The Foundations
of Statistics
LEONARD J. SAVAGE
Late Eugene Higgins Professor of Statistics
Yale University
SECOND REVISED EDITION
DOVER PUBLICATIONS, INC.
NEW YORK
Contents
Postulates of a personalistic theory of decision
End papers
1. INTRODUCTION
1. The role of foundations
2. Historical background
3. General outline of this book
1
1
4
2. PRELIMINARY CONSIDERATIONS ON DECISION IN THE FACE OF UNCERTAINTY
1.
2.
3.
4.
5.
6.
7.
Introduction
The person
The world, and states of the world
Events
Consequences, acts, and decisions
The simple ordering of acts with respect to preference
The sure-thing principle
6
7
8
10
13
17
21
3. PERSONAL PROBABILITY
1.
2.
3.
4.
5.
6.
7.
Introduction
Qualitative personal probability
Quantitative personal probability
Some mathematical details
Conditional probability, qualitative and quantitative
The approach to certainty through experience
Symmetric sequences of events
27
30
33
40
43
46
50
'. .
4. CRITICAL COMMENTS ON PERSONAL PROBABILITY
1.
2.
3.
4.
5.
6.
Introduction
Some shortcomings of the personalistic view
Connection with other views
Criticism of other views
The role of symmetry in probability
How can science use a personalistic view of probability?
•
56
57
60
60
63
67
5. UTILITY
1.
2.
3.
4.
5.
6.
Introduction
Gambles
Utility, and preference among gambles
The extension of utility to more general acts
Small worlds
Historical and critical comments on utility
xiii
69
70
73
76
82
91
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
105
106
111
.114
7. PARTITION PROBLEMS
120
Introduction
121
Structure of (twofold) partition problems
The value of observation
125
Extension of observations, and sufficient statistics
Likelihood ratios
134
Repeated observations
140
Sequential probability ratio procedures
142
Standard form, and absolute comparison between observations . . 148
1.
2.
3.
4.
5.
6.
7.
8.
8. STATISTICS PROPER
1.
2.
3.
4.
154
154
154
154
Introduction
What is statistics proper?
Multipersonal problems
The minimax theory
156
9. INTRODUCTION TO THE MINIMAX THEORY
1. Introduction
2. The behavioralistic outlook
3. Mixed acts
A
T
4.
5.
6.
7.
8.
Income and loss
The minimax rule, and the principle of admissibility
Illustrations of the minimax rule
Objectivistic motivation of the minimax rule
Loss as opposed to negative income in the minimax rule
158
159
162
J 1
162
163
' 164
165
. . • • •
10. A PERSONALISTIC REINTERPRETATION OF THE MINIMAX THEORY
1.
2.
3.
4.
Introduction
* 172
A model of group decision
• ' 172
The group minimax rule, and the group principle of admissibility •
Critique of the group minimax rule
*
11. THE PARALLELISM BETWEEN THE MINIMAX THEORY AND THE THEORY of
TWO-PERSON GAMES
1.
2.
3.
4.
Introduction
Standard games
Minimax play
Parallelism and contrast with the minimax theories
• •
•
•
*
12. THE MATHEMATICS OP MINIMAX PROBLEMS
1. Introduction
2. Abstract games
l84
l84
CONTENTS
xv
3. Bilinear games
4. An example of a bilinear game
5. Bilinear games exhibiting symmetry
186
189
193
13. OBJECTIONS TO THE MINIMAX RULES
1.
2.
3.
4.
5.
Introduction
A confusion between loss and negative income
Utility and the minimax rule
Almost sub-minimax acts
The minimax rule does not generate a simple ordering
200
200
201
203
205
14. T H E MINIMAX THEORY APPLIED TO OBSERVATIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction
Recapitulation of partition problems
Sufficient statistics
Simple dichotomy, an example
The approach to certainty
Cost of observation
Sequential probability ratio procedures
Randomization
Mixed acts in statistics
208
208
212
212
214
214
215
216
217
15. POINT ESTIMATION
1.
2.
3.
4.
5.
6.
7.
16.
Introduction
The verbalistic concept of point estimation
Examples of problems of point estimation
Criteria that have been proposed for point estimates
A behavioralistic review of the criteria for point estimation
A behavioralistic review, continued
A behavioralistic review, concluded
. . . .
220
221
221
223
229
234
244
TESTING
1. Introduction
2. A theory of testing
3. Testing in practice
246
247
252
17. INTERVAL ESTIMATION AND RELATED TOPICS
1.
2.
3.
4.
Estimates of the accuracy of estimates
Interval estimation and confidence intervals
Tolerance intervals
Fiducial probability
APPENDIX 1.
257
259
262
262
EXPECTED VALUE
263
APPENDIX 2.
CONVEX FUNCTIONS
266
APPENDIX 3.
BIBLIOGRAPHIC MATERIAL
270
APPENDIX 4.
BIBLIOGRAPHIC SUPPLEMENT
283
TECHNICAL SYMBOLS
299
AUTHOR INDEX
301
GENERAL INDEX
305