Approximation Methods for Efficient Learning of Bayesian Networks

148 Pages·2008·1.27 MB·English

by  C. Riggelsen

APPROXIMATION METHODS FOR EFFICIENT
LEARNING OF BAYESIAN NETWORKS
Frontiers in Artificial Intelligence and Applications
Volume 168
Published in the subseries
Dissertations in Artificial Intelligence

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ISSN 0922-6389
Approximation Methods for Efficient
Learning of Bayesian Networks

Carsten Riggelsen
Institute of Geosciences, University of Potsdam, Golm b. Potsdam, Germany

Amsterdam (cid:129) Berlin (cid:129) Oxford (cid:129) Tokyo (cid:129) Washington, DC
© 2008 The author and IOS Press.
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, without prior written permission from the publisher.
ISBN 978-1-58603-821-2
Library of Congress Control Number: 2007942192
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Contents
Foreword ix
1. INTRODUCTION 1
2. PRELIMINARIES 5
1 Random variables and conditional independence 5
2 Graph theory 6
3 Markov properties 7
3.1 The Markov blanket 11
3.2 Equivalence of DAGs 12
4 Bayesian networks 14
5 Bayesian network specification 14
5.1 Bayesian parameter specification 15
3. LEARNING BAYESIAN NETWORKS
FROM DATA 19
1 The basics 19
2 Learning parameters 20
2.1 The Maximum-Likelihood approach 20
2.2 The Bayesian approach 21
3 Learning models 24
3.1 The penalised likelihood approach 25
3.2 The Bayesian approach 26
3.2.1 Learning via the marginal likelihood 26
3.2.2 Determining the hyper parameter 29
3.3 Marginal and penalised likelihood 32
vi EFFICIENT LEARNING OF BAYESIAN NETWORKS
3.4 Search methodologies 33
3.4.1 Model search space 34
3.4.2 Traversal strategy 35
4. MONTE CARLO METHODS AND MCMC SIMULATION 37
1 Monte Carlo methods 37
1.1 Importance sampling 38
1.1.1 Choice of the sampling distribution 40
2 Markov chain Monte Carlo—MCMC 41
2.1 Markov chains 42
2.1.1 The invariant target distribution 43
2.1.2 Reaching the invariant distribution 43
2.1.3 Metropolis-Hastings sampling 44
2.1.4 Gibbs sampling 46
2.1.5 Mixing, burn-in and convergence of MCMC 49
2.1.6 The importance of blocking 51
3 Learning models via MCMC 53
3.1 Sampling models 54
3.2 Sampling edges 54
3.3 Blocking edges 56
3.3.1 Blocks and Markov blankets 58
3.3.2 Sampling blocks 59
3.3.3 Validity of the sampler 62
3.3.4 The MB-MCMC model sampler 62
3.3.5 Evaluation 64
3.3.6 Conclusion 69
5. LEARNING FROM INCOMPLETE DATA 71
1 The concept of incomplete data 72
1.1 Missing data mechanisms 73
2 Learning from incomplete data 75
2.1 Likelihood decomposition 76
2.1.1 Complications for learning parameters 78
2.1.2 Bayesian sequential updating 79
2.1.3 Complications for learning models 80
3 Principled iterative methods 81
3.1 Expectation Maximisation—EM 81
Contents vii
3.1.1 Structural EM—SEM 85
3.2 Data Augmentation—DA 87
3.2.1 DA and eliminating the P-step—DA-P 90
3.3 DA-P and model learning—MDA-P 92
3.4 Efficiency issues of MDA-P 93
3.4.1 Properties of the sub-MCMC samplers 93
3.4.2 Interdependence between samplers 95
3.5 Imputation via importance sampling 97
3.5.1 The general idea 97
3.5.2 Importance sampling in the I-step—ISMDA-P 98
3.5.3 Generating new population vs. re-weighing 101
3.5.4 The marginal likelihood as predictive distribution 102
3.5.5 The eMC4 sampler 103
3.5.6 Evaluation—proof of concept 104
3.5.7 Conclusion 109
4 Ad-hoc and heuristic methods 110
4.1 Available cases analysis 110
4.2 Bound and Collapse—BC 111
4.3 Markov Blanket Predictor—MBP 114
4.3.1 The general idea 115
4.3.2 Approximate predictive distributions 115
4.3.3 Parameter estimation 117
4.3.4 Prediction and missing parents 119
4.3.5 Predictive quality 120
4.3.6 Selecting predictive variables 120
4.3.7 Implementation of MBP 123
4.3.8 Parameter estimation 124
4.3.9 Model learning 126
4.3.10 Conclusion and discussion 128
6. CONCLUSION 131
References 133
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Foreword
This dissertation is the result of 4 years at Utrecht University as a
Ph.D-student at the Department of Information and Computing Sci-
ences. The work presented in this thesis is mainly based on the research
published in various papers during that time. However, it is not merely
a bundle of research articles. In order to provide a coherent treatment
of matters, thereby helping the reader to gain a thorough understanding
of the whole concept of learning Bayesian networks from (in)complete
data, this thesis combines in a clarifying way all the issues presented in
the papers with previously unpublished work. I hope that my efforts
have been worthwhile!
Mygratitudegoestomysupervisorandco-promotor,dr.AdFeelders.
I am thankful to my promotor, prof.dr. Arno Siebes and to Jeroen De
Knijf, Edwin de Jong and all the other colleagues in the Algorithmic
Data Analysis group, and at the computer science department.
I would like to thank the members of the outstanding reading com-
mittee: prof.dr.ir. Linda van der Gaag, prof.dr. Richard D. Gill, prof.dr.
Finn V. Jensen, prof.dr. Bert Kappen and prof.dr. Pedro Larran˜aga.
The present dissertation was successfully defended on October 23,
2006 at Utrecht University. For publication in the series “Frontiers in
Artificial Intelligence and Applications”, IOS Press, only minor details
have been corrected and very few additions have been made.
Carsten Riggelsen
Berlin, October 2007
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