The Likelihood Principle Author(s): James O. Berger, Robert L. Wolpert, M. J. Bayarri, M. H. DeGroot, Bruce M. Hill, David A. Lane and Lucien LeCam Source: Lecture Notes-Monograph Series, Vol. 6, The Likelihood Principle (1988), pp. iii-v+vii- xii+1-199 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/4355509 . Accessed: 28/05/2013 20:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to Lecture Notes-Monograph Series. http://www.jstor.org This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions Institute of Mathematical Statistics LECTURE NOTES-MONOGRAPH SERIES Shanti S. Gupta, Series Editor Volume 6 The Likelihood Principle (Second Edition) James O. Berger Purdue University Robert L. Wolpert Duke University Institute of Mathematical Statistics Hayward, California This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions Institute of Mathematical Statistics Lecture Notes-Monograph Series Series Editor, Shanti S. Gupta, Purdue University The production of the IMS Lecture Notes-Monograph Series is managed by the IMS Business Office: Nicholas P. Jewell, IMS Treasurer, and Jose L. Gonzalez, IMS Business Manager. Library of Congress Catalog Card Number: 88-81456 International Standard Book Number 0-940600-13-7 Copyright ? 1988 Institute of Mathematical Statistics All rights reserved Printed in the United States of America This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions To my parents, Orvis and Thelma James Berger To rr\y wife, Ruta Robert Wolpert This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions PREFACE This monograph began with research designed to provide a generali- zation of the Likelihood Principle (LP) to quite arbitrary statistical situations. The purpose of seeking such a generalization was to partially answer certain criticisms that had been levied against the LP, criticisms which seemed to prevent many statisticians from seriously considering the LP and its implications. The research effort seemed worthwhile because of the simplicity, central importance, and far reaching implications of the LP. Background reading for the research revealed a wider than expected range of published criticisms of the LP. In an attempt to be complete and address all such criticisms, the research paper expanded considerably. Eventually it seemed sensible to enlarge the paper to a monograph. This also allowed for discussion of conditioning ideas in general and for a review of the implications of the LP. It was decided, however, to stop short of a general review of conditional methods in statistics. In particular, the monograph does not discuss the many likelihood based statistical methodologies that have been developed, although references to these methodologies will be given. This limitation was, in part, because such an endeavor would be far too ambitious, and, in part, because we feel (and indeed argue in Chapter 5) that Bayesian implementation of the LP is the correct conditional methodology. The mathematical level of the monograph is, for the most part, kept at a nontechnical level. The main exception is the generalization of the LP in Section 3.4, which is (necessarily) presented at a measure-theoretic level, but can be skipped with no loss in continuity. Also, the monograph vn This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions viii PREFACE presupposes no familiarity with conditioning concepts. Indeed Chapter 2 provides an elementary review of conditioning, with many examples. This second edition was produced under the rather severe constraint that the original manuscript, used for photo-offset printing, was inadvertantly destroyed; only the photos were kept. Thus changes could only be made by retyping entire pages or inserting new pages. A list of corrections that were too minor to justify the retyping of an entire page is given at the end of the monograph. Inserted pages received decimal page numbers: e.g. 74.1, 74.2. A list of additional references was added, and new discussions were kindly contri- buted by M. J. Bayarri and M. H. DeGroot, Bruce Hill, and Lucien Le Cam. Substantial changes or additions were made in Sections 3.1, 3.5, 4.2.1, 4.4, and 4.5. The changes in Section 4.4 correct a glaring oversight in the first edition: the failure to emphasize the misleading conclusions that can result from violation of the Likelihood Principle in significance testing of a precise hypothesis. Another very weak part of the first edition was Section 3.5, which discussed prediction, design, and nuisance parameters. The new material incorporates recent substantive insights from the literature. Numerous other minor changes and literature updatings were made throughout the monograph. We did not attempt complete coverage of recent lit- erature, however. We are grateful to a number of people for valuable discussions on this subject and/or for comments and suggestions on original drafts or the first edition of the monograph. In particular, we would like to thank George Barnard, M. J. Bayarri, Mark Berliner, Lawrence Brown, George Casella, Morris DeGroot, J. L. Foulley, Leon Gleser, Prem Goel, Clyde Hardin, Bruce Hill, Jiunn Hwang, Rajeev Karandikar, Lucien Le Cam, Ker-Chau Li, Dennis Lindley, George McCabe, Georges Monette, John Pratt, Don Rubin, Herman Rubin, Myra Samuels, Steve Samuels, and Tom Sellke. We are especially grateful to M. J. Bayarri and M. H. DeGroot for an exceptionally complete and insightful set of corrections and comments on the first edition. We are also grateful to Shanti Gupta for the encouragement to turn the material into a monograph. This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions PREFACE ix Thanks are also due to the Alfred P. Sloan Foundation, the National Science Foundation (Grants MCS-7801737, MCS-8101670A1, and DMS-8702620), and the Center for Stochatic Processes at the University of North Carolina for support of the research in the monograph. Finally, we are extremely grateful to Norma Lucas, Teena Chase, and Betty Gick for excellent typing of the manuscript. March, 1988 JAMES 0. BERGER Purdue University, West Lafayette ROBERTWOLPERT Duke University, Durham This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions TABLE OF CONTENTS CHAPTER1. INTRODUCTION . 1 CHAPTER2. CONDITIONING . 5 2.1. Simple Examples . 5 2.2. Relevant Subsets . 11 2.3. Ancillarity. 13 2.4. Conditional Frequentisi Procedures . 14 2.4.1. Conditional Confidence . 14 2.4.2. Estimated Confidence. 16 2.5. Criticisms of Partial Conditioning . 16 CHAPTER3. THE LIKELIHOOD PRINCIPLE AND GENERALIZATIONS . 19 3.1. Introduction. 19 3.2. History of the Likelihood Principle . 22 3.3. Birnbaum's Development - the Discrete Case. 24 3.3.1. Evidence y Conditionally > and Sufficiency. 24 3.3.2. Axiomatic Development. 26 3.4. Generalizations Beyond the Discrete Case . 28 3.4.1. Difficulties in the Nondiscrete Case. 29 3.4.2. Evidence? Conditionally3 and Sufficiency . 31 3.4.3. The Relative Likelihood Principle . 32 3.5. Prediction, Design, Nuisance Parameters, and the LP . 36 3.5.1. Introduction . 36 3.5.2. Unobserved Variables: Prediction and Design . 37 3.5.3. Nuisance Variables and Parameters . 41 This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions TABLE OF CONTENTS xi 3.6. Criticisms of Birnbaum's Axiomatic Development . 42 3.6.1. The Model Assumption. 43 3.6.2. The Evidence Assumption . 45 3.6.3. The Weak Conditionality Principle . 45 3.6.4. The Sufficiency Principle. 46 3.7. Violation of the Likelihood Principle: Inadmissibility and Incoherency. 50 3.7.1. Introduction . 50 3.7.2. Decision Theoretic Evaluation . 52 3.7.3. Betting Evaluation . 59 CHAPTER4. CONSEQUENCESAND CRITICISMS OF THE LIKELIHOOD PRINCIPLE AND THE RELATIVE LIKELIHOOD PRINCIPLE . 65 4.1. Incompatibility with Frequentist Concepts . 65 4.1.1. Introduction . 65 4.1.2. Objectivity . 67 4.1.3. Procedures for Nonspecialists .............. 68 4.1.4. Repeatability . 70 4.1.5. The Confidence Principle. 71 4.2. The Irrelevance of Stopping Rules. 74 4.2.1. Introduction . 74 4.2.2. The (Discrete) Stopping Rule Principle. 75 4.3.2. Positive Implications . 77 4.2.4. Criticisms. 80 4.2.5. Stopping Rules and Inadmissibility . 83 4.2.6. The General Stopping Rule Principle. 86 4.2.7. Informative Stopping Rules. 88 4.3. The Irrelevance of Censoring Mechanisms 4.3.1. Introduction . 90 4.3.2. Fixed Censoring and Equivalent Censoring Mechanisms ... 92 4.3.3. Random Censoring. 95 4.3.4. Informative Censoring . 103 This content downloaded from 128.173.127.127 on Tue, 28 May 2013 20:57:09 PM All use subject to JSTOR Terms and Conditions xi i TABLE OF CONTENTS 4.4.