Statistical Reasoning with Imprecise Probabilities

719 Pages·1991·17.409 MB·English

by  Peter Walley (auth.)

MONOGRAPHS ON
STATISTICS AND APPLIED PROBABILITY
General Editors
D.R. Cox, D.V. Hinkley, D. Rubin and B.W. Silverman
Stochastic Population Models in Ecology and Epidemiology
M.S. Bart/ett (1960)
2 Queues D.R. Cox and w.L. Smith (1961)
3 Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964)
4 The Statistical Analysis of Series of Events D.R. Cox and
P.A. W. Lewis (1966)
5 Population Genetics w.J. Ewens (1969)
6 Probability, Statistics and Time M.S. Bartlett (1975)
7 Statistical Inference S.D. Silvey (1975)
8 The Analysis of Contingency Tables B.S. Everitt (1977)
9 Multivariate Analysis in Behavioural Research A.E. Maxwell (1977)
10 Stochastic Abundance Models S. Engen (1978)
11 Some Basic Theory for Statistical Inference E.J.G. Pitman (1979)
12 Point Processes D.R. Cox and V. Isham (1980)
13 Identification of Outliers D.M. Hawkins (1980)
14 Optimal Design S.D. Silvey (1980)
15 Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981)
16 Classification A.D. Gordon (1981)
17 Distribution-free Statistical Methods J.S. Maritz (1981)
18 Residuals and Influence in Regression R.D. Cook and S. Weisberg (1982)
19 Applications of Queueing Theory G.F. Newell (1982)
20 Risk Theory, 3rd edition R.E. Beard, T. Pentikainen and
E. Pesonen (1984)
21 Analysis of Survival Data D.R. Cox and D. Oakes (1984)
22 An Introduetion to Latent Variable Models B.S. Everitt (1984)
23 Bandit Problems D.A. Berry and B. Fristedt (1985)
24 Stoehastie Modelling and Control M.H.A. Davis and R Vinter (1985)
25 The Statistieal Analysis of Compositional Data J. Aitchison (1986)
26 Density Estimation for Statistieal and Data Analysis
B.W Silverman (1986)
27 Regression Analysis with Applieations G.B. Wetherill (1986)
28 Sequential Methods in Statisties, 3rd edition G.B. Wetherill (1986)
29 Tensor Methods in Statisties P. McCullagh (1987)
30 Transformation and Weighting in Regression RJ. Carroll and
D. Ruppert (1988)
31 Asymptotie Teehniques for Use in Statisties O.E. Barndoff-Nielsen
and D.R Cox (1989)
32 Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989)
33 Analysis of Infeetious Disease Data N.G. Becker (1989)
34 Design and Analysis of Cross-Over Trials B. Jones and
M.G. Kenward (1989)
35 Empirieal Bayes Methods, 2nd edition J.S. Maritz and T. Lwin (1989)
36 Symmetrie M ultivariate and Related Distributions K.-T. Fang,
S. Kotz and K. Ng (1989)
37 Generalized Linear Models, 2nd edition P. McCullagh and
J.A. Neider (1989)
38 Cyc1ie Designs J.A. John (1987)
39 Analog Estimation Methods in Eeonometrics C.F. Manski (1988)
40 Subset Selection in Regression A.J. Miller (1991)
41 Analysis of Repeated Measures M.J. Crowder and
D.J. Hand (1990)
42 Statistieal Reasoning with Impreeise Probabilities P. Walley (1991)
(Full details concerning this series are available from the publishers)
Statistical Reasoning with
Imprecise Probabilities
PETER W ALLEY
Department of M athematics
The University of Western Australia
N edlands. W A 6009
Australia
Springer-Science+Business Media, B.Y.
First edition 1991
© 1991 Peter Walley
Originally published by Chapman and Hall in 1991.
Softcover reprint of the hardcover 1s t edition 1991
ISBN 978-0-412-28660-5 ISBN 978-1-4899-3472-7 (eBook)
DOI 10.1007/978-1-4899-3472-7
All rights reserved. No part of this pub1ication may be reproduced or
transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording or otherwise, or store in any retrieval system
of any nature, without the written permission of the copyright holder
and the publisher, application for which shall be made to the publisher.
British Library Cataloguing in Publication Data
Walley, Peter
Statistical reasoning with imprecise probabilities.
1. Probabilities & statistical mathematics
I. Title
519.2
Library of Congress Cataloging-in-Publication Data
Walley, Peter.
Statistical reasoning with imprecise probabilities / Peter Walley.
-1st ed
p. cm. -(Monographs on statistics and applied probability)
Includes bibliographical references and index.
1. Mathematical statistics. 2. Probabilities. I. Title.
11. Series.
QA276.W299 1990
519.5-dc20 90-42753
CIP
Contents
Preface ix
1 Reasoning and behaviour 1
1.1 Introduction 1
1.2 Outline of the book 9
1.3 Interpretations of prob ability 13
1.4 Beliefs and behaviour 17
1.5 Inference and decision 21
1.6 Reasoning and rationality 26
1.7 Assessment strategies 32
1.8 Survey of related work 42
2 Coherent previsions 52
2.1 Possibility 54
2.2 Probability currency 58
2.3 U pper and lower previsions 61
2.4 A voiding sure loss 67
2.5 Coherence 72
2.6 Basic properties of coherent previsions 76
2.7 Coherent probabilities 81
2.8 Linear previsions and additive probabilities 86
2.9 Examples of coherent previsions 92
2.10 Interpretations of lower previsions 101
2.11 Objections to behavioural theories of probability 109
3 Extensions, envelopes and decisions 120
3.1 Natural extension 121
3.2 Extension from a field 127
3.3 Lower envelopes of linear previsions 132
3.4 Linear extension 136
3.5 Invariant linear previsions 139
3.6 Compactness and extreme points of A(f) 145
vi CONTENTS
3.7 Desirability and preference 150
3.8 Equivalent models for beliefs 156
3.9 Decision making 160
4 Assessment and elicitation 167
4.1 A general elicitation procedure 168
4.2 Finitely generated models and simplex representations 174
4.3 Steps in the assessment process 177
4.4 Classificatory probability 188
4.5 Comparative probability 191
4.6 Other types of assessment 197
5 The importance of imprecision 207
5.1 Uncertainty, indeterminacy and imprecision 209
5.2 Sources of imprecision 212
5.3 Information from Bernoulli trials 217
5.4 Prior-data conflict 222
5.5 Bayesian noninformative priors 226
5.6 Indecision 235
5.7 Axioms of precision 241
5.8 Practical reasons for precision 248
5.9 Bayesian sensitivity analysis 253
5.10 Second-order probabilities 258
5.11 Fuzzy sets 261
5.12 Maximum entropy 266
5.13 The Dempster-Shafer theory of belief functions 272
6 Conditional previsions 282
6.1 Updated and contingent previsions 284
6.2 Separate coherence 289
6.3 Coherence with unconditional previsions 293
6.4 The generalized Bayes rule 297
6.5 Coherence axioms 301
6.6 Examples of conditional previsions 308
6.7 Extension of conditional and marginal previsions 313
6.8 Conglomerability 317
6.9 Countable additivity 323
6.10 Conditioning on events of probability zero 328
6.11 Updating beliefs 334
7 Coherent statistical models 341
7.1 General concepts of coherence 342
7.2 Sampling models 349
CONTENTS vii
7.3 Coherence of sampling model and posterior previsions 362
7.4 Inferences from improper priors 367
7.5 Confidence intervals and relevant subsets 377
7.6 Proper prior previsions 388
7.7 Standard Bayesian inference 393
7.8 Inferences from imprecise priors 397
7.9 Joint prior previsions 403
8 Statistical reasoning 406
8.1 A general theory of natural extension 408
8.2 Extension to prior previsions 415
8.3 Extension to predictive previsions 421
8.4 Extension to posterior previsions 424
8.5 Posteriors for imprecise sampling models 430
8.6 The likelihood principle 434
9 Structural judgements 442
9.1 Independent events 443
9.2 Independent experiments 448
9.3 Constructing joint previsions from independent marginals 452
9.4 Permutability 457
9.5 Exchangeability 460
9.6 Robust Bernoulli models 467
9.7 Structural judgements 472
Notes 478
Chapter 1 478
Chapter 2 490
Chapter 3 502
Chapter 4 509
Chapter 5 519
Chapter 6 545
Chapter 7 561
Chapter 8 576
Chapter 9 595
Appendices
A. Verifying coherence 596
B. n-Coherence 599
C. Win-and-place betting on horses 602
.se
D. Topological structure of and g; 608
E. Separating hyperplane theorems 611
viii CONTENTS
F. Desirability 614
G. Upper and lower variances 617
H. Operationa l measurement procedures 622
I. The World CUp football experiment 632
J. Regular extension 639
K. W -coherence 642
References 645
Glossary of notation 674
Index ofaxioms 680
Author index 682
Subject index 688
Preface
When I started writing this book, my mind was full of ignorance and
uncertainty, particularly about how best to deal with ignorance and
uncertainty. Much of the ignorance and uncertainty remains, now that the
book is finished, but it is organized more coherently. I see that as progress.
As the title indicates, the book is about methods of reasoning and
statistical inference using imprecise probabilities. The methods are based
on a behavioural interpretation of probability and principles of coherence.
The idea for such a book originated in 1982, after I had written two long
reports on the mathematics and elicitation of upper and lower probabilities.
My experience in teaching and applying the existing theories of statistical
inference had convinced me that each of them was inadequate. The Bayesian
theory is inadequate, despite its great virtues of coherence, because it requires
all probability assessments to be precise yet gives little guidance on how to
make them. It seemed natural to investigate whether the Bayesian theory
could be modified by admitting imprecise probabilities as models for partial
ignorance. Is it possible to reconcile imprecision with coherence, vagueness
with rationality? Fortunately the ans wer is yes!
The basic ideas of the book appeared in the two technical reports (1981,
1982). The coherence principles for conditional probabilities and statistical
models were worked out in New Zealand, in 1983. A draft of the book was
written in 1984-85, under the title 'Rationality and vagueness', but at that
point I decided that the results were interesting enough to be given a more
careful examination. That has taken me four years. The result is that I say
much more than I originally intended about other ways of modelling
ignorance and uncertainty, and about issues such as finite versus countable
additivity, conditioning on continuous variables, the likelihood principle, and
the incoherence of standard statistical methods. To keep the book to a
manageable length, I have omitted a great deal of material from earlier
versions, particularly concerning the aggregation of beliefs, decision making
with imprecise utilities, and stronger properties of coherence.
A few comments are necessary on the aims, style and limitations of the
book. In writing the book I have tried to work out the implications of a