OPTIMUM INDUCTIVE METHODS - Springer978-94-015-8131-8/1.pdf · ROBERTO FESTA Fellow of the Department of Philosophy of Science, University of Groningen, The Netherlands OPTIMUM INDUCTIVE

OPTIMUM INDUCTIVE METHODS

SYNTHESE LIBRARY

STUDIES IN EPISTEMOLOGY,

LOGIC, METHODOLOGY, AND PHll...OSOPHY OF SCIENCE

Managing Editor:

JAAKKO HINTIKKA, Boston University

Editors:

DIRK VAN DALEN, University of Utrecht, The Netherlands DONALD DAVIDSON, University of California, Berkeley

THEO A.F. KUIPERS, University ofGroningen, The Netherlands PATRICK SUPPES, Stanford University, California

JAN WOLENSKI, Jagiellonian University, Krakow. Poland

VOLUME232

ROBERTO FESTA Fellow of the Department of Philosophy of Science,

University of Groningen, The Netherlands

OPTIMUM INDUCTIVE METHODS

A Study in Inductive Probability, Bayesian Statistics, and Verisimilitude

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data

Festa, Roberto. Optimum inductive methods a study in inductive probability,

Bayesian statistics, and verisimi litude / by Roberto Festa. p. cm. -- (Synthese 1 ibrary ; v. 232)

Includes bibl iographical references and indexes. ISBN 978-90-481-4318-4 ISBN 978-94-015-8131-8 (eBook)

1. Bayesian statistical decision theory. 2. Probabilities. 3. Induction (Mathematics) 4. Truth. r. Title. II. Series. QA279.5.F47 1993 519.5'42--dc20 93-11840

ISBN 978-90-481-4318-4

Printed on acid-free paper

AH Rights Reserved © 1993 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1993 Softcover reprint ofthe hardcover Ist edition 1993

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical,

including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

DOI 10.1007/978-94-015-8131-8

To my parents

TABLE OF CONTENTS

ACKNOWLEDGMENTS x1

1. INTRODUCfiON 1

1. An outline of issues and objectives 1 2. The probabilistic and the verisimilitude views: two fallibilistic

methodological traditions 3 3. Bayesian statistics and the theory of inductive probabilities 4 4. Optimum prior probabilities: the contextual approach 6 5. The layout of the book 7

PART I. INDUCTIVE PROBABILITIES, BAYESIAN STATISTICS, AND VERISIMILITUDE

2. THE THEORY OF INDUCfiVE PROBABILITIES: BASIC FEATURES AND APPLICATIONS 13

1. Inductive methods 13 2. Multicategorical inferences 16 3. Multinomial contexts 17

3. BAYESIAN STATISTICS AND MULTINOMIAL INFERENCES: BASIC FEATURES 20

1. What is Bayesian statistics? 20 2. Probability distributions 23 3. Bayesian statistical inferences 29 4. Probability distributions for multinomial contexts 30 5. Bayesian multinomial inferences 35

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viii TABLE OF CON1ENTS

4. BAYESIAN POINT ESTIMATION, VERISIMILITUDE, AND IMMODESTY 38

1. Bayesian point estimation and verisimilitude 38 2. The principle of immodesty 40 3. The basic issues of the verisimilitude theory 45

PART II. DE FINETTI'S THEOREM, GC-SYSTEMS, AND DIRICHLET DISTRIBUTIONS

5. EXCHANGEABLE INDUCTIVE METHODS, BAYESIAN STATISTICS, AND CONVERGENCE TOWARDS THE TRUTH 51

1. De Finetti's representation theorem 51 2. Convergence of opinion and convergence towards the truth 53

6. GC-SYSTEMS AND DIRICHLET DISTRIBUTIONS 57

1. GC-systems 57 2. Dirichlet distributions 60 3. The equivalence between GC-systems and Dirichlet distributions 65 4. Extreme GC-systems and extreme Dirichlet distributions 66 5. The axiomatization of GC-systems 68 6. The axiomatization of Dirichlet distributions 70

PART III. VERISIMILITUDE, DISORDER, AND OPTIMUM PRIOR PROBABILITIES

7. THE CHOICE OF PRIOR PROBABILmES: THE SUBJECTIVE, APRIORISTIC, AND CONTEXTUAL APPROACHES 75

1. The choice of priors in Bayesian statistics: the subjective approach 76

TABLE OF CONTENTS ix

2. The choice of priors in Bayesian statistics: the aprioristic approach 79

3. The subjective interpretation of the theory of inductive probabilities 88

4. The aprioristic interpretation of the theory of inductive probabilities 90

5. The contextual view of prior probabilities 95 6. A contextual justification of Dirichlet distributions

and GC-systems 100

8. THE EPISTEMIC PROBLEM OF OPTIMALITY (EPO): A CONTEXTUAL APPROACH 103

1. The optimum prior vector 104 2. GO-contexts 105 3. The CC-solution to EPO 108 4. The logical problem of optimality 109 5. The V-solution to EPO 116 6. The equivalence between the V -solution and the CC-solution

to EPO 118 7. Camap's optimum inductive methods 121

9. THE CONTEXTUAL APPROACH TO EPO: COMPARISONS WITH OTHER VIEWS 123

1. The universalistic view 123 2. The hyperempiricist view 127 3. The presupposition view 132 4. The verisimilitude view 136

10. DISORDERED UNIVERSES: DIVERSITY MEASURES IN STATISTICS AND THE EMPIRICAL SCIENCES 139

1. Gini diversity 140 2. Explicating diversity 141 3. Diversity measures in the empirical sciences 147

11. CONCLUDING REMARKS 150

X TABLE OF CON1ENTS

NOTES REFERENCES INDEX OF NAMES INDEX OF SUBJECTS List of requirements and acronyms

154 177 185 188 192

ACKNOWLEDGMENTS

My views on scientific inference have been decisively influenced by the works of Karl Popper and Rudolf Camap. In particular, my research in this area has been inspired by Popper's view- that verisimilitude, or truthlikeness, is the main cognitive goal of science - and by Carnap's theory of inductive probabilities (and, more generally, by his Bayesian approach to scientific methodology).

In spite of the well known disputes between these two great masters of contemporary epistemology, I believe that a 'reconciliation' between Sir Karl Popper and Rudolf Carnap is possible. This book, indeed, is inspired by the conviction that Popper's verisimilitude view can be fruitfully embedded within the Bayesian approach.

Unfortunately there are very few epistemologists interested in inductive probabilities and verisimilitude. In particular, the set of students who have given important contributions to both fields seems to include only two members: Professor I. Niiniluoto and Professor T.A.F. Kuipers. Thanks to scholarships offered by the Italian Ministry of Foreign Affairs, I had the opportunity to work under the supervision of both.

In 1981-82 I was at the Department of Philosophy at the University of Helsinki where I worked on verisimilitude under Professor Niiniluoto's supervision (see Festa, 1982, 1983). I have been deeply influenced by Professor Niiniluoto's views on scientific inference and, more generally, on the methods and tools of philosophy of science. Among other things, I am grateful to Professor Niiniluoto for giving a sharp exposition of my unpublished proposals on verisimilitude in monadic first-order languages (Festa, 1982) in his book Truthlikeness (1987, pp. 319-321).

Afterwards, in 1983-84, I was at the Department of Philosophy at the University of Groningen where I continued my research into verisimilitude under Professor Kuipers's supervision (see Festa, 1986, 1987a).

In 1986-87 I started working, under Professor Kuipers's supervision, on the subject of this book (see Festa, 1987b and Festa and Buttasi, 1987). My personal debt to Professor Kuipers for the research leading to the completion of this book is enormous. His continuous support and encouragement during many years have been of the utmost importance to me. Indeed he was much

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xii ACKNOWLEDGMENTS

more than the supervisor of this research: without him this book would never have been written.

At the beginning of 1990 Professor Kuipers suggested inviting Professor W. Molenaar, from the Department of Statistics and Measurement Theory of the Faculty of Behavioral and Social Sciences (University of Groningen), to join him in the supervision of my work. In spite of the clumsy statistical terminology used in the first draft of this book, Professor Molenaar very kindly accepted the invitation. His help in improving the form and the content of the earlier drafts has been invaluable. If a Bayesian statistician can appreciate.the proposals advanced here without rejecting them immediately the credit is largely due to him.

Although I have profited enormously from the time and care that both my supervisors devoted to reading and criticizing the earlier drafts, this does not mean that they will be satisfied with everything that is in the final version. Of course any mistakes in it are mine, not theirs.

I also wish to thank Professor W. K B. Hofstee, Professor I. Niiniluoto and Professor W. Schaafsma for reading the manuscript and suggesting several improvements.

I am most indebted to Professor A. Pasquinelli, Professor G. Sandri and Professor M. C. Galavotti of the Department of Philosophy at the University of Bologna. From all of them I received stimulus and encouragement while studying for my degree in philosophy and later.

My interest in Bayesian statistics and its relationships with the theory of inductive probabilities was strongly stimulated by Professor D. Costantini through his works, numerous talks and his comments on my first paper on the subject of this book (Festa, 1984).

I also have an intellectual debt to several friends and colleagues who have -through publications, correspondence, and personal contacts - influenced my work. In particular, I would like to mention Dario Antiseri, Alberto Artosi, Roger Cooke, Paolo Garbolino, Pierdaniele Giaretta, Giulio Giorello, Risto Hilpinen, Isaac Levi, Marco Mondadori, David Pearce, Marcello Pera, Gianguido Piazza, Claudio Pizzi, Yao H. Tan, Raimo Tuomela, Bas van Fraassen, Henk Zandvoort.

I am grateful to Professor L. Nauta of the Department of Philosophy at the University of Groningen, for his help and kindness. I also owe my thanks to Ms Mariette Elzenga and Mr Nico de Jager for their assistance while I was working at the library of the Department of Philosophy at the University of Groningen.

ACKNOWLEDGMENTS xiii

For practical help in preparing the manuscript, I wish to thank especially Ms Cheryl Gwyther (for improving the English style) and Mr Massimo Bonra (for printing the camera-ready copy and drawing the figures).

The time I spent in Holland would not have been so enjoyable without the friendship of Hans, Simone, Annet, Claar, and Jos Mooij, Inge de Wilde, Barend van Heusden, Harrie Jonkman, Ineke Siersema and Jos Griffioen. I am particularly grateful to Claar Mooij and Barend van Heusden for the enormous practical help they gave me while I was working on the completion of this book, not only during my presence in Holland but also, and especially, during my absence.