface [Seite 6]
2 - Contents [Seite 8]
3 - Introduction [Seite 11]
4 - Part I On the Meaning of Probability [Seite Part I On the Meaning of Probability]
- 12 [Seite 12]
5 - 1 Interpretations of Probability [Seite 13]
5.1 - 1.1.Variety of Theories [Seite 14]
5.2 - 1.2.Pluralists [Seite 16]
5.3 - References [Seite 18]
6 - 2 A Mathematical Approach to the Interpretation Problem [Seite 19]
6.1 - 2.1.Why I am a Pluralist [Seite 19]
6.2 - 2.2.Guidelines [Seite 20]
6.3 - 2.3.An Analytical Approach [Seite 21]
6.3.1 - 2.3.1 Theorem of Large Numbers (Strong) [Seite 21]
6.3.2 - 2.3.2 Theorem of a Single Number [Seite 22]
6.4 - 2.4.Two Remarks on Mathematics and Science [Seite 22]
6.4.1 - 2.4.1 Physical Mapping [Seite 22]
6.4.2 - 2.4.2 Division of Labor [Seite 26]
6.5 - 2.5.Different Arguments, Different Probabilities [Seite 27]
6.6 - References [Seite 31]
7 - 3 Probability Validation [Seite 33]
7.1 - 3.1.A Parameter that Cannot be Validated Does Not Exist [Seite 33]
7.2 - 3.2.Theoretical Proof and Empirical Validation [Seite 34]
7.3 - 3.3.From Controllability to Realism [Seite 35]
7.3.1 - 3.3.1 Statistics of Long-Term Events [Seite 36]
7.4 - 3.4.From Uncontrollability to Unrealism [Seite 37]
7.4.1 - 3.4.1 Statistics of Single Events [Seite 40]
7.5 - 3.5.Sorry Science [Seite 41]
7.5.1 - 3.5.1 Severe Constraint [Seite 42]
7.5.2 - 3.5.2 Opponents to Unrealism [Seite 43]
7.5.3 - 3.5.3 Things are No Longer the Same [Seite 44]
7.6 - 3.6.Closing Remarks [Seite 45]
7.7 - References [Seite 46]
8 - 4 About the Compatibility of the Diverging Interpretations [Seite 48]
8.1 - 4.1.Not Contradictory Approaches [Seite 48]
8.2 - 4.2.Method of Working [Seite 49]
8.2.1 - 4.2.1 Lemma of Obligatory Subjectivism [Seite 49]
8.2.2 - 4.2.2 Mutually Exclusive Tactics [Seite 50]
8.3 - References [Seite 53]
9 - 5 Criticism on the Philosophical Pollution [Seite 54]
9.1 - 5.1.Analytical Assumptions [Seite 54]
9.2 - 5.2.Separation of Areas [Seite 55]
9.3 - 5.3.Inclusiveness [Seite 55]
9.4 - 5.4.Terms of Comparison [Seite 56]
9.5 - 5.5.Moderated Debates [Seite 59]
9.5.1 - 5.5.1 Death of Dogmatism [Seite 59]
9.5.2 - 5.5.2 Empirical Problem Solving [Seite 61]
9.6 - 5.6.Open Doors [Seite 63]
9.7 - References [Seite 64]
10 - Part II Considerations Around the Probability Axiomatization [Seite Part II Considerations Around the Probability Axiomatization]
- 65 [Seite 65]
11 - 6 Some Remarks on the Argument of Probability [Seite 66]
11.1 - 6.1.Abstract Axioms [Seite 66]
11.2 - 6.2.System Modeling [Seite 67]
11.2.1 - 6.2.1 DFDs [Seite 68]
11.2.2 - 6.2.2 ERD [Seite 69]
11.3 - 6.3.A Negligible Topic? [Seite 69]
11.3.1 - 6.3.1 Famous Correspondence [Seite 70]
11.3.2 - 6.3.2 Pascal's Conjecture [Seite 70]
11.4 - References [Seite 73]
12 - 7 Classical Modeling of the Probability Argument [Seite 75]
12.1 - 7.1.Subset [Seite 75]
12.2 - 7.2.Sentence [Seite 77]
12.3 - 7.3.Precise and Generic Arguments [Seite 80]
13 - 8 Structural Modeling of the Probability Argument [Seite 83]
13.1 - 8.1.Structural Model [Seite 83]
13.1.1 - 8.1.1 Graph Model [Seite 84]
13.1.2 - 8.1.2 Algebraic Model [Seite 84]
13.1.3 - 8.1.3 Use of the Structural Model [Seite 86]
13.2 - 8.2.Compound Structures [Seite 87]
13.2.1 - 8.2.1 Meta-Relationships [Seite 87]
13.2.2 - 8.2.2 Basic Structural Forms [Seite 89]
13.3 - 8.3.Structure of Levels [Seite 91]
13.3.1 - 8.3.1 A Case [Seite 93]
13.4 - References [Seite 94]
14 - 9 Some Topics on Probability and Structural Models [Seite 96]
14.1 - 9.1.Demanding Structural Analysis [Seite 96]
14.2 - 9.2.Illustration of Events [Seite 97]
14.2.1 - 9.2.1 Complete Structure [Seite 98]
14.2.2 - 9.2.2 Incomplete Structure [Seite 99]
14.2.3 - 9.2.3 Ignorance, Uncertainty and Perfect Knowledge [Seite 100]
14.2.4 - 9.2.4 Structures and Probability [Seite 101]
14.3 - 9.3.In Search of a Precise Argument [Seite 102]
14.3.1 - 9.3.1 Experiments [Seite 102]
14.3.2 - 9.3.2 Classical Definition [Seite 103]
14.3.3 - 9.3.3 Definitional Arguments of Probability [Seite 105]
14.3.4 - 9.3.4 Useful Arguments of Probability [Seite 105]
14.4 - References [Seite 107]
15 - 10 Exploring into the Essence of Events [Seite 108]
15.1 - 10.1.The Core [Seite 109]
15.1.1 - 10.1.1 Equivocal Description [Seite 109]
15.1.2 - 10.1.2 More or Less Steady [Seite 111]
15.1.3 - 10.1.3 Multiplication Law [Seite 113]
15.1.4 - 10.1.4 Addition Law [Seite 114]
15.2 - 10.2.About Event's Outcome [Seite 115]
15.3 - 10.3.Final Remark [Seite 116]
15.4 - References [Seite 117]
16 - Appendix A Interpretations of Probability-An Outline [Seite Appendix A Interpretations of Probability-An Outline]
- 118 [Seite 118]
17 - Appendix B: Pluralist Works-A Partial Bibliography [Seite Appendix B: Pluralist Works-A Partial Bibliography]
- 140 [Seite 140]
18 - Appendix C: Law of Large Numbers-A Proof [Seite 143]
19 - Index [Seite 148]
2 - Contents [Seite 8]
3 - Introduction [Seite 11]
4 - Part I On the Meaning of Probability [Seite Part I On the Meaning of Probability]
- 12 [Seite 12]
5 - 1 Interpretations of Probability [Seite 13]
5.1 - 1.1.Variety of Theories [Seite 14]
5.2 - 1.2.Pluralists [Seite 16]
5.3 - References [Seite 18]
6 - 2 A Mathematical Approach to the Interpretation Problem [Seite 19]
6.1 - 2.1.Why I am a Pluralist [Seite 19]
6.2 - 2.2.Guidelines [Seite 20]
6.3 - 2.3.An Analytical Approach [Seite 21]
6.3.1 - 2.3.1 Theorem of Large Numbers (Strong) [Seite 21]
6.3.2 - 2.3.2 Theorem of a Single Number [Seite 22]
6.4 - 2.4.Two Remarks on Mathematics and Science [Seite 22]
6.4.1 - 2.4.1 Physical Mapping [Seite 22]
6.4.2 - 2.4.2 Division of Labor [Seite 26]
6.5 - 2.5.Different Arguments, Different Probabilities [Seite 27]
6.6 - References [Seite 31]
7 - 3 Probability Validation [Seite 33]
7.1 - 3.1.A Parameter that Cannot be Validated Does Not Exist [Seite 33]
7.2 - 3.2.Theoretical Proof and Empirical Validation [Seite 34]
7.3 - 3.3.From Controllability to Realism [Seite 35]
7.3.1 - 3.3.1 Statistics of Long-Term Events [Seite 36]
7.4 - 3.4.From Uncontrollability to Unrealism [Seite 37]
7.4.1 - 3.4.1 Statistics of Single Events [Seite 40]
7.5 - 3.5.Sorry Science [Seite 41]
7.5.1 - 3.5.1 Severe Constraint [Seite 42]
7.5.2 - 3.5.2 Opponents to Unrealism [Seite 43]
7.5.3 - 3.5.3 Things are No Longer the Same [Seite 44]
7.6 - 3.6.Closing Remarks [Seite 45]
7.7 - References [Seite 46]
8 - 4 About the Compatibility of the Diverging Interpretations [Seite 48]
8.1 - 4.1.Not Contradictory Approaches [Seite 48]
8.2 - 4.2.Method of Working [Seite 49]
8.2.1 - 4.2.1 Lemma of Obligatory Subjectivism [Seite 49]
8.2.2 - 4.2.2 Mutually Exclusive Tactics [Seite 50]
8.3 - References [Seite 53]
9 - 5 Criticism on the Philosophical Pollution [Seite 54]
9.1 - 5.1.Analytical Assumptions [Seite 54]
9.2 - 5.2.Separation of Areas [Seite 55]
9.3 - 5.3.Inclusiveness [Seite 55]
9.4 - 5.4.Terms of Comparison [Seite 56]
9.5 - 5.5.Moderated Debates [Seite 59]
9.5.1 - 5.5.1 Death of Dogmatism [Seite 59]
9.5.2 - 5.5.2 Empirical Problem Solving [Seite 61]
9.6 - 5.6.Open Doors [Seite 63]
9.7 - References [Seite 64]
10 - Part II Considerations Around the Probability Axiomatization [Seite Part II Considerations Around the Probability Axiomatization]
- 65 [Seite 65]
11 - 6 Some Remarks on the Argument of Probability [Seite 66]
11.1 - 6.1.Abstract Axioms [Seite 66]
11.2 - 6.2.System Modeling [Seite 67]
11.2.1 - 6.2.1 DFDs [Seite 68]
11.2.2 - 6.2.2 ERD [Seite 69]
11.3 - 6.3.A Negligible Topic? [Seite 69]
11.3.1 - 6.3.1 Famous Correspondence [Seite 70]
11.3.2 - 6.3.2 Pascal's Conjecture [Seite 70]
11.4 - References [Seite 73]
12 - 7 Classical Modeling of the Probability Argument [Seite 75]
12.1 - 7.1.Subset [Seite 75]
12.2 - 7.2.Sentence [Seite 77]
12.3 - 7.3.Precise and Generic Arguments [Seite 80]
13 - 8 Structural Modeling of the Probability Argument [Seite 83]
13.1 - 8.1.Structural Model [Seite 83]
13.1.1 - 8.1.1 Graph Model [Seite 84]
13.1.2 - 8.1.2 Algebraic Model [Seite 84]
13.1.3 - 8.1.3 Use of the Structural Model [Seite 86]
13.2 - 8.2.Compound Structures [Seite 87]
13.2.1 - 8.2.1 Meta-Relationships [Seite 87]
13.2.2 - 8.2.2 Basic Structural Forms [Seite 89]
13.3 - 8.3.Structure of Levels [Seite 91]
13.3.1 - 8.3.1 A Case [Seite 93]
13.4 - References [Seite 94]
14 - 9 Some Topics on Probability and Structural Models [Seite 96]
14.1 - 9.1.Demanding Structural Analysis [Seite 96]
14.2 - 9.2.Illustration of Events [Seite 97]
14.2.1 - 9.2.1 Complete Structure [Seite 98]
14.2.2 - 9.2.2 Incomplete Structure [Seite 99]
14.2.3 - 9.2.3 Ignorance, Uncertainty and Perfect Knowledge [Seite 100]
14.2.4 - 9.2.4 Structures and Probability [Seite 101]
14.3 - 9.3.In Search of a Precise Argument [Seite 102]
14.3.1 - 9.3.1 Experiments [Seite 102]
14.3.2 - 9.3.2 Classical Definition [Seite 103]
14.3.3 - 9.3.3 Definitional Arguments of Probability [Seite 105]
14.3.4 - 9.3.4 Useful Arguments of Probability [Seite 105]
14.4 - References [Seite 107]
15 - 10 Exploring into the Essence of Events [Seite 108]
15.1 - 10.1.The Core [Seite 109]
15.1.1 - 10.1.1 Equivocal Description [Seite 109]
15.1.2 - 10.1.2 More or Less Steady [Seite 111]
15.1.3 - 10.1.3 Multiplication Law [Seite 113]
15.1.4 - 10.1.4 Addition Law [Seite 114]
15.2 - 10.2.About Event's Outcome [Seite 115]
15.3 - 10.3.Final Remark [Seite 116]
15.4 - References [Seite 117]
16 - Appendix A Interpretations of Probability-An Outline [Seite Appendix A Interpretations of Probability-An Outline]
- 118 [Seite 118]
17 - Appendix B: Pluralist Works-A Partial Bibliography [Seite Appendix B: Pluralist Works-A Partial Bibliography]
- 140 [Seite 140]
18 - Appendix C: Law of Large Numbers-A Proof [Seite 143]
19 - Index [Seite 148]
