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Interpreting Probability Controversies and Developments in the Early Twentieth Century

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Interpreting Probability Controversies and Developments in the Early Twentieth Century Interpreting Probability Controversies and Developments in the Early Twentieth Century DAVID HOWIE PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org C David Howie 2002 This book is in copyright.Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2002 Printed in the United Kingdom at the University Press, Cambridge Typeface Times Roman 10.25/13 pt. System LATEX2ε [TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging in Publication Data Howie, David, 1970– Interpreting probability : controversies and developments in the early twentieth century / David Howie. p.cm.–(Cambridge studies in probability, induction, and decision theory) Includes bibliographical references and index. ISBN 0-521-81251-8 1.Probabilities. 2.Bayesian statistical decision theory. 3.Jeffreys, Harold, Sir, 1891–1989 4.Fisher, Ronald Aylmer, Sir, 1890–1962. I.Title. II.Series. QA273.A4 H69 2002 2001052430 ISBN 0 521 81251 8 hardback Contents Acknowledgments page xi 1 Introduction 1 1.1 The meaning of probability 1 1.2 The history of probability 2 1.3 Scope of this book 4 1.4 Methods and argument 5 1.5 Synopsis and aims 11 2 Probability up to the Twentieth Century 14 2.1 Introduction 14 2.2 Early applications of the probability calculus 15 2.3 Resistance to the calculation of uncertainty 17 2.4 The doctrine of chances 19 2.5 Inverse probability 23 2.6 Laplacean probability 27 2.7 The eclipse of Laplacean probability 28 2.8 Social statistics 33 2.9 The rise of the frequency interpretation of probability 36 2.10 Opposition to social statistics and probabilistic methods 38 2.11 Probability theory in the sciences: evolution and biometrics 41 2.12 The interpretation of probability around the end of the nineteenth century 47 3 R.A. Fisher and Statistical Probability 52 3.1 R.A. Fisher’s early years 52 3.2 Evolution – the biometricians versus the Mendelians 53 3.3 Fisher’s early work 56 3.4 The clash with Pearson 59 vii 3.5 Fisher’s rejection of inverse probability 61 3.5.1 Fisher’s new version of probability 61 3.5.2 The papers of 1921 and 1922 62 3.5.3 The Pearson–Fisher feud 65 3.6 The move to Rothamsted: experimental design 70 3.7 The position in 1925 – Statistical Methods for Research Workers 72 3.8 The development of fiducial probability 75 3.9 Fisher’s position in 1932 79 4 Harold Jeffreys and Inverse Probability 81 4.1 Jeffreys’s background and early career 81 4.2 The Meteorological Office 83 4.3 Dorothy Wrinch 85 4.4 Broad’s 1918 paper 87 4.5 Wrinch and Jeffreys tackle probability 89 4.6 After the first paper 92 4.6.1 General relativity 92 4.6.2 The Oppau explosion 94 4.6.3 New work on probability – John Maynard Keynes 96 4.6.4 Other factors 101 4.7 Probability theory extended 103 4.7.1 The Simplicity Postulate 103 4.7.2 The papers of 1921 and 1923 107 4.8 The collaboration starts to crumble 109 4.9 Jeffreys becomes established 111 4.10 Probability and learning from experience – Scientific Inference 113 4.10.1 Science and probability 113 4.10.2 Scientific Inference 114 4.11 Jeffreys and prior probabilities 119 4.11.1 The status of prior probabilities 119 4.11.2 J.B.S. Haldane’s paper 121 4.12 Jeffreys’s position in 1932 126 5 The Fisher–Jeffreys Exchange, 1932–1934 128 5.1 Errors of observation and seismology 128 5.2 Fisher responds 133 5.3 Outline of the dispute 137 viii 5.4 The mathematics of the dispute 139 5.5 Probability and science 143 5.5.1 The status of prior probabilities 144 5.5.2 The Principle of Insufficient Reason 148 5.5.3 The definition of probability 150 5.5.4 Logical versus epistemic probabilities 152 5.5.5 Role of science: inference and estimation 154 5.6 Conclusions 162 6 Probability During the 1930s 171 6.1 Introduction 171 6.2 Probability in statistics 172 6.2.1 The position of the discipline to 1930 172 6.2.2 Mathematical statistics 173 6.2.3 The Neyman–Fisher dispute 176 6.2.4 The Royal Statistical Society 180 6.2.5 The reading of Fisher’s 1935 paper 183 6.2.6 Statisticians and inverse probability 187 6.3 Probability in the social sciences 191 6.3.1 Statistics in the social sciences 191 6.3.2 Statistics reformed for the social sciences 194 6.3.3 The social sciences reformed for statistics 197 6.4 Probability in physics 199 6.4.1 General remarks 199 6.4.2 Probability and determinism: statistical physics 199 6.4.3 Probability at the turn of the century 202 6.4.4 The rejection of causality 204 6.4.5 The view in the 1930s 206 6.4.6 The interpretation of probability in physics 209 6.4.7 Quantum mechanics and inverse probability 211 6.5 Probability in biology 213 6.6 Probability in mathematics 216 6.6.1 Richard von Mises’s theory 216 6.6.2 Andrei Kolmogorov’s theory 219 6.7 Conclusions 220 7 Epilogue and Conclusions 222 7.1 Epilogue 222 7.2 Conclusions 226 ix Appendix 1 Sources for Chapter 2 231 Appendix 2 Bayesian Conditioning as a Model of Scientific Inference 235 Appendix 3 Abbreviations Used in the Footnotes 237 Bibliography 239 Index 253 x 1 Introduction “It is curious how often the most acute and powerful intellects have gone astray in the calculation of probabilities.” William Stanley Jevons 1.1. the meaning of probability The single term ‘probability’can be used in several distinct senses. These fall into two main groups. A probability can be a limiting ratio in a sequence of repeatable events. Thus the statement that a coin has a 50% probability of landing heads is usually taken to mean that approximately half of a series of tosses will be heads, the ratio becoming ever more exact as the series is extended. But a probability can also stand for something less tangible: a de- gree of knowledge or belief. In this case, the probability can apply not just to sequences, but also to single events. The weather forecaster who predicts rain 1 tomorrow with a probability of 2 is not referring to a sequence of future days. He is concerned more to make a reliable forecast for tomorrow than to spec- ulate further ahead; besides, the forecast is based on particular atmospheric conditions that will never be repeated. Instead, the forecaster is expressing his confidence of rain, based on all the available information, as a value on a scale on which 0 and 1 represent absolute certainty of no rain and rain respectively. The former is called the frequency interpretation of probability, the lat- ter the epistemic or ‘degree of belief’or Bayesian interpretation, after the Reverend Thomas Bayes, an eighteenth-century writer on probability.1 A fre- quency probability is a property of the world. It applies to chance events. A Bayesian probability, in contrast, is a mental construct that represents uncer- tainty. It applies not directly to events, but to our knowledge of them, and can 1 I shall use the terms “Bayesian” and “Bayesianism” when discussing the epistemic inter- pretation, though they were not common until after World War II. 1 thus be used in determinate situations. A Bayesian can speak of the probability of a tossed coin, for example, even if he believes that with precise knowledge of the physical conditions of the toss – the coin’s initial position and mass distribution, the force and direction of the flick, the state of the surrounding air – he could predict exactly how it would land. The difference between the two interpretations is not merely semantic, but reflects different attitudes to what constitutes relevant knowledge. A fre- quentist would take the probability of throwing a head with a biased coin as =1 some value P, where P 2 . This would express his expectation that more heads than tails, or vice versa, would be found in a long sequence of results. A Bayesian, however, would continue to regard the probability of a head 1 as 2 , since unless he suspected the direction of the bias, he would have no more reason to expect the next throw to be a head than a tail. The frequentist would estimate a value for P from a number of trial tosses; the Bayesian would revise his initial probability assessment with each successive result. As the number of trial tosses is increased, the values of the two probabil- ities will tend to converge. But the interpretations of these values remain distinct. 1.2. the history of probability The mathematical theory of probability can be traced back to the mid- seventeenth century. Early probabilists such as Fermat and Pascal focused on games of chance. Here the frequency interpretation is natural: the roll of a die, or the selection of cards from a deck, is a well-defined and re- peatable event for which exact sampling ratios can easily be calculated.
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