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 2007
Edition:	Corr. print.
Series:	Information science and statistics
Subjects:	Makine öğrenimi Örüntü algısı Örüntü tanıma sistemleri Machine learning Pattern perception Pattern recognition systems Mustererkennung Maschinelles Lernen
Online Access:	Inhaltsverzeichnis
Physical Description:	XX, 738 S. Ill., graph. Darst.
ISBN:	0387310738 9780387310732

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Record in the Search Index

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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 xiii xiv CONTENTS 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 Student s 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 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 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 Fisher s linear discriminant .................. 186 4.1.5 Relation to least squares ................... 189 4.1.6 Fisher s 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 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 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 AT-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 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 Lowerbound ......................... 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 SampHng-importance-resarnpling .............. 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 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 В Probability Distributions 685 Appendix С Properties of Matrices 695 xx CONTENTS Appendix D Calculus of Variations 703 Appendix E Lagrange Multipliers 707 References 711 Index 729
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 xiii xiv CONTENTS 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 Student's 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 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 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 Fisher's linear discriminant . 186 4.1.5 Relation to least squares . 189 4.1.6 Fisher's 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 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 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 AT-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 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 Lowerbound . 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 SampHng-importance-resarnpling . 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 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 В Probability Distributions 685 Appendix С Properties of Matrices 695 xx CONTENTS Appendix D Calculus of Variations 703 Appendix E Lagrange Multipliers 707 References 711 Index 729
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id	DE-604.BV023326998
illustrated	Illustrated
index_date	2024-07-02T20:56:00Z
indexdate	2024-07-09T21:15:57Z
institution	BVB
isbn	0387310738 9780387310732
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016510988
oclc_num	228415823
open_access_boolean
owner	DE-91G DE-BY-TUM DE-824 DE-20 DE-355 DE-BY-UBR DE-703 DE-19 DE-BY-UBM DE-384 DE-91 DE-BY-TUM DE-739 DE-83
owner_facet	DE-91G DE-BY-TUM DE-824 DE-20 DE-355 DE-BY-UBR DE-703 DE-19 DE-BY-UBM DE-384 DE-91 DE-BY-TUM DE-739 DE-83
physical	XX, 738 S. Ill., graph. Darst.
publishDate	2007
publishDateSearch	2007
publishDateSort	2007
publisher	Springer
record_format	marc
series2	Information science and statistics
spelling	Bishop, Christopher M. 1959- Verfasser (DE-588)120454165 aut Pattern recognition and machine learning Christopher M. Bishop Corr. print. New York, NY Springer 2007 XX, 738 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Information science and statistics Makine öğrenimi Örüntü algısı Örüntü tanıma sistemleri Machine learning Pattern perception Pattern recognition systems 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 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016510988&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Bishop, Christopher M. 1959- Pattern recognition and machine learning Makine öğrenimi Örüntü algısı Örüntü tanıma sistemleri Machine learning Pattern perception Pattern recognition systems 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	Makine öğrenimi Örüntü algısı Örüntü tanıma sistemleri Machine learning Pattern perception Pattern recognition systems Mustererkennung (DE-588)4040936-3 gnd Maschinelles Lernen (DE-588)4193754-5 gnd
topic_facet	Makine öğrenimi Örüntü algısı Örüntü tanıma sistemleri Machine learning Pattern perception Pattern recognition systems Mustererkennung Maschinelles Lernen
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016510988&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT bishopchristopherm patternrecognitionandmachinelearning

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