Probability and Bayesian Statistics Probability and Bayesian Statistics Edited by R. Vietti Technical University of Vienna Vienna, Austria Plenum Press • New York and London Library of Congress Cataloging in Publication Data International Symposium on Probability and Bayesian Statistics (1986: Innsbruck, Austria) Probability and Bayesian statistics. "Based on the proceedings of the International Symposium on Probability and Bayesian Statistics, held September 23-26, 1986, in Innsbruck, Austria" - T.p. verso. "Dedicated to the memory of Bruno de Finetti"-Pref. Bibliography: p. Includes index. 1. Probabilities-Congresses. 2. Bayesian statistical decision theory-Congresses. 3. De Finetti, Bruno-Congresses. 1. Viertl, R. (Reinhard) II. De Finetti, Bruno. III. Title. QA273.A1l58 1986 519.2 87·15298 ISBN-13: 978-1-4612-9050-6 e-ISBN-13: 978-1-4613-1885-9 DOl: 10.1007/ 978-1-4613-1885-9 Based on the proceedings of the International Symposium on Probability and Bayesian Statistics, held September 23-26, 1986, in Innsbruck, Austria © 1987 Plenum Press, New York Softcover reprint ofthe hardcover 1st edition 1987 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher To the memory of Bruno de Finetti (1906-1985) PREFACE This book contains selected and refereed contributions to the "Inter national Symposium on Probability and Bayesian Statistics" which was orga nized to celebrate the 80th birthday of Professor Bruno de Finetti at his birthplace Innsbruck in Austria. Since Professor de Finetti died in 1985 the symposium was dedicated to the memory of Bruno de Finetti and took place at Igls near Innsbruck from 23 to 26 September 1986. Some of the pa pers are published especially by the relationship to Bruno de Finetti's scientific work. The evolution of stochastics shows growing importance of probability as coherent assessment of numerical values as degrees of believe in certain events. This is the basis for Bayesian inference in the sense of modern statistics. The contributions in this volume cover a broad spectrum ranging from foundations of probability across psychological aspects of formulating sub jective probability statements, abstract measure theoretical considerations, contributions to theoretical statistics and stochastic processes, to real applications in economics, reliability and hydrology. Also the question is raised if it is necessary to develop new techniques to model and analyze fuzzy observations in samples. The articles are arranged in alphabetical order according to the family name of the first author of each paper to avoid a hierarchical ordering of importance of the different topics. Readers interested in special topics can use the index at the end of the book as guide. The editor wants to thank the referees for their anonymous work. Some of them are also authors in the present volume and their names are not given here. The following scientists who where not participants at the symposium or made no contribution to this Proceedings volume were so glad to act as referees: J.O. Berger, Lafayette, D. Blackwell, Berkeley, L.D. Broemeling, Arlington, L. Crisma, Trieste, I. Csiszar, Budapest, M. Deistler, Wien, P. Diaconis, Stanford, L.E. Dubins, Berkeley, R. Dutter, Wien, W. Ettl, Wien, T.L. Fine, Ithaca, D. Furst, Roma, P. Hackl, Wien, W. Jammernegg, Graz, A. Kandel, Tallahassee, F. Konecny, Wien, D.V. Lindley, Somerset, M. Luptacik, Wien, G. Marinell, Innsbruck, B. Natvig, Oslo, T. Postelnicu, Bucuresti, H. Rauch, Wien, P. Revesz, Wien, M. Schemper, Wien, K.D. Schmidt, Mannheim, A.F.M. Smith, Nottingham, F. Spizzichino, Roma, H. Stadler, Wien, H. Strasser, Bayreuth, S. Weber, Mainz, G.A. Whitmore, Montreal. It is the intention of this volume to make Bayesian ideas available for a broader audience and to present different recent developments in pro bability and statistics. I want to thank PLENUM for publishing this volume in short time which makes it possible to produce an up to date contribution and especially Ms. M. Carter for her kind advice. R. Viertl CONTENTS Stochastic Linear Programming with Recourse under Partial Information •••••••••••••••••••••••••••••••••••• 1 P. Abel Applied GEWR (n,p,q) Normal Discount Bayesian Model: An Austrian Economic Case Study........................ 7 M. Akram Using Influence Diagrams to Solve a Calibration Problem •••••••••• 17 R.E. Barlow, R.W. Mensing and N.G. Smiriga Reliability of a Complex System from Bayesian Viewpoint 31 A. Basu and G. Tarmast Information in Selection Models •••••••••••••••••••••••••••••••••• 39 M.J. Bayarri amd M.H. DeGroot Approximations in Statistics from a Decision-Theoretical Viewpoint .............................................. 53 J.M. Bernardo Restricted Bayes Estimates for Binomial Parameters ••••••••••••••• 61 J.D. Broffitt Bayesian Prevision of Time Series by Transfer Function Models •••••••••••••••••••••••••••••••••••••••• 73 F. Carlucci and G. Zornitta Priors for Exponential Families which maximize the Association between Past and Future Observations 83 D.M. Cifarelli and E. Regazzini Calibrating and Combining Precipitation Probability Forecasts ••••••••••••••••••.•••.••••••••••••••••••••.•• 97 R.T. Clemen and R.L. Winkler Coherent Distributions and Lindley's Paradox ••••••••••••••••.•••• III G. Consonni and P. Veronese Mean Variance, Expected Utility and Ruin Probability in Reinsurance Decisions: Suggestions and Comments on the Line of de Finetti's Seminal Work ••••••••••••••• 121 L. Daboni and F. Pres sacco ix A Product of Multivariate T Densities as Upper Bound for the Posterior Kernel of Simultaneous Equation Model Parameters ••••••••••••••••••••••••••••••••••••••• 129 H.K. van Dijk A Common Model Selection Criterion 139 N.R. Draper and I. Guttman Predictive Screening Methods in Binary Response Models •••••••.••• 151 I.R. Dunsmore and R.J. Boys De Finetti's Probabilistic Approach and the Theory of Expectations in Economics •••••••••••••••••••••••••••••• 159 M. de Felice and G. Pelloni Some Characteristics of Bayesian Designs 169 K. Felsenstein The Analysis of Multiple Choice Tests in Educational Assessment............................................. 175 S. French Dynamic Inference on Survival Functions •••••••••••••••••••••••••• 183 D. Gamerman The Role of Probability and Statistics in Physics •••••••••••••••• 193 G. Gerlich Can we build a Subjectivist Statistical Package? ••••••••••••••••• 203 M. Goldstein Life Time Distributions and Stochastic Dynamical Systems ••••••••• 219 J. Grasman Calibration of a Radiation Detector: Chromosome Dosimetry for Neutrons ••••••••••••••••••••••••••••••••••••••••••• 225 P.G. Groer and C.A. De B. Pereira On Some Bayes and Empirical Bayes Selection Procedures ••••••••••• 233 S.S. Gupta and T. Liang Bayesian Aspects in the Theory of Comparison of Statistical Experiments ••••••••••••.•••••••••••••.••••• 247 H. Heyer Maximal Semigroups and the Support of Gauss - Semigroups ••••••••• 257 J. Hilgert Sufficiency Completeness Principle ••••••••••••••••••••••••••••••• 263 J. de la Horra On the Interpretation of Hypothesis Tests following Neyman and Pearson ••••••••••••••••••••••••••.••.••••••• 267 D. Johnstone De Finetti's Methods of Elicitation 279 J.B. Kadane and R.L. Winkler Bayesian Estimation of Design Floods under Regional and Subjective Prior Information ••••••••••••••••••••.•••••• 285 R. Kirnbauer, S. Schnatter and D. Gutknecht x Bayesian Methods in Multiperiod Financial Decision Ma.king .•..........••.........•.........•.........•..... 295 P. Kischka Cognitive Representation of Incomplete Knownledge •••••••••••••••• 305 G.D. Kleiter Comparison of some Statistical Methods for Counting Process Observations ••••••••••••••••••••••••••••••••••• 321 G. Koch Bayes Inference in Life Tests When Samples Sizes Are Fixed or Random •••••••••••••••••••••••••••••••••••••••• 335 G.S. Lingappaiah On Cox's Confidence Distribution 347 J. Mau A Bayesian Analysis of a Generalized Slope Rario Bioassay........ 357 M. Mendoza On Absolute Continuity of Measures Due to Gaussian Locally Stationary Processes ••••••••••••••••••••••••••• 365 J. Mich~l1ek Bayesian Adaptive Decision Theory Versus Dynamic Games as Models for Economic Planning and Policy-Making under Uncertainty........................ 369 R. Neck Remarks on Foundations of Bayesian Statistics and Econometrics 379 W. Polasek HPD-Regions for the Linear Regression Model...................... 395 K. Potzelberger A Very General De Finetti-Type Theorem ••••••••••••••••••••••••••• 403 P. Ressel A Bayesian Approach to Estimating the Parameters of a Hydrological Forecasting System ••••••.••.•••••••.• 415 S. Schnatter, D. Gutknecht and R. Kirnbauer The Extended Bayes-Postulate, Its Potential Effect on Statistical Methods and Some Historical Aspects 423 F. Schreiber The Analysis of Weibull Lifetime Data Incorporating Expert Opinion ••••••••••••••••.•••••••••••••••••••••••• 431 N.D. Singpurwalla and M. Shi Song Robust Tests for Trend in Binomial Proportions 443 R.T. Smythe, D. Krewski and A. Dewanji Decomposition of Weibul1 Mixture-Distributions in Accelerated Life Testing by Bayesian Methods ••••••••.•••••••••••••• 455 H. Strelec Robust Bayesian Methods 463 D. Thorburn xi Is It Necessary to Develop a Fuzzy Bayesian Inference? ••.••..•••• 471 R. Viert1 A Predictive Density Criterion for Selecting Non-Nested Linear Models and Comparison with Other Criteria 477 H. Wago and H. Tsurumi Bayesian Models and Methods for Binary Time Series ••••.••.•••••• 487 M. West and J. Mortera Semi-Sufficiency in Accelerated Life Testing •••••.•••••••••••••• 497 R. Willing INDEX ••••••••••••••••.•••••••••••••••.•••••••••••••••••••••.•••• 505 STOCHASTIC LINEAR PROGRAMMING WITH RECOURSE UNDER PARTIAL INFORMATION Peter Abel Fegro Praunheimer Str. 5-11 D-6236 Eschborn/Ost, F.R.G. 1. INTRODUCTION Stochastic progranuning models with random variables with only incompletely known distributions were up to now comparatively seldom analysed, although an attempt to declare probability distribution not always gives a satisfactory description of factors of influence in a decision model: "In any specific problem the selection of a definite probability distribution is made on the basis of a number of factors, such as the sequence of past demands, judgements about trends, etc. For various reasons, however, these factors may be insuffi cient to estimate the future distribution. As an example, the sample size of the past demands may be quite small, or we have reason to suspect that the future demand will come from a distribution which differs from that governing past history in an unpredictable way" (Scarf, 1958). One of the reasons for the rarely use of stochastic programming models with only incomplete informations about the distribution of the states in practise may be that often no numerical methods are available for the compu tation of the optimal solutions of such a model. In this paper we present a stochastic linear programming model with par tial informations about the probability distribution of the random coeffi cients and have a special look at the possibilities of the numerical compu tation of all optimal solutions of the presented model. 2. THE PROBLEM The stochastic linear programming model with recourse (two-stage model) with known common probability distribution of the random coefficients is of the form (see Dantzig, 1955) inf Ep{C'(W)x + inf{q'(w)yl W(w)y = b(w) - A(w)x, y ~ on (1) subject to x E X with random variable (A,b,c,W,q): Q -+ mmn+m+n+mn'+n' and probability dis tribution P.