The Nature of Statistical Evidence (eBook)

Bill Thompson (Autor)

eBook Download: PDF

2007 | 2007
X, 152 Seiten
Springer New York (Verlag)
978-0-387-40054-9 (ISBN)

Infos

The purpose of this book is to discuss whether statistical methods make sense. The present volume begins the task of providing interpretations and explanations of several theories of statistical evidence. It should be relevant to anyone interested in the logic of experimental science. Have we achieved a true Foundation of Statistics? We have made the link with one widely accepted view of science and we have explained the senses in which Bayesian statistics and p-values allow us to draw conclusions. This book has substantial implications for all users of Statistical methods.

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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 "e;truth"e; is not absolute. Take, for example, the "e;truths"e; 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, "e;Surely it is better, postmodems tell us, to encourage insights that flow from many different perspectives. . . "e; 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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 21.12.2007
• Reihe/Serie: Lecture Notes in Statistics
• Reihe/Serie: Lecture Notes in Statistics
• Zusatzinfo: X, 152 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• Schlagworte: Bayesian Statistics • Evidence • experimental science • Probability • p-values
• ISBN-10: 0-387-40054-0 / 0387400540
• ISBN-13: 978-0-387-40054-9 / 9780387400549