Gaussian Processes for Machine Learning

1 C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of Processes for Machine LearningC. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of Computation and Machine LearningThomas Dietterich, EditorChristopher Bishop, David Heckerman, Michael Jordan, and Michael Kearns, Associate EditorsBioinformatics: The Machine Learning Approach,Pierre Baldi and S ren BrunakReinforcement Learning : an introduction ,Richard S. Sutton and Andrew G. BartoGraphical Models for Machine Learning and Digital Communication,Brendan J. FreyLearning in Graphical Models,Michael I. JordanCausation, Prediction, and Search, second edition,Peter Spirtes, Clark Glymour, and Richard ScheinesPrinciples of Data Mining,David Hand, Heikki Mannila, and Padhraic SmythBioinformatics: The Machine Learning Approach, second edition,Pierre Baldi and S ren BrunakLearning Kernel Classifiers: Theory and Algorithms,Ralf HerbrichLearning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond,Bernhard Sch olkopf and Alexander J.

2 SmolaIntroduction to Machine Learning ,Ethem AlpaydinGaussian Processes for Machine Learning ,Carl Edward Rasmussen and Christopher K. I. WilliamsC. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of Processes for Machine LearningCarl Edward RasmussenChristopher K. I. WilliamsThe MIT PressCambridge, MassachusettsLondon, EnglandC. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of 2006 Massachusetts Institute of TechnologyAll rights reserved. No part of this book may be reproduced in any form by any electronic or mechanicalmeans (including photocopying, recording, or information storage and retrieval) without permission inwriting from the Press books may be purchased at special quantity discounts for business or sales promotional information, please write to Special Sales Department,The MIT Press, 55 Hayward Street, Cambridge, MA by the authors using LATEX 2.

3 This book was printed and bound in the United States of of Congress Cataloging-in-Publication DataRasmussen, Carl Processes for Machine Learning / Carl Edward Rasmussen, Christopher K. I. cm. (Adaptive computation and Machine Learning )Includes bibliographical references and 0-262-18253-X1. Gaussian Processes Data processing. 2. Machine Learning Mathematical Williams, Christopher K. I. II. Title. III. '3 dc22200505343310 9 8 7 6 5 4 3 2C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of actual science of logic is conversant at present only with things eithercertain, impossible, or entirely doubtful, none of which (fortunately) we have toreason 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 tobe, in a reasonable man s mind.

4 James Clerk Maxwell [1850]C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of Foreword ..xiPreface ..xiiiSymbols and Notation ..xvii1 A Pictorial introduction to Bayesian Modelling .. Roadmap ..52 Weight-space View .. The Standard Linear Model .. Projections of Inputs into Feature Space .. Function-space View .. Varying the Hyperparameters .. Decision Theory for Regression .. An Example Application .. Smoothing, Weight Functions and Equivalent Kernels ..24 Incorporating Explicit Basis Functions .. Marginal Likelihood .. History and Related Work .. Exercises ..303 Classification Problems .. Decision Theory for Classification.

5 Linear Models for Classification .. Gaussian Process Classification .. The Laplace Approximation for the Binary GP Classifier .. Posterior .. Predictions .. Implementation .. Marginal Likelihood ..47 Multi-class Laplace Approximation .. Implementation .. Expectation Propagation .. Predictions .. Marginal Likelihood .. Implementation .. Experiments .. A Toy Problem .. One-dimensional Example .. Binary Handwritten Digit Classification Example .. 10-class Handwritten Digit Classification Example .. Discussion ..72 Sections marked by an asterisk contain advanced material that may be omitted on a first E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of Appendix: Moment Derivations .. Exercises ..754 Covariance Preliminaries.

6 79 Mean Square Continuity and Differentiability .. Examples of Covariance Functions .. Stationary Covariance Functions .. Dot Product Covariance Functions .. Other Non-stationary Covariance Functions .. Making New Kernels from Old .. Eigenfunction Analysis of Kernels ..96 An Analytic Example .. Numerical Approximation of Eigenfunctions .. Kernels for Non-vectorial Inputs .. String Kernels .. Fisher Kernels .. Exercises ..1025 Model Selection and Adaptation of The Model Selection Problem .. Bayesian Model Selection .. Cross-validation .. Model Selection for GP Regression .. Marginal Likelihood .. Cross-validation .. Examples and Discussion .. Model Selection for GP Classification ..124 Derivatives of the Marginal Likelihood for Laplace s Approximation125 Derivatives of the Marginal Likelihood for EP.

7 Cross-validation .. Example .. Exercises ..1286 Relationships between GPs and Other Reproducing Kernel Hilbert Spaces .. Regularization ..132 Regularization Defined by Differential Operators .. Obtaining the Regularized Solution .. The Relationship of the Regularization View to Gaussian ProcessPrediction .. Spline Models ..136 A 1-d Gaussian Process Spline Construction ..138 Support Vector Machines .. Support Vector Classification .. Support Vector Regression ..145 Least-squares Classification .. Probabilistic Least-squares Classification ..147C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of Relevance Vector Machines .. Exercises ..1507 Theoretical The Equivalent Kernel .. Some Specific Examples of Equivalent Kernels.

8 153 Asymptotic Analysis .. Consistency .. Equivalence and Orthogonality ..157 Average-case Learning Curves ..159 PAC-Bayesian Analysis .. The PAC Framework .. PAC-Bayesian Analysis .. PAC-Bayesian Analysis of GP Classification .. Comparison with Other Supervised Learning Methods ..165 Appendix: Learning Curve for the Ornstein-Uhlenbeck Process .. Exercises ..1698 Approximation Methods for Large Reduced-rank Approximations of the Gram Matrix .. Greedy Approximation .. Approximations for GPR with Fixed Hyperparameters .. Subset of Regressors .. The Nystr om Method .. Subset of Datapoints .. Projected Process Approximation .. Bayesian Committee Machine .. Iterative Solution of Linear Systems .. Comparison of Approximate GPR Methods .. Approximations for GPC with Fixed Hyperparameters ..185 Approximating the Marginal Likelihood and its Derivatives.

9 185 Appendix: Equivalence of SR and GPR Using the Nystr om ApproximateKernel .. Exercises ..1879 Further Issues and Multiple Outputs .. Noise Models with Dependencies .. Non- Gaussian Likelihoods .. Derivative Observations .. Prediction with Uncertain Inputs .. Mixtures of Gaussian Processes .. Global Optimization .. Evaluation of Integrals .. Student stProcess .. Invariances .. Latent Variable Models .. Conclusions and Future Directions ..196C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of A Mathematical Joint, Marginal and Conditional Probability .. Gaussian Identities .. Matrix Identities .. Matrix Derivatives .. Matrix Norms .. Cholesky Decomposition .. Entropy and Kullback-Leibler Divergence .. Limits .. Measure and Integration.

10 Fourier Transforms .. Convexity ..206 Appendix B Gaussian Markov Fourier Analysis .. Sampling and Periodization .. Continuous-time Gaussian Markov Processes .. Continuous-time GMPs onR.. The Solution of the Corresponding SDE on the Circle .. Discrete-time Gaussian Markov Processes .. Discrete-time GMPs onZ.. The Solution of the Corresponding Difference Equation onPN.. The Relationship Between Discrete-time and Sampled Continuous-timeGMPs .. Markov Processes in Higher Dimensions ..218 Appendix C Datasets and Code221 Bibliography223 Author Index239 Subject Index245C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning , the MIT Press, 2006,ISBN 2006 Massachusetts Institute of ForewordThe goal of building systems that can adapt to their environments and learnfrom their experience has attracted researchers from many fields, including com-puter science, engineering, mathematics, physics, neuroscience, and cognitivescience.