Subjective Probability : The Real Thing

148 Pages·2004·5.927 MB·English

by  Richard C. Jeffrey

Subjective Probability
This book offers a concise survey of basic probability theory from a thor¬
oughly subjective point of view whereby probability is a mode of judg¬
ment. Written by one of the greatest figures in the field of probability
theory the book is both a summation and a synthesis of a lifetime of
wrestling with these problems and issues.
After an introduction to basic probability theory, there are chapters
on scientific hypothesis-testing, on changing your mind in response to
generally uncertain observations, on expectations of the values of random
variables, on de Finetti’s dissolution of the so-called problem of induction,
and on decision theory.
Richard Jeffrey was Emeritus Professor in the Department of Philosophy,
Princeton University.
Digitized by the Internet Archive
in 2019 with funding from
Kahle/Austin Foundation
https://archive.org/details/subjectiveprobabOOOOjeff
Subjective Probability
The Real Thing
RICHARD JEFFREY
Princeton University
1 homas J. Bate Library
TRENT UNIVERSITY
PETERBOROUGH, ONTARIO
CAMBRIDGE
UNIVERSITY PRESS
fcc.i4d 'tj-iK aoo4
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 Alarcon 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http: // www.cambridge.org
© Richard Jeffrey 2004
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 2004
Printed in the United States of America
Typeface Computer Modern 10/12 pt. System MfcjK 2e [TB]
A catalog record for this book is available from the British Library.
Library of Congress Cataloging in Publication Data
Jeffrey, Richard C.
Subjective probability : the real thing / Richard Jeffrey.
p. cm.
Includes bibliographical references and index.
ISBN 0-521-82971-2 - ISBN 0-521-53668-5 (pbk.)
1. Probability. 2. Judgment. I. Title.
BC141.J445 2004
12T.63-dc22 2003065689
ISBN 0 521 82971 2 hardback
ISBN 0 521 53668 5 paperback
For Edith

Contents
Preface page xi
1 Probability Primer 1
1.1 Bets and Probabilities 1
1.2 Why Probabilities Are Additive 4
1.2.1 Dutch Book Argument for Finite Additivity 5
1.2.2 Dutch Book Argument for Countable Additivity 8
1.3 Probability Logic 9
1.4 Conditional Probability 12
1.5 Why “|” Cannot Be a Connective 15
1.6 Bayes’s Theorem 16
1.7 Independence 17
1.8 Objective Chance 19
1.9 Supplements 22
2 Testing Scientific Theories 29
2.1 Quantifications of Confirmation 30
2.2 Observation and Sufficiency 32
2.3 Leverrier on Neptune 34
2.4 Dorling on the Duhem Problem 35
2.4.1 Einstein/Newton, 1919 37
2.4.2 Bell’s Inequalities: Holt/Clauser 38
2.4.3 Laplace/Adams 41
2.4.4 Dorling’s Conclusions 42
2.5 Old News Explained 44
2.6 Supplements 48
3 Probability Dynamics; Collaboration 51
3.1 Conditioning 61
vii
viii Contents
3.2 Generalized Conditioning 53
3.3 Probabilistic Observation Reports 55
3.4 Updating Twice: Commutativity 57
3.4.1 Updating on Alien Probabilities for Diagnoses 58
3.4.2 Updating on Alien Factors for Diagnoses 58
3.5 Softcore Empiricism 59
4 Expectation Primer 62
4.1 Probability and Expectation 62
4.2 Conditional Expectation 63
4.3 Laws of Expectation 65
4.4 Median and Mean 67
4.5 Variance 70
4.6 A Law of Large Numbers 72
4.7 Supplements 73
4.7.1 Markov’s Inequality: pr(X > e) < if e > o
and X > 0 6 73
4.7.2 Chebyshev’s Inequality: pr(\X — ex(X)\ > e) < iar^ )
if e > 0 6 73
4.7.3 Many Pairwise Uncorrelated R.V.’s 73
4.7.4 Corollary: Bernoulli Trials 74
4.7.5 Noncorrelation and Independence 75
5 Updating on Statistics 76
5.1 Where Do Probabilities Come from? 76
5.1.1 Probabilities from Statistics: Minimalism 76
5.1.2 Probabilities from Statistics: Exchangeability 78
5.1.3 Exchangeability: Urn Examples 79
5.1.4 Supplements 82
5.2 Exchangeability Itself 83
5.3 Two Species of Partial Exchangeability 85
5.3.1 2x2 Tables 86
5.3.2 Markov Dependency 86
5.4 Finite Forms of de Finetti’s Theorem on Partial
Exchangeability 88
5.5 Technical Interpolation: Infinite Forms 90
5.6 Concluding Remarks 93
6 Choosing gg
6.1 Preference Logic g8
6.1.1 Denial Reverses Preferences: Ay B\—B >—A 99
6.1.2 If A A B FT and Ay B then Ay A V B y B 100