高斯过程回归与分类学习的经典书籍.pdf

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Gaussian Processes for Machine Learning Carl Edward Rasmussen and Christopher K. I. Williams

Gaussian Processes for Machine Learning

Adaptive Computation and Machine Learning Thomas Dietterich, Editor Christopher Bishop, David Heckerman, Michael Jordan, and Michael Kearns, Associate Editors Bioinformatics: The Machine Learning Approach, Pierre Baldi and Søren Brunak Reinforcement Learning: An Introduction, Richard S. Sutton and Andrew G. Barto Graphical Models for Machine Learning and Digital Communication, Brendan J. Frey Learning in Graphical Models, Michael I. Jordan Causation, Prediction, and Search, second edition, Peter Spirtes, Clark Glymour, and Richard Scheines Principles of Data Mining, David Hand, Heikki Mannila, and Padhraic Smyth Bioinformatics: The Machine Learning Approach, second edition, Pierre Baldi and Søren Brunak Learning Kernel Classiﬁers: Theory and Algorithms, Ralf Herbrich Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, Bernhard Sch¨olkopf and Alexander J. Smola Introduction to Machine Learning, Ethem Alpaydin Gaussian Processes for Machine Learning, Carl Edward Rasmussen and Christopher K. I. Williams

Gaussian Processes for Machine Learning Carl Edward Rasmussen Christopher K. I. Williams The MIT Press Cambridge, Massachusetts London, England

c 2006 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. MIT Press books may be purchased at special quantity discounts for business or sales promotional use. For information, please email special sales@mitpress.mit.edu or write to Special Sales Department, The MIT Press, 55 Hayward Street, Cambridge, MA 02142. Typeset by the authors using LATEX 2ε. This book printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Rasmussen, Carl Edward. Gaussian processes for machine learning / Carl Edward Rasmussen, Christopher K. I. Williams. p. cm. —(Adaptive computation and machine learning) Includes bibliographical references and indexes. ISBN 0-262-18253-X 1. Gaussian processes—Data processing. 2. Machine learning—Mathematical models. I. Williams, Christopher K. I. II. Title. III. Series. QA274.4.R37 2006 519.2’3—dc22 10 9 8 7 6 5 4 3 2 1 2005053433

The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man’s mind. — James Clerk Maxwell [1850]

Contents xi Series Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Symbols and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 1 Introduction 1.1 A Pictorial Introduction to Bayesian Modelling . . . . . . . . . . . . . . . 1.2 Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Regression 2.1 Weight-space View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The Standard Linear Model . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Projections of Inputs into Feature Space . . . . . . . . . . . . . . . 2.2 Function-space View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Varying the Hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Decision Theory for Regression . . . . . . . . . . . . . . . . . . . . . . . . 2.5 An Example Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Smoothing, Weight Functions and Equivalent Kernels . . . . . . . . . . . Incorporating Explicit Basis Functions . . . . . . . . . . . . . . . . . . . . 2.7.1 Marginal Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 History and Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ 2.7 1 3 5 7 7 8 11 13 19 21 22 24 27 29 29 30 3 Classiﬁcation 3.1 Classiﬁcation Problems 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.1 Decision Theory for Classiﬁcation . . . . . . . . . . . . . . . . . . 35 3.2 Linear Models for Classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Gaussian Process Classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 The Laplace Approximation for the Binary GP Classiﬁer . . . . . . . . . . 41 3.4.1 Posterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.2 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4.4 Marginal Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ∗ 3.5 Multi-class Laplace Approximation . . . . . . . . . . . . . . . . . . . . . . 48 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 Expectation Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6.1 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.6.2 Marginal Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.7 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.7.1 A Toy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.7.2 One-dimensional Example . . . . . . . . . . . . . . . . . . . . . . 62 3.7.3 Binary Handwritten Digit Classiﬁcation Example . . . . . . . . . . 63 3.7.4 10-class Handwritten Digit Classiﬁcation Example . . . . . . . . . 70 3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ∗Sections marked by an asterisk contain advanced material that may be omitted on a ﬁrst reading. 3.5.1

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