Thompson The Nature of Statistical Evidence

1. Auflage 2007
ISBN: 978-0-387-40054-9
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 152 Seiten

Our motivation for writing this book was a dissatisfaction with the many books with titles hke Foundations of Statistics. These books provide a needed description of the subject with examples of various statistical methods; but they do not satisfy the discerning reader because they do not explain why certain conclusions may be drawn from certain data and they do not discuss how statistics, the subject, meshes with the scientific process. We naively set out to fill in these gaps, but the situation is not so simple. What is desired is a tool—the one true statistics—which can be applied to data with certainty to justify conclusions, but what is available, in fact, are various competing theories of statistics. The careful reader may correctly point out the relation of this manuscript to po- modem epistemology, a philosophy which emphasizes that all human knowing—in fields as diverse as religion and science—is culture dependent and that therefore "truth" is not absolute. Take, for example, the "truths" of Christianity and Islam— wars have been fought over their differences. In the field of statistics there are, for example, Bayesians and sampling theorists. Carson (2003) states, "Surely it is better, postmodems tell us, to encourage insights that flow from many different perspectives. . . " In statistics, postmodernism is manifest at two levels. First, the findings of statistical applications are not about the true state of nature but about models of experiments on nature.

Autoren/Hrsg.

Thompson, Bill

Inhaltsverzeichnis

1;Contents;5
2;Overview;10
3;The Context;15
3.1;Mathematics and Its Applications;16
3.1.1;2.1. Axiomatic Method and Mathematics;16
3.1.2;2.2. Prepositional Calculus as an Axiomatic System;19
3.1.3;2.3. The Euclidean Geometry Paradigm^;20
3.1.4;2.4. Formal Science and Applied Mathematics;22
3.1.5;Notes on the Literature;24
3.2;The Evolution of Natural Scientists and Their Theories;25
3.2.1;3.1. Background;25
3.2.2;3.2. "Prediction = Science"?;28
3.2.3;3.3. External Aspects of Science;31
3.2.4;3.4. Conclusion Theory;36
3.3;Law and Learning;40
3.3.1;4.1. The Law;40
3.3.2;4.2. Learning Theory;42
4;Interpreting Probability;45
4.1;Introduction to Probability;46
4.1.1;5.1. The Axiomatic Theory;47
4.1.2;5.2. Interpreting the Axiomatic Theory;52
4.1.3;5.3. Enumerative Probability;54
4.1.4;5.4. Frequency Probability;55
4.1.5;5.5. Propensity Probability—"Tendency To Happen";56
4.2;The Fair Betting Utility Interpretation of Probability;59
4.2.1;6.1. Probability as Personal Price;59
4.2.2;6.2. Probability as Personal Degree of Belief;63
4.2.3;6.3. Conditional Subjective Probability;64
4.3;Attitudes Toward Chance;66
4.3.1;7.1. Indeterminism and Chance;66
4.3.2;7.2. The Location of Probability—Mind or Matter?;67
4.3.3;7.3. Equally Likely;68
4.3.4;7.4. Frequency and the Law of Large Numbers;72
4.3.5;7.5. The Single Instance;74
4.3.6;7.6. Locations of Some Kinds of Probabilities;75
5;Statistical Models of Induction;77
5.1;A Framework for Statistical Evidence;78
5.1.1;8.1. Introduction;78
5.1.2;8.2. What Is an Experiment?;79
5.1.3;8.3. Ey{ E, y) Does Not Exist;80
5.1.4;8.4. The Fitted Parameter Model;81
5.1.5;8.5. Interlaboratory Experimentation;82
5.1.6;8.6. Conclusion;86
5.2;A Critique of Bayesian Inference;87
5.2.1;9.1. Randomness Needs Explaining;87
5.2.2;9.2. How to Adjust Your Belief;89
5.2.3;9.3. The Economic Approach to Group BeHef;91
5.2.4;9.4. Objective Bayesian Statistics;97
5.3;The Long-Run Consequence of Behavior;100
5.3.1;10.1. Decision Theory;100
5.3.2;10.2. The Accept-Reject Theory;101
5.3.3;10.3. Frequency Interpretation of Confidence Intervals;102
5.4;A Critique of p-Values;105
5.4.1;11.1. The Context;105
5.4.2;11.2. Simple Hypotheses;107
5.4.3;11.3. Composite Hypotheses;111
5.4.4;11.4. Test of a Direction;113
5.4.5;11.5. Asymptotic Considerations;115
5.4.6;11.6. Intuitive Test Statistics;116
5.4.7;11.7. Discussion;118
5.5;The Nature of Statistical Evidence;119
5.5.1;12.1. Introduction;119
5.5.2;12.2. Birnbaum's Theorem;120
5.5.3;12.3. Counterexamples;123
5.6;The Science of Statistics;125
5.6.1;13.1. Meanings of " Science";125
5.6.2;13.2. Concepts of Statistics;127
5.6.3;13.3. Is Statistics a Science?;127
5.7;Comparison of Evidential Theories;129
5.7.1;14.1. Evidential Models and Their Criteria;129
5.7.2;14.2. Two Interpretations of the Mathematics of Testing Are Needed;129
5.7.3;14.3. Evidential Meaning;130
5.7.4;14.4. Comparison of Bayesian, Likelihood, and " Show- Me" Evidence;130
5.7.5;14.5. An Example—Probability Evidence and the Law;133
5.8;Appendix A Deductive Logic;136
5.8.1;A. l. The Propositional Calculus;136
5.8.2;A.2. Truth Tables;138
5.8.3;A. 3. Deductive Arguments and Conditional Statements;139
5.8.4;A.4. Critique of Classical Logic;142
5.8.5;A. 5. Many-Valued Logic;143
5.8.6;Notes on the Literature;145
6;Author Index;151
7;Subject Index;153