The Foundations of Statistics: (Dover Books on Mathematics)

Classic analysis of the foundations of statistics and development of personal probability, one of the greatest controversies in modern statistical thought. Revised edition. Calculus, probability, statistics, and Boolean algebra are recommended. With the 1954 publication of his Foundations of Statistics, in which he proposed a basis that takes into account not only strictly objective and repetitive events, but also vagueness and interpersonal differences, Leonard J. Savage opened the greatest controversy in modern statistical thought. His theory of the foundations, connected with the personalistic interpretation of probability, challenged the then dominant frequentist school. In the first seven chapters of his book, Professor Savage is concerned with the foundations at a relatively deep level. To explain and defend his theory of the behavior of a highly idealized person faced with uncertainty, he considers decision making, the sure-thing principle, qualitative and quantitative personal probability, the approach to certainty through experience, symmetric sequences of events, critical comments on personal probability, utility, observations as they affect the decision, and partition problems. In chapters eight through seventeen he discusses statistics proper - the actual devices of the discipline - from the personalistic view. He concentrates on minimax problems and on the theories of estimation and testing. Exercises are included throughout to reinforce and supplement the text. The mathematical techniques used are quite elementary, some calculus and elementary probability theory being presupposed. Understanding of all the material calls for some mathematical maturity on the part of the reader. Professor Savage had reevaluated his position somewhat during the decade and a half since the work was first published. While reaffirming the material in the first seven chapters, he had reconsidered the appropriateness of many frequentistic applications. To explain these recent developments, he added a new preface, new footnotes, and a supplementary 180-item, annotated bibliography. Because of Professor Savage's death, the revisions that he made for this edition are his final analysis of the situation. As he says on page one, "the foundations are the most controversial parts of many, if not all, sciences." In statistics, the foundation of probability is "as controversial a subject as one could name." In 1954, the controversy was very great, and although it has quieted since, the problem has yet to be resolved. A new generation of readers who have missed Savage's analysis have here an opportunity to study firsthand what his important foundation of statistics - personal probability - is, and what it means to statistical thought.

Table of Contents:
1. INTRODUCTION 1. The role of foundations 2. Historical background 3. General outline of this book 2. PRELIMINARY CONSIDERATIONS ON DECISION IN THE FACE OF UNCERTAINTY 1. Introduction 2. The person 3. "The world, and states of the world" 4. Events 5. "Consequences, acts, and decisions" 6. The simple ordering of acts with respect to preference 7. The sure-thing principle 3. PERSONAL PROBABILITY 1. Introduction 2. Qualitative personal probability 3. Quantitative personal probability 4. Some mathematical details 5. "Conditional probability, qualitative and quantitative" 6. The approach to certainty through experience 7. Symmetric sequences of events 4. CRITICAL COMMENTS ON PERSONAL PROBABILITY 1. Introduction 2. Some shortcomings of the personalistic view 3. Connection with other views 4. Criticism of other views 5. The role of symmetry in probability 6. How can science use a personalistic view of probability? 5. UTILITY 1. Introduction 2. Gambles 3. "Utility, and preference among gambles" 4. The extension of utility to more general acts 5. Small worlds 6. Historical and critical comments on utility 6. OBSERVATION 1. Introduction 2. What an observation is 3. "Multiple observations, and extensions of observations and of sets of acts" 4. Dominance and admissibility 5. Outline of the design of experiments 7. PARTITION PROBLEMS 1. Introduction 2. Structure of (twofold) partition problems 3. The value of observation 4. "Extension of observations, and sufficient statistics" 5. Likelihood ratios 6. Repeated observations 7. Sequential probability ratio procedures 8. "Standard form, and absolute comparison between observations" 8. STATISTICS PROPER 1. Introduction 2. What is statistics proper? 3. Multipersonal problems 4. The minimax theory 9. INTRODUCTION TO THE MINIMAX THEORY 1. Introduction 2. The behavioralistic outlook 3. Mixed acts 4. Income and loss 5. "The minimax rule, and the principle of admissibility" 6. Illustrations of the minimax rule 7. Objectivistic motivation of the minimax rule 8. Loss as opposed to negative income in the minimax rule 10. A PERSONALISTIC REINTERPRETATION OF THE MINIMAX THEORY 1. Introduction 2. A model of group decision 3. "The group minimax rule, and the group principle of admissibility" 4. Critique of the group minimax rule 11. THE PARALLELISM BETWEEN THE MINIMAX THEORY AND THE THEORY OF TWO-PERSON GAMES 1. Introduction 2. Standard games 3. Minimax play 4. Parallelism and contrast with the minimax theories 12. THE MATHEMATICS OF MINIMAX PROBLEMS 1. Introduction 2. Abstract games 3. Bilinear games 4. An example of a bilinear game 5. Bilinear games exhibiting symmetry 13. OJBECTIONS TO THE MINIMAX RULES 1. Introduction 2. A confusion between loss and negative income 3. Utility and the minimax rule 4. Almost sub-minimax acts 5. The minimax rule does not generate a simple ordering 14. THE MINIMAX THEORY APPLIED TO OBSERVATIONS 1. Introduction 2. Recapitulation of partition problems 3. Sufficient statistics 4. "Simple dichotomy, an example" 5. The approach to certainty 6. Cost of observation 7. Sequential probability ratio procedures 8. Randomization 9. Mixed acts in statistics 15. POINT ESTIMATION 1. Introduction 2. The verbalistic concept of point estimation 3. Examples of problems of point estimation 4. Criteria that have been proposed for point estimates 5. A behavioralistic review of the criteria for point estimation 6. "A behavioralistic review, continued" 7. "A behavioralistic review, concluded" 16. TESTING 1. Introduction 2. A theory of testing 3. Testing in practice 17. INTERVAL ESTIMATION AND RELATED TOPICS 1. Estimates of the accuracy of estimates 2. Interval estimation and confidence intervals 3. Tolerance intervals 4. Fiducial probability APPENDIX 1. EXPECTED VALUE APPENDIX 2. CONVEX FUNCTIONS APPENDIX 3. BIBLIOGRAPHIC MATERIAL APPENDIX 4. BIBLIOGRAPHIC SUPPLEMENT TECHNICAL SYMBOLS AUTHOR INDEX GENERAL INDEX