## The Foundations of StatisticsWith 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. |

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### Contents

Postulates of a personalistic theory of decision End papers | 1 |

PsasoNAL PnoBABILITY | 27 |

irs on PnnsoxaL Paosaamrrr | 53 |

UnLrrY | 71 |

10 | 109 |

PARTITIoN PnoaLsus | 116 |

13 | 132 |

17 | 139 |

Mixed acts in statistics | 217 |

Ponrr ESTIMATIoN 1 Introduction | 220 |

27 | 221 |

Criteria that have been proposed for point estimates | 223 |

A behavioralistic review of the criteria for point estimation I | 229 |

A behavioralistic review continued | 234 |

A behavioralistic review concluded | 244 |

Tnsrmo 1 Introduction | 246 |

Srarrsrrcs | 154 |

PaaaLLausu BETWEEN THE Mrsruax TaroRY AND ran TuaoRr | 178 |

Tan IAIHEhlAlICS or MiNruax PaoaLaus | 184 |

Bilinear games | 186 |

An example of a bilinear game | 189 |

Bilinear games exhibiting symmetry | 193 |

Mr m x Rows 1 Introduction | 200 |

Utility and the minimax rule | 201 |

Almost subminimax acts | 203 |

The minimax rule does not generate a simple ordering | 205 |

0 T0 OnsnnvxrroNs 1 Introduction | 208 |

Suﬁcient statistics | 212 |

The approach to certainty | 214 |

Sequential probability ratio procedues | 215 |

Randomization | 216 |

A theory of testing | 247 |

Testing in practice | 252 |

IrrrznvaL EsrlmanoN AND RELATED Torres 1 Estimates of the accuracy of estimates | 257 |

Interval estimation and conﬁdence intervals | 259 |

Tolerance intervals | 262 |

cnn VALUE | 263 |

Convex FUNcrIoNS | 266 |

aLu | 270 |

43 | 271 |

Bmuocaarnrc SoreLzmsrvr | 283 |

TzcnmcaL SYMBoLS | 299 |

Arrrnoa Irmax | 301 |

305 | |

### Common terms and phrases

act f analysis apply basic acts behavior behavioralistic Bernoulli bilinear games called chapter concept condition conditional probability conﬁned consequences considered context convex convex set Criterion decision problem deﬁned deﬁnition derived act discussion dominates elements equal equivalent example Exercise expected value f 5 g f and g ﬁnd Finetti ﬁnite number ﬁnite set ﬁrst ﬁve ﬁxed formal g given Gamble hypothesis idea implies income inﬁnite interpretation interval estimation justiﬁed least likelihood ratio likelihood-ratio test linear logic loss function lottery max L(f maximin minimax rule minimax theory mixed acts n-tuples notion null objectivistic objectivistic decision problem objectivistic views observation particular partition problems personal probability personalistic view possible postulate preference probability measure qualitative probability random variables real numbers satisﬁes seems sequence situation small world sufﬁcient sufficient statistics suﬁicient Suppose sure-thing principle symmetry testing Theorem tion typically utility verbalistic veriﬁed Wald