Probability Theory: The Logic of Science.pdf

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Title Page

Contents

Editor’s foreword

Preface

Part 1 Principles and elementary applications

1 Plausible reasoning

2 The quantitative rules

3 Elementary sampling theory

4 Elementary hypothesis testing

5 Queer uses for probability theory

6 Elementary parameter estimation

7 The central, Gaussian or normal distribution

8 Sufficiency, ancillarity, and all that

9 Repetitive experiments: probability and frequency

10 Physics of ‘random experiments’

Part 2 Advanced applications

11 Discrete prior probabilities: the entropy principle

12 Ignorance priors and transformation groups

13 Decision theory, historical background

14 Simple applications of decision theory

15 Paradoxes of probability theory

16 Orthodox methods: historical background

17 Principles and pathology of orthodox statistics

18 The Apdistribution and rule of succession

19 Physical measurements

20 Model comparison

21 Outliers and robustness

22 Introduction to communication theory

Appendix A Other approaches to probability theory

Appendix B Mathematical formalities and style

Appendix C Convolutions and cumulants

References

Bibliography

Author index

Subject index

PROBABILITY THEORY THE LOGIC OF SCIENCE

PROBABILITY THEORY THE LOGIC OF SCIENCE E . T . Jaynes edited by G. Larry Bretthorst

   Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521592710 © E. T. Jaynes 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 ISBN-13 978-0-511-06589-7 eBook (NetLibrary) ISBN-10 0-511-06589-2 eBook (NetLibrary) ISBN-13 978-0-521-59271-0 hardback ISBN-10 0-521-59271-2 hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Dedicated to the memory of Sir Harold Jeffreys, who saw the truth and preserved it.

Contents Part I Editor’s foreword Preface Principles and elementary applications 1 Plausible reasoning 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Deductive and plausible reasoning Analogies with physical theories The thinking computer Introducing the robot Boolean algebra Adequate sets of operations The basic desiderata Comments 1.8.1 1.8.2 Common language vs. formal logic Nitpicking 2 The quantitative rules 2.1 2.2 2.3 2.4 2.5 2.6 The product rule The sum rule Qualitative properties Numerical values Notation and ﬁnite-sets policy Comments 2.6.1 2.6.2 2.6.3 2.6.4 ‘Subjective’ vs. ‘objective’ G¨odel’s theorem Venn diagrams The ‘Kolmogorov axioms’ 3 Elementary sampling theory 3.1 3.2 3.3 3.4 3.5 Sampling without replacement Logic vs. propensity Reasoning from less precise information Expectations Other forms and extensions vii page xvii xix 3 3 6 7 8 9 12 17 19 21 23 24 24 30 35 37 43 44 44 45 47 49 51 52 60 64 66 68

viii Contents 3.6 3.7 3.8 Probability as a mathematical tool The binomial distribution Sampling with replacement 3.8.1 Correction for correlations Digression: a sermon on reality vs. models 3.9 3.10 Simpliﬁcation 3.11 Comments 3.11.1 A look ahead 4 Elementary hypothesis testing Prior probabilities Testing binary hypotheses with binary data Nonextensibility beyond the binary case 4.1 4.2 4.3 4.4 Multiple hypothesis testing 4.5 4.6 4.7 4.8 Digression on another derivation 4.4.1 Continuous probability distribution functions Testing an inﬁnite number of hypotheses 4.6.1 Simple and compound (or composite) hypotheses Comments 4.8.1 4.8.2 What have we accomplished? Historical digression Etymology 5 Queer uses for probability theory Extrasensory perception 5.1 5.2 Mrs Stewart’s telepathic powers 5.3 5.4 5.5 Digression on the normal approximation Back to Mrs Stewart 5.2.1 5.2.2 Converging and diverging views Visual perception – evolution into Bayesianity? The discovery of Neptune 5.5.1 5.5.2 Horse racing and weather forecasting Discussion 5.6.1 Paradoxes of intuition Bayesian jurisprudence Comments 5.9.1 What is queer? 6 Elementary parameter estimation Digression on alternative hypotheses Back to Newton 5.6 5.7 5.8 5.9 6.1 6.2 6.3 6.4 Inversion of the urn distributions Both N and R unknown Uniform prior Predictive distributions 68 69 72 73 75 81 82 84 86 87 90 97 98 101 107 109 112 115 116 116 117 119 119 120 122 122 126 132 133 135 137 140 142 143 144 146 148 149 149 150 152 154

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