Bayesian Networks WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg, Harvey Goldstein Editors Emeriti: Vic Barnett, J. Stuart Hunter, Jozef L. Teugels A complete list of the titles in this series appears at the end of this volume. Bayesian Networks An Introduction Timo Koski Institutionen f¨or Matematik, Kungliga Tekniska H¨ogskolan, Stockholm, Sweden John M. Noble Matematiska Institutionen, Link¨opings Tekniska H¨ogskola, Link¨opings universitet, Link¨oping, Sweden A John Wiley and Sons, Ltd., Publication This edition first published 2009 2009, John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. 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If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Koski, Timo. Bayesian networks : an introduction / Timo Koski, John M. Noble. p. cm. – (Wiley series in probability and statistics) Includes bibliographical references and index. ISBN 978-0-470-74304-1 (cloth) 1. Bayesian statistical decision theory. 2. Neural networks (Computer science) I. Noble, John M. II. Title. QA279.5.K68 2009 519.542–dc22 2009031404 A catalogue record for this book is available from the British Library. ISBN: 978-0-470-74304-1 TypeSet in 10/12pt Times by Laserwords Private Limited, Chennai, India. Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall. Contents Preface ix 1 Graphical models and probabilistic reasoning 1 1.1 Introduction 1 1.2 Axioms of probability and basic notations 4 1.3 The Bayes update of probability 9 1.4 Inductive learning 11 1.4.1 Bayes’ rule 12 1.4.2 Jeffrey’s rule 13 1.4.3 Pearl’s method of virtual evidence 13 1.5 Interpretations of probability and Bayesian networks 14 1.6 Learning as inference about parameters 15 1.7 Bayesian statistical inference 17 1.8 Tossing a thumb-tack 20 1.9 Multinomial sampling and the Dirichlet integral 24 Notes 28 Exercises: Probabilistic theories of causality, Bayes’ rule, multinomial sampling and the Dirichlet density 31 2 Conditional independence, graphs and d-separation 37 2.1 Joint probabilities 37 2.2 Conditional independence 38 2.3 Directed acyclic graphs and d-separation 41 2.3.1 Graphs 41 2.3.2 Directed acyclic graphs and probability distributions 45 2.4 The Bayes ball 50 2.4.1 Illustrations 51 2.5 Potentials 53 2.6 Bayesian networks 58 2.7 Object oriented Bayesian networks 63 2.8 d-Separation and conditional independence 66 vi CONTENTS 2.9 Markov models and Bayesian networks 67 2.10 I-maps and Markov equivalence 69 2.10.1 The trek and a distribution without a faithful graph 72 Notes 73 Exercises: Conditional independence and d-separation 75 3 Evidence, sufficiency and Monte Carlo methods 81 3.1 Hard evidence 82 3.2 Soft evidence and virtual evidence 85 3.2.1 Jeffrey’s rule 86 3.2.2 Pearl’s method of virtual evidence 87 3.3 Queries in probabilistic inference 88 3.3.1 The chest clinic problem 89 3.4 Bucket elimination 89 3.5 Bayesian sufficient statistics and prediction sufficiency 92 3.5.1 Bayesian sufficient statistics 92 3.5.2 Prediction sufficiency 95 3.5.3 Prediction sufficiency for a Bayesian network 97 3.6 Time variables 98 3.7 A brief introduction to Markov chain Monte Carlo methods 100 3.7.1 Simulating a Markov chain 103 3.7.2 Irreducibility, aperiodicity and time reversibility 104 3.7.3 The Metropolis-Hastings algorithm 108 3.7.4 The one-dimensional discrete Metropolis algorithm 111 Notes 112 Exercises: Evidence, sufficiency and Monte Carlo methods 113 4 Decomposable graphs and chain graphs 123 4.1 Definitions and notations 124 4.2 Decomposable graphs and triangulation of graphs 127 4.3 Junction trees 131 4.4 Markov equivalence 133 4.5 Markov equivalence, the essential graph and chain graphs 138 Notes 144 Exercises: Decomposable graphs and chain graphs 145 5 Learning the conditional probability potentials 149 5.1 Initial illustration: maximum likelihood estimate for a fork connection 149 5.2 The maximum likelihood estimator for multinomial sampling 151 5.3 MLE for the parameters in a DAG: the general setting 155 5.4 Updating, missing data, fractional updating 160 Notes 161 Exercises: Learning the conditional probability potentials 162 6 Learning the graph structure 167 6.1 Assigning a probability distribution to the graph structure 168 CONTENTS vii 6.2 Markov equivalence and consistency 171 6.2.1 Establishing the DAG isomorphic property 173 6.3 Reducing the size of the search 176 6.3.1 The Chow-Liu tree 177 6.3.2 The Chow-Liu tree: A predictive approach 179 6.3.3 The K2 structural learning algorithm 183 6.3.4 The MMHC algorithm 184 6.4 Monte Carlo methods for locating the graph structure 186 6.5 Women in mathematics 189 Notes 191 Exercises: Learning the graph structure 192 7 Parameters and sensitivity 197 7.1 Changing parameters in a network 198 7.2 Measures of divergence between probability distributions 201 7.3 The Chan-Darwiche distance measure 202 7.3.1 Comparison with the Kullback-Leibler divergence and euclidean distance 209 7.3.2 Global bounds for queries 210 7.3.3 Applications to updating 212 7.4 Parameter changes to satisfy query constraints 216 7.4.1 Binary variables 218 7.5 The sensitivity of queries to parameter changes 220 Notes 224 Exercises: Parameters and sensitivity 225 8 Graphical models and exponential families 229 8.1 Introduction to exponential families 229 8.2 Standard examples of exponential families 231 8.3 Graphical models and exponential families 233 8.4 Noisy ‘or’ as an exponential family 234 8.5 Properties of the log partition function 237 8.6 Fenchel Legendre conjugate 239 8.7 Kullback-Leibler divergence 241 8.8 Mean field theory 243 8.9 Conditional Gaussian distributions 246 8.9.1 CG potentials 249 8.9.2 Some results on marginalization 249 8.9.3 CG regression 250 Notes 251 Exercises: Graphical models and exponential families 252 9 Causality and intervention calculus 255 9.1 Introduction 255 9.2 Conditioning by observation and by intervention 257 9.3 The intervention calculus for a Bayesian network 258 viii CONTENTS 9.3.1 Establishing the model via a controlled experiment 262 9.4 Properties of intervention calculus 262 9.5 Transformations of probability 265 9.6 A note on the order of ‘see’ and ‘do’ conditioning 267 9.7 The ‘Sure Thing’ principle 268 9.8 Back door criterion, confounding and identifiability 270 Notes 273 Exercises: Causality and intervention calculus 275 10 The junction tree and probability updating 279 10.1 Probability updating using a junction tree 279 10.2 Potentials and the distributive law 280 10.2.1 Marginalization and the distributive law 283 10.3 Elimination and domain graphs 284 10.4 Factorization along an undirected graph 288 10.5 Factorizing along a junction tree 290 10.5.1 Flow of messages initial illustration 292 10.6 Local computation on junction trees 294 10.7 Schedules 296 10.8 Local and global consistency 302 10.9 Message passing for conditional Gaussian distributions 305 10.10 Using a junction tree with virtual evidence and soft evidence 311 Notes 313 Exercises: The junction tree and probability updating 314 11 Factor graphs and the sum product algorithm 319 11.1 Factorization and local potentials 319 11.1.1 Examples of factor graphs 320 11.2 The sum product algorithm 323 11.3 Detailed illustration of the algorithm 329 Notes 332 Exercise: Factor graphs and the sum product algorithm 333 References 335 Index 343 Preface This book evolved from courses developed at Linkoping¨ Institute of Technology and KTH, given by the authors, starting with a graduate course given by Timo Koski in 2002, who was the Professor of Mathematical Statistics at LiTH at the time and subsequently developed by both authors. The book has been aimed at senior undergraduate, masters and beginning Ph.D. students in computer engineering. The students are expected to have a first course in probability and statistics, a first course in discrete mathematics and a first course in algorithmics.