Holdings: Pattern recognition and machine learning

Pattern recognition and machine learning:

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Bibliographic Details
Main Author:	Bishop, Christopher M. 1959- (Author)
Format:	Book
Language:	English
Published:	New York, NY Springer 2006
Series:	Information science and statistics
Subjects:	Maschinelles Lernen Mustererkennung Machine learning Pattern perception
Online Access:	Beschreibung Publisher description Table of contents only Inhaltsverzeichnis Inhaltsverzeichnis
Item Description:	Literaturverzeichnis Seite 711 - 728 Hier auch später erschienene, unveränderte Nachdrucke
Physical Description:	XX, 738 S. Ill., graph. Darst.
ISBN:	0387310738 9780387310732

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MARC


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650		4	\|a Mustererkennung
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adam_text	CONTENTS PREFACE VII MATHEMATICAL NOTATION XI CONTENTS XIII 1 INTRODUCTION 1 1.1 EXAMPLE: POLYNOMIAL CURVE FITTING . . . . . . . . . . . . . . . . . 4 1.2 PROBABILITY THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 PROBABILITY DENSITIES . . . . . . . . . . . . . . . . . . . . . 17 1.2.2 EXPECTATIONS AND COVARIANCES . . . . . . . . . . . . . . . . 19 1.2.3 BAYESIAN PROBABILITIES . . . . . . . . . . . . . . . . . . . . 21 1.2.4 THE GAUSSIAN DISTRIBUTION . . . . . . . . . . . . . . . . . . 24 1.2.5 CURVE FITTING RE-VISITED . . . . . . . . . . . . . . . . . . . . 28 1.2.6 BAYESIAN CURVE FITTING . . . . . . . . . . . . . . . . . . . . 30 1.3 MODEL SELECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4 THE CURSE OF DIMENSIONALITY . . . . . . . . . . . . . . . . . . . . . 33 1.5 DECISION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5.1 MINIMIZING THE MISCLASSIFICATION RATE . . . . . . . . . . . . 39 1.5.2 MINIMIZING THE EXPECTED LOSS . . . . . . . . . . . . . . . . 41 1.5.3 THE REJECT OPTION . . . . . . . . . . . . . . . . . . . . . . . 42 1.5.4 INFERENCE AND DECISION . . . . . . . . . . . . . . . . . . . . 42 1.5.5 LOSS FUNCTIONS FOR REGRESSION . . . . . . . . . . . . . . . . . 46 1.6 INFORMATION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.6.1 RELATIVE ENTROPY AND MUTUAL INFORMATION . . . . . . . . . . 55 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML XIII XIV CONTENTS 2 PROBABILITY DISTRIBUTIONS 67 2.1 BINARY VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.1.1 THE BETA DISTRIBUTION . . . . . . . . . . . . . . . . . . . . . 71 2.2 MULTINOMIAL VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . 74 2.2.1 THE DIRICHLET DISTRIBUTION . . . . . . . . . . . . . . . . . . . 76 2.3 THE GAUSSIAN DISTRIBUTION . . . . . . . . . . . . . . . . . . . . . . 78 2.3.1 CONDITIONAL GAUSSIAN DISTRIBUTIONS . . . . . . . . . . . . . . 85 2.3.2 MARGINAL GAUSSIAN DISTRIBUTIONS . . . . . . . . . . . . . . . 88 2.3.3 BAYES* THEOREM FOR GAUSSIAN VARIABLES . . . . . . . . . . . . 90 2.3.4 MAXIMUM LIKELIHOOD FOR THE GAUSSIAN . . . . . . . . . . . . 93 2.3.5 SEQUENTIAL ESTIMATION . . . . . . . . . . . . . . . . . . . . . 94 2.3.6 BAYESIAN INFERENCE FOR THE GAUSSIAN . . . . . . . . . . . . . 97 2.3.7 STUDENTS T-DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 102 2.3.8 PERIODIC VARIABLES . . . . . . . . . . . . . . . . . . . . . . . 105 2.3.9 MIXTURES OF GAUSSIANS . . . . . . . . . . . . . . . . . . . . 110 2.4 THE EXPONENTIAL FAMILY . . . . . . . . . . . . . . . . . . . . . . . 113 2.4.1 MAXIMUM LIKELIHOOD AND SUFFICIENT STATISTICS . . . . . . . . 116 2.4.2 CONJUGATE PRIORS . . . . . . . . . . . . . . . . . . . . . . . 117 2.4.3 NONINFORMATIVE PRIORS . . . . . . . . . . . . . . . . . . . . 117 2.5 NONPARAMETRIC METHODS . . . . . . . . . . . . . . . . . . . . . . . 120 2.5.1 KERNEL DENSITY ESTIMATORS . . . . . . . . . . . . . . . . . . . 122 2.5.2 NEAREST-NEIGHBOUR METHODS . . . . . . . . . . . . . . . . . 124 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3 LINEAR MODELS FOR REGRESSION 137 3.1 LINEAR BASIS FUNCTION MODELS . . . . . . . . . . . . . . . . . . . . 138 3.1.1 MAXIMUM LIKELIHOOD AND LEAST SQUARES . . . . . . . . . . . . 140 3.1.2 GEOMETRY OF LEAST SQUARES . . . . . . . . . . . . . . . . . . 143 3.1.3 SEQUENTIAL LEARNING . . . . . . . . . . . . . . . . . . . . . . 143 3.1.4 REGULARIZED LEAST SQUARES . . . . . . . . . . . . . . . . . . . 144 3.1.5 MULTIPLE OUTPUTS . . . . . . . . . . . . . . . . . . . . . . . 146 3.2 THE BIAS-VARIANCE DECOMPOSITION . . . . . . . . . . . . . . . . . . 147 3.3 BAYESIAN LINEAR REGRESSION . . . . . . . . . . . . . . . . . . . . . 152 3.3.1 PARAMETER DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 152 3.3.2 PREDICTIVE DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 156 3.3.3 EQUIVALENT KERNEL . . . . . . . . . . . . . . . . . . . . . . . 159 3.4 BAYESIAN MODEL COMPARISON . . . . . . . . . . . . . . . . . . . . . 161 3.5 THE EVIDENCE APPROXIMATION . . . . . . . . . . . . . . . . . . . . 165 3.5.1 EVALUATION OF THE EVIDENCE FUNCTION . . . . . . . . . . . . . 166 3.5.2 MAXIMIZING THE EVIDENCE FUNCTION . . . . . . . . . . . . . . 168 3.5.3 EFFECTIVE NUMBER OF PARAMETERS . . . . . . . . . . . . . . . 170 3.6 LIMITATIONS OF FIXED BASIS FUNCTIONS . . . . . . . . . . . . . . . . 172 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 C CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML CONTENTS XV 4 LINEAR MODELS FOR CLASSIFICATION 179 4.1 DISCRIMINANT FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . 181 4.1.1 TWO CLASSES . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.1.2 MULTIPLE CLASSES . . . . . . . . . . . . . . . . . . . . . . . . 182 4.1.3 LEAST SQUARES FOR CLASSIFICATION . . . . . . . . . . . . . . . . 184 4.1.4 FISHERS LINEAR DISCRIMINANT . . . . . . . . . . . . . . . . . . 186 4.1.5 RELATION TO LEAST SQUARES . . . . . . . . . . . . . . . . . . . 189 4.1.6 FISHERS DISCRIMINANT FOR MULTIPLE CLASSES . . . . . . . . . . 191 4.1.7 THE PERCEPTRON ALGORITHM . . . . . . . . . . . . . . . . . . . 192 4.2 PROBABILISTIC GENERATIVE MODELS . . . . . . . . . . . . . . . . . . . 196 4.2.1 CONTINUOUS INPUTS . . . . . . . . . . . . . . . . . . . . . . 198 4.2.2 MAXIMUM LIKELIHOOD SOLUTION . . . . . . . . . . . . . . . . 200 4.2.3 DISCRETE FEATURES . . . . . . . . . . . . . . . . . . . . . . . 202 4.2.4 EXPONENTIAL FAMILY . . . . . . . . . . . . . . . . . . . . . . 202 4.3 PROBABILISTIC DISCRIMINATIVE MODELS . . . . . . . . . . . . . . . . . 203 4.3.1 FIXED BASIS FUNCTIONS . . . . . . . . . . . . . . . . . . . . . 204 4.3.2 LOGISTIC REGRESSION . . . . . . . . . . . . . . . . . . . . . . 205 4.3.3 ITERATIVE REWEIGHTED LEAST SQUARES . . . . . . . . . . . . . . 207 4.3.4 MULTICLASS LOGISTIC REGRESSION . . . . . . . . . . . . . . . . . 209 4.3.5 PROBIT REGRESSION . . . . . . . . . . . . . . . . . . . . . . . 210 4.3.6 CANONICAL LINK FUNCTIONS . . . . . . . . . . . . . . . . . . . 212 4.4 THE LAPLACE APPROXIMATION . . . . . . . . . . . . . . . . . . . . . 213 4.4.1 MODEL COMPARISON AND BIC . . . . . . . . . . . . . . . . . 216 4.5 BAYESIAN LOGISTIC REGRESSION . . . . . . . . . . . . . . . . . . . . 217 4.5.1 LAPLACE APPROXIMATION . . . . . . . . . . . . . . . . . . . . 217 4.5.2 PREDICTIVE DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 218 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5 NEURAL NETWORKS 225 5.1 FEED-FORWARD NETWORK FUNCTIONS . . . . . . . . . . . . . . . . . . 227 5.1.1 WEIGHT-SPACE SYMMETRIES . . . . . . . . . . . . . . . . . . 231 5.2 NETWORK TRAINING . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.2.1 PARAMETER OPTIMIZATION . . . . . . . . . . . . . . . . . . . . 236 5.2.2 LOCAL QUADRATIC APPROXIMATION . . . . . . . . . . . . . . . . 237 5.2.3 USE OF GRADIENT INFORMATION . . . . . . . . . . . . . . . . . 239 5.2.4 GRADIENT DESCENT OPTIMIZATION . . . . . . . . . . . . . . . . 240 5.3 ERROR BACKPROPAGATION . . . . . . . . . . . . . . . . . . . . . . . . 241 5.3.1 EVALUATION OF ERROR-FUNCTION DERIVATIVES . . . . . . . . . . . 242 5.3.2 A SIMPLE EXAMPLE . . . . . . . . . . . . . . . . . . . . . . 245 5.3.3 EFFICIENCY OF BACKPROPAGATION . . . . . . . . . . . . . . . . 246 5.3.4 THE JACOBIAN MATRIX . . . . . . . . . . . . . . . . . . . . . 247 5.4 THE HESSIAN MATRIX . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.4.1 DIAGONAL APPROXIMATION . . . . . . . . . . . . . . . . . . . 250 5.4.2 OUTER PRODUCT APPROXIMATION . . . . . . . . . . . . . . . . . 251 5.4.3 INVERSE HESSIAN . . . . . . . . . . . . . . . . . . . . . . . . 252 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML XVI CONTENTS 5.4.4 FINITE DIFFERENCES . . . . . . . . . . . . . . . . . . . . . . . 252 5.4.5 EXACT EVALUATION OF THE HESSIAN . . . . . . . . . . . . . . . 253 5.4.6 FAST MULTIPLICATION BY THE HESSIAN . . . . . . . . . . . . . . 254 5.5 REGULARIZATION IN NEURAL NETWORKS . . . . . . . . . . . . . . . . . 256 5.5.1 CONSISTENT GAUSSIAN PRIORS . . . . . . . . . . . . . . . . . . 257 5.5.2 EARLY STOPPING . . . . . . . . . . . . . . . . . . . . . . . . 259 5.5.3 INVARIANCES . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.5.4 TANGENT PROPAGATION . . . . . . . . . . . . . . . . . . . . . 263 5.5.5 TRAINING WITH TRANSFORMED DATA . . . . . . . . . . . . . . . . 265 5.5.6 CONVOLUTIONAL NETWORKS . . . . . . . . . . . . . . . . . . . 267 5.5.7 SOFT WEIGHT SHARING . . . . . . . . . . . . . . . . . . . . . . 269 5.6 MIXTURE DENSITY NETWORKS . . . . . . . . . . . . . . . . . . . . . . 272 5.7 BAYESIAN NEURAL NETWORKS . . . . . . . . . . . . . . . . . . . . . . 277 5.7.1 POSTERIOR PARAMETER DISTRIBUTION . . . . . . . . . . . . . . . 278 5.7.2 HYPERPARAMETER OPTIMIZATION . . . . . . . . . . . . . . . . 280 5.7.3 BAYESIAN NEURAL NETWORKS FOR CLASSIFICATION . . . . . . . . . 281 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 6 KERNEL METHODS 291 6.1 DUAL REPRESENTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.2 CONSTRUCTING KERNELS . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.3 RADIAL BASIS FUNCTION NETWORKS . . . . . . . . . . . . . . . . . . . 299 6.3.1 NADARAYA-WATSON MODEL . . . . . . . . . . . . . . . . . . . 301 6.4 GAUSSIAN PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . 303 6.4.1 LINEAR REGRESSION REVISITED . . . . . . . . . . . . . . . . . . 304 6.4.2 GAUSSIAN PROCESSES FOR REGRESSION . . . . . . . . . . . . . . 306 6.4.3 LEARNING THE HYPERPARAMETERS . . . . . . . . . . . . . . . . 311 6.4.4 AUTOMATIC RELEVANCE DETERMINATION . . . . . . . . . . . . . 312 6.4.5 GAUSSIAN PROCESSES FOR CLASSIFICATION . . . . . . . . . . . . . 313 6.4.6 LAPLACE APPROXIMATION . . . . . . . . . . . . . . . . . . . . 315 6.4.7 CONNECTION TO NEURAL NETWORKS . . . . . . . . . . . . . . . . 319 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 7 SPARSE KERNEL MACHINES 325 7.1 MAXIMUM MARGIN CLASSIFIERS . . . . . . . . . . . . . . . . . . . . 326 7.1.1 OVERLAPPING CLASS DISTRIBUTIONS . . . . . . . . . . . . . . . . 331 7.1.2 RELATION TO LOGISTIC REGRESSION . . . . . . . . . . . . . . . . 336 7.1.3 MULTICLASS SVMS . . . . . . . . . . . . . . . . . . . . . . . 338 7.1.4 SVMS FOR REGRESSION . . . . . . . . . . . . . . . . . . . . . 339 7.1.5 COMPUTATIONAL LEARNING THEORY . . . . . . . . . . . . . . . . 344 7.2 RELEVANCE VECTOR MACHINES . . . . . . . . . . . . . . . . . . . . . 345 7.2.1 RVM FOR REGRESSION . . . . . . . . . . . . . . . . . . . . . . 345 7.2.2 ANALYSIS OF SPARSITY . . . . . . . . . . . . . . . . . . . . . . 349 7.2.3 RVM FOR CLASSIFICATION . . . . . . . . . . . . . . . . . . . . 353 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML CONTENTS XVII 8 GRAPHICAL MODELS 359 8.1 BAYESIAN NETWORKS . . . . . . . . . . . . . . . . . . . . . . . . . . 360 8.1.1 EXAMPLE: POLYNOMIAL REGRESSION . . . . . . . . . . . . . . . 362 8.1.2 GENERATIVE MODELS . . . . . . . . . . . . . . . . . . . . . . 365 8.1.3 DISCRETE VARIABLES . . . . . . . . . . . . . . . . . . . . . . . 366 8.1.4 LINEAR-GAUSSIAN MODELS . . . . . . . . . . . . . . . . . . . 370 8.2 CONDITIONAL INDEPENDENCE . . . . . . . . . . . . . . . . . . . . . . 372 8.2.1 THREE EXAMPLE GRAPHS . . . . . . . . . . . . . . . . . . . . 373 8.2.2 D-SEPARATION . . . . . . . . . . . . . . . . . . . . . . . . . 378 8.3 MARKOV RANDOM FIELDS . . . . . . . . . . . . . . . . . . . . . . . 383 8.3.1 CONDITIONAL INDEPENDENCE PROPERTIES . . . . . . . . . . . . . 383 8.3.2 FACTORIZATION PROPERTIES . . . . . . . . . . . . . . . . . . . 384 8.3.3 ILLUSTRATION: IMAGE DE-NOISING . . . . . . . . . . . . . . . . 387 8.3.4 RELATION TO DIRECTED GRAPHS . . . . . . . . . . . . . . . . . . 390 8.4 INFERENCE IN GRAPHICAL MODELS . . . . . . . . . . . . . . . . . . . . 393 8.4.1 INFERENCE ON A CHAIN . . . . . . . . . . . . . . . . . . . . . 394 8.4.2 TREES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 8.4.3 FACTOR GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . 399 8.4.4 THE SUM-PRODUCT ALGORITHM . . . . . . . . . . . . . . . . . . 402 8.4.5 THE MAX-SUM ALGORITHM . . . . . . . . . . . . . . . . . . . 411 8.4.6 EXACT INFERENCE IN GENERAL GRAPHS . . . . . . . . . . . . . . 416 8.4.7 LOOPY BELIEF PROPAGATION . . . . . . . . . . . . . . . . . . . 417 8.4.8 LEARNING THE GRAPH STRUCTURE . . . . . . . . . . . . . . . . . 418 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 9 MIXTURE MODELS AND EM 423 9.1 K -MEANS CLUSTERING . . . . . . . . . . . . . . . . . . . . . . . . . 424 9.1.1 IMAGE SEGMENTATION AND COMPRESSION . . . . . . . . . . . . 428 9.2 MIXTURES OF GAUSSIANS . . . . . . . . . . . . . . . . . . . . . . . . 430 9.2.1 MAXIMUM LIKELIHOOD . . . . . . . . . . . . . . . . . . . . . 432 9.2.2 EM FOR GAUSSIAN MIXTURES . . . . . . . . . . . . . . . . . . 435 9.3 AN ALTERNATIVE VIEW OF EM . . . . . . . . . . . . . . . . . . . . . 439 9.3.1 GAUSSIAN MIXTURES REVISITED . . . . . . . . . . . . . . . . . 441 9.3.2 RELATION TO K -MEANS . . . . . . . . . . . . . . . . . . . . . 443 9.3.3 MIXTURES OF BERNOULLI DISTRIBUTIONS . . . . . . . . . . . . . . 444 9.3.4 EM FOR BAYESIAN LINEAR REGRESSION . . . . . . . . . . . . . . 448 9.4 THE EM ALGORITHM IN GENERAL . . . . . . . . . . . . . . . . . . . . 450 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 10 APPROXIMATE INFERENCE 461 10.1 VARIATIONAL INFERENCE . . . . . . . . . . . . . . . . . . . . . . . . . 462 10.1.1 FACTORIZED DISTRIBUTIONS . . . . . . . . . . . . . . . . . . . . 464 10.1.2 PROPERTIES OF FACTORIZED APPROXIMATIONS . . . . . . . . . . . 466 10.1.3 EXAMPLE: THE UNIVARIATE GAUSSIAN . . . . . . . . . . . . . . 470 10.1.4 MODEL COMPARISON . . . . . . . . . . . . . . . . . . . . . . 473 10.2 ILLUSTRATION: VARIATIONAL MIXTURE OF GAUSSIANS . . . . . . . . . . . . 474 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML XVIII CONTENTS 10.2.1 VARIATIONAL DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 475 10.2.2 VARIATIONAL LOWER BOUND . . . . . . . . . . . . . . . . . . . 481 10.2.3 PREDICTIVE DENSITY . . . . . . . . . . . . . . . . . . . . . . . 482 10.2.4 DETERMINING THE NUMBER OF COMPONENTS . . . . . . . . . . . 483 10.2.5 INDUCED FACTORIZATIONS . . . . . . . . . . . . . . . . . . . . 485 10.3 VARIATIONAL LINEAR REGRESSION . . . . . . . . . . . . . . . . . . . . 486 10.3.1 VARIATIONAL DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 486 10.3.2 PREDICTIVE DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 488 10.3.3 LOWER BOUND . . . . . . . . . . . . . . . . . . . . . . . . . 489 10.4 EXPONENTIAL FAMILY DISTRIBUTIONS . . . . . . . . . . . . . . . . . . 490 10.4.1 VARIATIONAL MESSAGE PASSING . . . . . . . . . . . . . . . . . 491 10.5 LOCAL VARIATIONAL METHODS . . . . . . . . . . . . . . . . . . . . . . 493 10.6 VARIATIONAL LOGISTIC REGRESSION . . . . . . . . . . . . . . . . . . . 498 10.6.1 VARIATIONAL POSTERIOR DISTRIBUTION . . . . . . . . . . . . . . . 498 10.6.2 OPTIMIZING THE VARIATIONAL PARAMETERS . . . . . . . . . . . . 500 10.6.3 INFERENCE OF HYPERPARAMETERS . . . . . . . . . . . . . . . . 502 10.7 EXPECTATION PROPAGATION . . . . . . . . . . . . . . . . . . . . . . . 505 10.7.1 EXAMPLE: THE CLUTTER PROBLEM . . . . . . . . . . . . . . . . 511 10.7.2 EXPECTATION PROPAGATION ON GRAPHS . . . . . . . . . . . . . . 513 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 11 SAMPLING METHODS 523 11.1 BASIC SAMPLING ALGORITHMS . . . . . . . . . . . . . . . . . . . . . 526 11.1.1 STANDARD DISTRIBUTIONS . . . . . . . . . . . . . . . . . . . . 526 11.1.2 REJECTION SAMPLING . . . . . . . . . . . . . . . . . . . . . . 528 11.1.3 ADAPTIVE REJECTION SAMPLING . . . . . . . . . . . . . . . . . 530 11.1.4 IMPORTANCE SAMPLING . . . . . . . . . . . . . . . . . . . . . 532 11.1.5 SAMPLING-IMPORTANCE-RESAMPLING . . . . . . . . . . . . . . 534 11.1.6 SAMPLING AND THE EM ALGORITHM . . . . . . . . . . . . . . . 536 11.2 MARKOV CHAIN MONTE CARLO . . . . . . . . . . . . . . . . . . . . . 537 11.2.1 MARKOV CHAINS . . . . . . . . . . . . . . . . . . . . . . . . 539 11.2.2 THE METROPOLIS-HASTINGS ALGORITHM . . . . . . . . . . . . . 541 11.3 GIBBS SAMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 11.4 SLICE SAMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 11.5 THE HYBRID MONTE CARLO ALGORITHM . . . . . . . . . . . . . . . . . 548 11.5.1 DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . 548 11.5.2 HYBRID MONTE CARLO . . . . . . . . . . . . . . . . . . . . . 552 11.6 ESTIMATING THE PARTITION FUNCTION . . . . . . . . . . . . . . . . . . 554 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 12 CONTINUOUS LATENT VARIABLES 559 12.1 PRINCIPAL COMPONENT ANALYSIS . . . . . . . . . . . . . . . . . . . . 561 12.1.1 MAXIMUM VARIANCE FORMULATION . . . . . . . . . . . . . . . 561 12.1.2 MINIMUM-ERROR FORMULATION . . . . . . . . . . . . . . . . . 563 12.1.3 APPLICATIONS OF PCA . . . . . . . . . . . . . . . . . . . . . 565 12.1.4 PCA FOR HIGH-DIMENSIONAL DATA . . . . . . . . . . . . . . . 569 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML CONTENTS XIX 12.2 PROBABILISTIC PCA . . . . . . . . . . . . . . . . . . . . . . . . . . 570 12.2.1 MAXIMUM LIKELIHOOD PCA . . . . . . . . . . . . . . . . . . 574 12.2.2 EM ALGORITHM FOR PCA . . . . . . . . . . . . . . . . . . . . 577 12.2.3 BAYESIAN PCA . . . . . . . . . . . . . . . . . . . . . . . . 580 12.2.4 FACTOR ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . 583 12.3 KERNEL PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 12.4 NONLINEAR LATENT VARIABLE MODELS . . . . . . . . . . . . . . . . . . 591 12.4.1 INDEPENDENT COMPONENT ANALYSIS . . . . . . . . . . . . . . . 591 12.4.2 AUTOASSOCIATIVE NEURAL NETWORKS . . . . . . . . . . . . . . . 592 12.4.3 MODELLING NONLINEAR MANIFOLDS . . . . . . . . . . . . . . . . 595 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 13 SEQUENTIAL DATA 605 13.1 MARKOV MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 13.2 HIDDEN MARKOV MODELS . . . . . . . . . . . . . . . . . . . . . . . 610 13.2.1 MAXIMUM LIKELIHOOD FOR THE HMM . . . . . . . . . . . . . 615 13.2.2 THE FORWARD-BACKWARD ALGORITHM . . . . . . . . . . . . . . 618 13.2.3 THE SUM-PRODUCT ALGORITHM FOR THE HMM . . . . . . . . . . 625 13.2.4 SCALING FACTORS . . . . . . . . . . . . . . . . . . . . . . . . 627 13.2.5 THE VITERBI ALGORITHM . . . . . . . . . . . . . . . . . . . . . 629 13.2.6 EXTENSIONS OF THE HIDDEN MARKOV MODEL . . . . . . . . . . . 631 13.3 LINEAR DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . 635 13.3.1 INFERENCE IN LDS . . . . . . . . . . . . . . . . . . . . . . . 638 13.3.2 LEARNING IN LDS . . . . . . . . . . . . . . . . . . . . . . . 642 13.3.3 EXTENSIONS OF LDS . . . . . . . . . . . . . . . . . . . . . . 644 13.3.4 PARTICLE FILTERS . . . . . . . . . . . . . . . . . . . . . . . . . 645 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 14 COMBINING MODELS 653 14.1 BAYESIAN MODEL AVERAGING . . . . . . . . . . . . . . . . . . . . . . 654 14.2 COMMITTEES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 14.3 BOOSTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 14.3.1 MINIMIZING EXPONENTIAL ERROR . . . . . . . . . . . . . . . . 659 14.3.2 ERROR FUNCTIONS FOR BOOSTING . . . . . . . . . . . . . . . . . 661 14.4 TREE-BASED MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . 663 14.5 CONDITIONAL MIXTURE MODELS . . . . . . . . . . . . . . . . . . . . . 666 14.5.1 MIXTURES OF LINEAR REGRESSION MODELS . . . . . . . . . . . . . 667 14.5.2 MIXTURES OF LOGISTIC MODELS . . . . . . . . . . . . . . . . . 670 14.5.3 MIXTURES OF EXPERTS . . . . . . . . . . . . . . . . . . . . . . 672 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 APPENDIX A DATA SETS 677 APPENDIX B PROBABILITY DISTRIBUTIONS 685 APPENDIX C PROPERTIES OF MATRICES 695 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML XX CONTENTS APPENDIX D CALCULUS OF VARIATIONS 703 APPENDIX E LAGRANGE MULTIPLIERS 707 REFERENCES 711 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML
adam_txt	CONTENTS PREFACE VII MATHEMATICAL NOTATION XI CONTENTS XIII 1 INTRODUCTION 1 1.1 EXAMPLE: POLYNOMIAL CURVE FITTING . . . . . . . . . . . . . . . . . 4 1.2 PROBABILITY THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 PROBABILITY DENSITIES . . . . . . . . . . . . . . . . . . . . . 17 1.2.2 EXPECTATIONS AND COVARIANCES . . . . . . . . . . . . . . . . 19 1.2.3 BAYESIAN PROBABILITIES . . . . . . . . . . . . . . . . . . . . 21 1.2.4 THE GAUSSIAN DISTRIBUTION . . . . . . . . . . . . . . . . . . 24 1.2.5 CURVE FITTING RE-VISITED . . . . . . . . . . . . . . . . . . . . 28 1.2.6 BAYESIAN CURVE FITTING . . . . . . . . . . . . . . . . . . . . 30 1.3 MODEL SELECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4 THE CURSE OF DIMENSIONALITY . . . . . . . . . . . . . . . . . . . . . 33 1.5 DECISION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5.1 MINIMIZING THE MISCLASSIFICATION RATE . . . . . . . . . . . . 39 1.5.2 MINIMIZING THE EXPECTED LOSS . . . . . . . . . . . . . . . . 41 1.5.3 THE REJECT OPTION . . . . . . . . . . . . . . . . . . . . . . . 42 1.5.4 INFERENCE AND DECISION . . . . . . . . . . . . . . . . . . . . 42 1.5.5 LOSS FUNCTIONS FOR REGRESSION . . . . . . . . . . . . . . . . . 46 1.6 INFORMATION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.6.1 RELATIVE ENTROPY AND MUTUAL INFORMATION . . . . . . . . . . 55 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML XIII XIV CONTENTS 2 PROBABILITY DISTRIBUTIONS 67 2.1 BINARY VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.1.1 THE BETA DISTRIBUTION . . . . . . . . . . . . . . . . . . . . . 71 2.2 MULTINOMIAL VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . 74 2.2.1 THE DIRICHLET DISTRIBUTION . . . . . . . . . . . . . . . . . . . 76 2.3 THE GAUSSIAN DISTRIBUTION . . . . . . . . . . . . . . . . . . . . . . 78 2.3.1 CONDITIONAL GAUSSIAN DISTRIBUTIONS . . . . . . . . . . . . . . 85 2.3.2 MARGINAL GAUSSIAN DISTRIBUTIONS . . . . . . . . . . . . . . . 88 2.3.3 BAYES* THEOREM FOR GAUSSIAN VARIABLES . . . . . . . . . . . . 90 2.3.4 MAXIMUM LIKELIHOOD FOR THE GAUSSIAN . . . . . . . . . . . . 93 2.3.5 SEQUENTIAL ESTIMATION . . . . . . . . . . . . . . . . . . . . . 94 2.3.6 BAYESIAN INFERENCE FOR THE GAUSSIAN . . . . . . . . . . . . . 97 2.3.7 STUDENTS T-DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 102 2.3.8 PERIODIC VARIABLES . . . . . . . . . . . . . . . . . . . . . . . 105 2.3.9 MIXTURES OF GAUSSIANS . . . . . . . . . . . . . . . . . . . . 110 2.4 THE EXPONENTIAL FAMILY . . . . . . . . . . . . . . . . . . . . . . . 113 2.4.1 MAXIMUM LIKELIHOOD AND SUFFICIENT STATISTICS . . . . . . . . 116 2.4.2 CONJUGATE PRIORS . . . . . . . . . . . . . . . . . . . . . . . 117 2.4.3 NONINFORMATIVE PRIORS . . . . . . . . . . . . . . . . . . . . 117 2.5 NONPARAMETRIC METHODS . . . . . . . . . . . . . . . . . . . . . . . 120 2.5.1 KERNEL DENSITY ESTIMATORS . . . . . . . . . . . . . . . . . . . 122 2.5.2 NEAREST-NEIGHBOUR METHODS . . . . . . . . . . . . . . . . . 124 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3 LINEAR MODELS FOR REGRESSION 137 3.1 LINEAR BASIS FUNCTION MODELS . . . . . . . . . . . . . . . . . . . . 138 3.1.1 MAXIMUM LIKELIHOOD AND LEAST SQUARES . . . . . . . . . . . . 140 3.1.2 GEOMETRY OF LEAST SQUARES . . . . . . . . . . . . . . . . . . 143 3.1.3 SEQUENTIAL LEARNING . . . . . . . . . . . . . . . . . . . . . . 143 3.1.4 REGULARIZED LEAST SQUARES . . . . . . . . . . . . . . . . . . . 144 3.1.5 MULTIPLE OUTPUTS . . . . . . . . . . . . . . . . . . . . . . . 146 3.2 THE BIAS-VARIANCE DECOMPOSITION . . . . . . . . . . . . . . . . . . 147 3.3 BAYESIAN LINEAR REGRESSION . . . . . . . . . . . . . . . . . . . . . 152 3.3.1 PARAMETER DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 152 3.3.2 PREDICTIVE DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 156 3.3.3 EQUIVALENT KERNEL . . . . . . . . . . . . . . . . . . . . . . . 159 3.4 BAYESIAN MODEL COMPARISON . . . . . . . . . . . . . . . . . . . . . 161 3.5 THE EVIDENCE APPROXIMATION . . . . . . . . . . . . . . . . . . . . 165 3.5.1 EVALUATION OF THE EVIDENCE FUNCTION . . . . . . . . . . . . . 166 3.5.2 MAXIMIZING THE EVIDENCE FUNCTION . . . . . . . . . . . . . . 168 3.5.3 EFFECTIVE NUMBER OF PARAMETERS . . . . . . . . . . . . . . . 170 3.6 LIMITATIONS OF FIXED BASIS FUNCTIONS . . . . . . . . . . . . . . . . 172 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 C CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML CONTENTS XV 4 LINEAR MODELS FOR CLASSIFICATION 179 4.1 DISCRIMINANT FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . 181 4.1.1 TWO CLASSES . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.1.2 MULTIPLE CLASSES . . . . . . . . . . . . . . . . . . . . . . . . 182 4.1.3 LEAST SQUARES FOR CLASSIFICATION . . . . . . . . . . . . . . . . 184 4.1.4 FISHERS LINEAR DISCRIMINANT . . . . . . . . . . . . . . . . . . 186 4.1.5 RELATION TO LEAST SQUARES . . . . . . . . . . . . . . . . . . . 189 4.1.6 FISHERS DISCRIMINANT FOR MULTIPLE CLASSES . . . . . . . . . . 191 4.1.7 THE PERCEPTRON ALGORITHM . . . . . . . . . . . . . . . . . . . 192 4.2 PROBABILISTIC GENERATIVE MODELS . . . . . . . . . . . . . . . . . . . 196 4.2.1 CONTINUOUS INPUTS . . . . . . . . . . . . . . . . . . . . . . 198 4.2.2 MAXIMUM LIKELIHOOD SOLUTION . . . . . . . . . . . . . . . . 200 4.2.3 DISCRETE FEATURES . . . . . . . . . . . . . . . . . . . . . . . 202 4.2.4 EXPONENTIAL FAMILY . . . . . . . . . . . . . . . . . . . . . . 202 4.3 PROBABILISTIC DISCRIMINATIVE MODELS . . . . . . . . . . . . . . . . . 203 4.3.1 FIXED BASIS FUNCTIONS . . . . . . . . . . . . . . . . . . . . . 204 4.3.2 LOGISTIC REGRESSION . . . . . . . . . . . . . . . . . . . . . . 205 4.3.3 ITERATIVE REWEIGHTED LEAST SQUARES . . . . . . . . . . . . . . 207 4.3.4 MULTICLASS LOGISTIC REGRESSION . . . . . . . . . . . . . . . . . 209 4.3.5 PROBIT REGRESSION . . . . . . . . . . . . . . . . . . . . . . . 210 4.3.6 CANONICAL LINK FUNCTIONS . . . . . . . . . . . . . . . . . . . 212 4.4 THE LAPLACE APPROXIMATION . . . . . . . . . . . . . . . . . . . . . 213 4.4.1 MODEL COMPARISON AND BIC . . . . . . . . . . . . . . . . . 216 4.5 BAYESIAN LOGISTIC REGRESSION . . . . . . . . . . . . . . . . . . . . 217 4.5.1 LAPLACE APPROXIMATION . . . . . . . . . . . . . . . . . . . . 217 4.5.2 PREDICTIVE DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 218 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5 NEURAL NETWORKS 225 5.1 FEED-FORWARD NETWORK FUNCTIONS . . . . . . . . . . . . . . . . . . 227 5.1.1 WEIGHT-SPACE SYMMETRIES . . . . . . . . . . . . . . . . . . 231 5.2 NETWORK TRAINING . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.2.1 PARAMETER OPTIMIZATION . . . . . . . . . . . . . . . . . . . . 236 5.2.2 LOCAL QUADRATIC APPROXIMATION . . . . . . . . . . . . . . . . 237 5.2.3 USE OF GRADIENT INFORMATION . . . . . . . . . . . . . . . . . 239 5.2.4 GRADIENT DESCENT OPTIMIZATION . . . . . . . . . . . . . . . . 240 5.3 ERROR BACKPROPAGATION . . . . . . . . . . . . . . . . . . . . . . . . 241 5.3.1 EVALUATION OF ERROR-FUNCTION DERIVATIVES . . . . . . . . . . . 242 5.3.2 A SIMPLE EXAMPLE . . . . . . . . . . . . . . . . . . . . . . 245 5.3.3 EFFICIENCY OF BACKPROPAGATION . . . . . . . . . . . . . . . . 246 5.3.4 THE JACOBIAN MATRIX . . . . . . . . . . . . . . . . . . . . . 247 5.4 THE HESSIAN MATRIX . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.4.1 DIAGONAL APPROXIMATION . . . . . . . . . . . . . . . . . . . 250 5.4.2 OUTER PRODUCT APPROXIMATION . . . . . . . . . . . . . . . . . 251 5.4.3 INVERSE HESSIAN . . . . . . . . . . . . . . . . . . . . . . . . 252 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML XVI CONTENTS 5.4.4 FINITE DIFFERENCES . . . . . . . . . . . . . . . . . . . . . . . 252 5.4.5 EXACT EVALUATION OF THE HESSIAN . . . . . . . . . . . . . . . 253 5.4.6 FAST MULTIPLICATION BY THE HESSIAN . . . . . . . . . . . . . . 254 5.5 REGULARIZATION IN NEURAL NETWORKS . . . . . . . . . . . . . . . . . 256 5.5.1 CONSISTENT GAUSSIAN PRIORS . . . . . . . . . . . . . . . . . . 257 5.5.2 EARLY STOPPING . . . . . . . . . . . . . . . . . . . . . . . . 259 5.5.3 INVARIANCES . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.5.4 TANGENT PROPAGATION . . . . . . . . . . . . . . . . . . . . . 263 5.5.5 TRAINING WITH TRANSFORMED DATA . . . . . . . . . . . . . . . . 265 5.5.6 CONVOLUTIONAL NETWORKS . . . . . . . . . . . . . . . . . . . 267 5.5.7 SOFT WEIGHT SHARING . . . . . . . . . . . . . . . . . . . . . . 269 5.6 MIXTURE DENSITY NETWORKS . . . . . . . . . . . . . . . . . . . . . . 272 5.7 BAYESIAN NEURAL NETWORKS . . . . . . . . . . . . . . . . . . . . . . 277 5.7.1 POSTERIOR PARAMETER DISTRIBUTION . . . . . . . . . . . . . . . 278 5.7.2 HYPERPARAMETER OPTIMIZATION . . . . . . . . . . . . . . . . 280 5.7.3 BAYESIAN NEURAL NETWORKS FOR CLASSIFICATION . . . . . . . . . 281 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 6 KERNEL METHODS 291 6.1 DUAL REPRESENTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.2 CONSTRUCTING KERNELS . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.3 RADIAL BASIS FUNCTION NETWORKS . . . . . . . . . . . . . . . . . . . 299 6.3.1 NADARAYA-WATSON MODEL . . . . . . . . . . . . . . . . . . . 301 6.4 GAUSSIAN PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . 303 6.4.1 LINEAR REGRESSION REVISITED . . . . . . . . . . . . . . . . . . 304 6.4.2 GAUSSIAN PROCESSES FOR REGRESSION . . . . . . . . . . . . . . 306 6.4.3 LEARNING THE HYPERPARAMETERS . . . . . . . . . . . . . . . . 311 6.4.4 AUTOMATIC RELEVANCE DETERMINATION . . . . . . . . . . . . . 312 6.4.5 GAUSSIAN PROCESSES FOR CLASSIFICATION . . . . . . . . . . . . . 313 6.4.6 LAPLACE APPROXIMATION . . . . . . . . . . . . . . . . . . . . 315 6.4.7 CONNECTION TO NEURAL NETWORKS . . . . . . . . . . . . . . . . 319 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 7 SPARSE KERNEL MACHINES 325 7.1 MAXIMUM MARGIN CLASSIFIERS . . . . . . . . . . . . . . . . . . . . 326 7.1.1 OVERLAPPING CLASS DISTRIBUTIONS . . . . . . . . . . . . . . . . 331 7.1.2 RELATION TO LOGISTIC REGRESSION . . . . . . . . . . . . . . . . 336 7.1.3 MULTICLASS SVMS . . . . . . . . . . . . . . . . . . . . . . . 338 7.1.4 SVMS FOR REGRESSION . . . . . . . . . . . . . . . . . . . . . 339 7.1.5 COMPUTATIONAL LEARNING THEORY . . . . . . . . . . . . . . . . 344 7.2 RELEVANCE VECTOR MACHINES . . . . . . . . . . . . . . . . . . . . . 345 7.2.1 RVM FOR REGRESSION . . . . . . . . . . . . . . . . . . . . . . 345 7.2.2 ANALYSIS OF SPARSITY . . . . . . . . . . . . . . . . . . . . . . 349 7.2.3 RVM FOR CLASSIFICATION . . . . . . . . . . . . . . . . . . . . 353 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML CONTENTS XVII 8 GRAPHICAL MODELS 359 8.1 BAYESIAN NETWORKS . . . . . . . . . . . . . . . . . . . . . . . . . . 360 8.1.1 EXAMPLE: POLYNOMIAL REGRESSION . . . . . . . . . . . . . . . 362 8.1.2 GENERATIVE MODELS . . . . . . . . . . . . . . . . . . . . . . 365 8.1.3 DISCRETE VARIABLES . . . . . . . . . . . . . . . . . . . . . . . 366 8.1.4 LINEAR-GAUSSIAN MODELS . . . . . . . . . . . . . . . . . . . 370 8.2 CONDITIONAL INDEPENDENCE . . . . . . . . . . . . . . . . . . . . . . 372 8.2.1 THREE EXAMPLE GRAPHS . . . . . . . . . . . . . . . . . . . . 373 8.2.2 D-SEPARATION . . . . . . . . . . . . . . . . . . . . . . . . . 378 8.3 MARKOV RANDOM FIELDS . . . . . . . . . . . . . . . . . . . . . . . 383 8.3.1 CONDITIONAL INDEPENDENCE PROPERTIES . . . . . . . . . . . . . 383 8.3.2 FACTORIZATION PROPERTIES . . . . . . . . . . . . . . . . . . . 384 8.3.3 ILLUSTRATION: IMAGE DE-NOISING . . . . . . . . . . . . . . . . 387 8.3.4 RELATION TO DIRECTED GRAPHS . . . . . . . . . . . . . . . . . . 390 8.4 INFERENCE IN GRAPHICAL MODELS . . . . . . . . . . . . . . . . . . . . 393 8.4.1 INFERENCE ON A CHAIN . . . . . . . . . . . . . . . . . . . . . 394 8.4.2 TREES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 8.4.3 FACTOR GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . 399 8.4.4 THE SUM-PRODUCT ALGORITHM . . . . . . . . . . . . . . . . . . 402 8.4.5 THE MAX-SUM ALGORITHM . . . . . . . . . . . . . . . . . . . 411 8.4.6 EXACT INFERENCE IN GENERAL GRAPHS . . . . . . . . . . . . . . 416 8.4.7 LOOPY BELIEF PROPAGATION . . . . . . . . . . . . . . . . . . . 417 8.4.8 LEARNING THE GRAPH STRUCTURE . . . . . . . . . . . . . . . . . 418 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 9 MIXTURE MODELS AND EM 423 9.1 K -MEANS CLUSTERING . . . . . . . . . . . . . . . . . . . . . . . . . 424 9.1.1 IMAGE SEGMENTATION AND COMPRESSION . . . . . . . . . . . . 428 9.2 MIXTURES OF GAUSSIANS . . . . . . . . . . . . . . . . . . . . . . . . 430 9.2.1 MAXIMUM LIKELIHOOD . . . . . . . . . . . . . . . . . . . . . 432 9.2.2 EM FOR GAUSSIAN MIXTURES . . . . . . . . . . . . . . . . . . 435 9.3 AN ALTERNATIVE VIEW OF EM . . . . . . . . . . . . . . . . . . . . . 439 9.3.1 GAUSSIAN MIXTURES REVISITED . . . . . . . . . . . . . . . . . 441 9.3.2 RELATION TO K -MEANS . . . . . . . . . . . . . . . . . . . . . 443 9.3.3 MIXTURES OF BERNOULLI DISTRIBUTIONS . . . . . . . . . . . . . . 444 9.3.4 EM FOR BAYESIAN LINEAR REGRESSION . . . . . . . . . . . . . . 448 9.4 THE EM ALGORITHM IN GENERAL . . . . . . . . . . . . . . . . . . . . 450 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 10 APPROXIMATE INFERENCE 461 10.1 VARIATIONAL INFERENCE . . . . . . . . . . . . . . . . . . . . . . . . . 462 10.1.1 FACTORIZED DISTRIBUTIONS . . . . . . . . . . . . . . . . . . . . 464 10.1.2 PROPERTIES OF FACTORIZED APPROXIMATIONS . . . . . . . . . . . 466 10.1.3 EXAMPLE: THE UNIVARIATE GAUSSIAN . . . . . . . . . . . . . . 470 10.1.4 MODEL COMPARISON . . . . . . . . . . . . . . . . . . . . . . 473 10.2 ILLUSTRATION: VARIATIONAL MIXTURE OF GAUSSIANS . . . . . . . . . . . . 474 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML XVIII CONTENTS 10.2.1 VARIATIONAL DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 475 10.2.2 VARIATIONAL LOWER BOUND . . . . . . . . . . . . . . . . . . . 481 10.2.3 PREDICTIVE DENSITY . . . . . . . . . . . . . . . . . . . . . . . 482 10.2.4 DETERMINING THE NUMBER OF COMPONENTS . . . . . . . . . . . 483 10.2.5 INDUCED FACTORIZATIONS . . . . . . . . . . . . . . . . . . . . 485 10.3 VARIATIONAL LINEAR REGRESSION . . . . . . . . . . . . . . . . . . . . 486 10.3.1 VARIATIONAL DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 486 10.3.2 PREDICTIVE DISTRIBUTION . . . . . . . . . . . . . . . . . . . . 488 10.3.3 LOWER BOUND . . . . . . . . . . . . . . . . . . . . . . . . . 489 10.4 EXPONENTIAL FAMILY DISTRIBUTIONS . . . . . . . . . . . . . . . . . . 490 10.4.1 VARIATIONAL MESSAGE PASSING . . . . . . . . . . . . . . . . . 491 10.5 LOCAL VARIATIONAL METHODS . . . . . . . . . . . . . . . . . . . . . . 493 10.6 VARIATIONAL LOGISTIC REGRESSION . . . . . . . . . . . . . . . . . . . 498 10.6.1 VARIATIONAL POSTERIOR DISTRIBUTION . . . . . . . . . . . . . . . 498 10.6.2 OPTIMIZING THE VARIATIONAL PARAMETERS . . . . . . . . . . . . 500 10.6.3 INFERENCE OF HYPERPARAMETERS . . . . . . . . . . . . . . . . 502 10.7 EXPECTATION PROPAGATION . . . . . . . . . . . . . . . . . . . . . . . 505 10.7.1 EXAMPLE: THE CLUTTER PROBLEM . . . . . . . . . . . . . . . . 511 10.7.2 EXPECTATION PROPAGATION ON GRAPHS . . . . . . . . . . . . . . 513 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 11 SAMPLING METHODS 523 11.1 BASIC SAMPLING ALGORITHMS . . . . . . . . . . . . . . . . . . . . . 526 11.1.1 STANDARD DISTRIBUTIONS . . . . . . . . . . . . . . . . . . . . 526 11.1.2 REJECTION SAMPLING . . . . . . . . . . . . . . . . . . . . . . 528 11.1.3 ADAPTIVE REJECTION SAMPLING . . . . . . . . . . . . . . . . . 530 11.1.4 IMPORTANCE SAMPLING . . . . . . . . . . . . . . . . . . . . . 532 11.1.5 SAMPLING-IMPORTANCE-RESAMPLING . . . . . . . . . . . . . . 534 11.1.6 SAMPLING AND THE EM ALGORITHM . . . . . . . . . . . . . . . 536 11.2 MARKOV CHAIN MONTE CARLO . . . . . . . . . . . . . . . . . . . . . 537 11.2.1 MARKOV CHAINS . . . . . . . . . . . . . . . . . . . . . . . . 539 11.2.2 THE METROPOLIS-HASTINGS ALGORITHM . . . . . . . . . . . . . 541 11.3 GIBBS SAMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 11.4 SLICE SAMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 11.5 THE HYBRID MONTE CARLO ALGORITHM . . . . . . . . . . . . . . . . . 548 11.5.1 DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . 548 11.5.2 HYBRID MONTE CARLO . . . . . . . . . . . . . . . . . . . . . 552 11.6 ESTIMATING THE PARTITION FUNCTION . . . . . . . . . . . . . . . . . . 554 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 12 CONTINUOUS LATENT VARIABLES 559 12.1 PRINCIPAL COMPONENT ANALYSIS . . . . . . . . . . . . . . . . . . . . 561 12.1.1 MAXIMUM VARIANCE FORMULATION . . . . . . . . . . . . . . . 561 12.1.2 MINIMUM-ERROR FORMULATION . . . . . . . . . . . . . . . . . 563 12.1.3 APPLICATIONS OF PCA . . . . . . . . . . . . . . . . . . . . . 565 12.1.4 PCA FOR HIGH-DIMENSIONAL DATA . . . . . . . . . . . . . . . 569 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML CONTENTS XIX 12.2 PROBABILISTIC PCA . . . . . . . . . . . . . . . . . . . . . . . . . . 570 12.2.1 MAXIMUM LIKELIHOOD PCA . . . . . . . . . . . . . . . . . . 574 12.2.2 EM ALGORITHM FOR PCA . . . . . . . . . . . . . . . . . . . . 577 12.2.3 BAYESIAN PCA . . . . . . . . . . . . . . . . . . . . . . . . 580 12.2.4 FACTOR ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . 583 12.3 KERNEL PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 12.4 NONLINEAR LATENT VARIABLE MODELS . . . . . . . . . . . . . . . . . . 591 12.4.1 INDEPENDENT COMPONENT ANALYSIS . . . . . . . . . . . . . . . 591 12.4.2 AUTOASSOCIATIVE NEURAL NETWORKS . . . . . . . . . . . . . . . 592 12.4.3 MODELLING NONLINEAR MANIFOLDS . . . . . . . . . . . . . . . . 595 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 13 SEQUENTIAL DATA 605 13.1 MARKOV MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 13.2 HIDDEN MARKOV MODELS . . . . . . . . . . . . . . . . . . . . . . . 610 13.2.1 MAXIMUM LIKELIHOOD FOR THE HMM . . . . . . . . . . . . . 615 13.2.2 THE FORWARD-BACKWARD ALGORITHM . . . . . . . . . . . . . . 618 13.2.3 THE SUM-PRODUCT ALGORITHM FOR THE HMM . . . . . . . . . . 625 13.2.4 SCALING FACTORS . . . . . . . . . . . . . . . . . . . . . . . . 627 13.2.5 THE VITERBI ALGORITHM . . . . . . . . . . . . . . . . . . . . . 629 13.2.6 EXTENSIONS OF THE HIDDEN MARKOV MODEL . . . . . . . . . . . 631 13.3 LINEAR DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . 635 13.3.1 INFERENCE IN LDS . . . . . . . . . . . . . . . . . . . . . . . 638 13.3.2 LEARNING IN LDS . . . . . . . . . . . . . . . . . . . . . . . 642 13.3.3 EXTENSIONS OF LDS . . . . . . . . . . . . . . . . . . . . . . 644 13.3.4 PARTICLE FILTERS . . . . . . . . . . . . . . . . . . . . . . . . . 645 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 14 COMBINING MODELS 653 14.1 BAYESIAN MODEL AVERAGING . . . . . . . . . . . . . . . . . . . . . . 654 14.2 COMMITTEES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 14.3 BOOSTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 14.3.1 MINIMIZING EXPONENTIAL ERROR . . . . . . . . . . . . . . . . 659 14.3.2 ERROR FUNCTIONS FOR BOOSTING . . . . . . . . . . . . . . . . . 661 14.4 TREE-BASED MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . 663 14.5 CONDITIONAL MIXTURE MODELS . . . . . . . . . . . . . . . . . . . . . 666 14.5.1 MIXTURES OF LINEAR REGRESSION MODELS . . . . . . . . . . . . . 667 14.5.2 MIXTURES OF LOGISTIC MODELS . . . . . . . . . . . . . . . . . 670 14.5.3 MIXTURES OF EXPERTS . . . . . . . . . . . . . . . . . . . . . . 672 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 APPENDIX A DATA SETS 677 APPENDIX B PROBABILITY DISTRIBUTIONS 685 APPENDIX C PROPERTIES OF MATRICES 695 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML XX CONTENTS APPENDIX D CALCULUS OF VARIATIONS 703 APPENDIX E LAGRANGE MULTIPLIERS 707 REFERENCES 711 C * CHRISTPHER M. BISHOP (20022006). SPRINGER, 2006. FIRST PRINTING. FURTHER INFORMATION AVAILABLE AT HTTP://RESEARCH.MICROSOFT.COM/ CMBISHOP/PRML
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id	DE-604.BV021648269
illustrated	Illustrated
index_date	2024-07-02T15:02:12Z
indexdate	2025-03-12T09:00:33Z
institution	BVB
isbn	0387310738 9780387310732
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-014862953
oclc_num	254896146
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owner	DE-91G DE-BY-TUM DE-703 DE-20 DE-473 DE-BY-UBG DE-355 DE-BY-UBR DE-739 DE-860 DE-384 DE-706 DE-573 DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-945 DE-898 DE-BY-UBR DE-861 DE-29T DE-521 DE-83 DE-634 DE-11 DE-525 DE-188 DE-M382
owner_facet	DE-91G DE-BY-TUM DE-703 DE-20 DE-473 DE-BY-UBG DE-355 DE-BY-UBR DE-739 DE-860 DE-384 DE-706 DE-573 DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-945 DE-898 DE-BY-UBR DE-861 DE-29T DE-521 DE-83 DE-634 DE-11 DE-525 DE-188 DE-M382
physical	XX, 738 S. Ill., graph. Darst.
publishDate	2006
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publisher	Springer
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series2	Information science and statistics
spelling	Bishop, Christopher M. 1959- Verfasser (DE-588)120454165 aut Pattern recognition and machine learning Christopher M. Bishop New York, NY Springer 2006 XX, 738 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Information science and statistics Literaturverzeichnis Seite 711 - 728 Hier auch später erschienene, unveränderte Nachdrucke Maschinelles Lernen Mustererkennung Machine learning Pattern perception Mustererkennung (DE-588)4040936-3 gnd rswk-swf Maschinelles Lernen (DE-588)4193754-5 gnd rswk-swf Mustererkennung (DE-588)4040936-3 s Maschinelles Lernen (DE-588)4193754-5 s DE-604 Äquivalent Druck-Ausgabe, Paperback 2016 978-1-4939-3843-8 (DE-604)BV044802275 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2718189&prov=M&dok_var=1&dok_ext=htm Beschreibung text/html http://www.loc.gov/catdir/enhancements/fy0818/2006922522-d.html Publisher description text/html http://www.loc.gov/catdir/enhancements/fy0818/2006922522-t.html Table of contents only http://www3.ub.tu-berlin.de/ihv/001716289.pdf Inhaltsverzeichnis SWBplus Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014862953&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Bishop, Christopher M. 1959- Pattern recognition and machine learning Maschinelles Lernen Mustererkennung Machine learning Pattern perception Mustererkennung (DE-588)4040936-3 gnd Maschinelles Lernen (DE-588)4193754-5 gnd
subject_GND	(DE-588)4040936-3 (DE-588)4193754-5
title	Pattern recognition and machine learning
title_auth	Pattern recognition and machine learning
title_exact_search	Pattern recognition and machine learning
title_exact_search_txtP	Pattern recognition and machine learning
title_full	Pattern recognition and machine learning Christopher M. Bishop
title_fullStr	Pattern recognition and machine learning Christopher M. Bishop
title_full_unstemmed	Pattern recognition and machine learning Christopher M. Bishop
title_short	Pattern recognition and machine learning
title_sort	pattern recognition and machine learning
topic	Maschinelles Lernen Mustererkennung Machine learning Pattern perception Mustererkennung (DE-588)4040936-3 gnd Maschinelles Lernen (DE-588)4193754-5 gnd
topic_facet	Maschinelles Lernen Mustererkennung Machine learning Pattern perception
url	http://deposit.dnb.de/cgi-bin/dokserv?id=2718189&prov=M&dok_var=1&dok_ext=htm http://www.loc.gov/catdir/enhancements/fy0818/2006922522-d.html http://www.loc.gov/catdir/enhancements/fy0818/2006922522-t.html http://www3.ub.tu-berlin.de/ihv/001716289.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014862953&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT bishopchristopherm patternrecognitionandmachinelearning

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