章节目录
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 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 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 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 . . . . . . . . . . . . . . . . . . . . 153 3.3.2 Predictive distribution . . . . . . . . . . . . . . . . . . . . 156 3.3.3 Equivalent kernel . . . . . . . . . . . . . . . . . . . . . . . 157 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 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 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 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 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 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 Appendix D Calculus of Variations 703 Appendix E LagrangeMultipliers 707 References 711
内容简介
The dramatic growth in practical applications for machine learning over the last ten years has been accompanied by many important developments in the underlying algorithms and techniques. For example, Bayesian methods have grown from a specialist niche to become mainstream, while graphical models have emerged as a general framework for describing and applying probabilistic techniques. The practical applicability of Bayesian methods has been greatly enhanced by the development of a range of approximate inference algorithms such as variational Bayes and expectation propagation, while new models based on kernels have had a significant impact on both algorithms and applications. This completely new textbook reflects these recent developments while providing a comprehensive introduction to the fields of pattern recognition and machine learning. It is aimed at advanced undergraduates or first-year PhD students, as well as researchers and practitioners. No previous knowledge of pattern recognition or machine learning concepts is assumed. Familiarity with multivariate calculus and basic linear algebra is required, and some experience in the use of probabilities would be helpful though not essential as the book includes a self-contained introduction to basic probability theory. The book is suitable for courses on machine learning, statistics, computer science, signal processing, computer vision, data mining, and bioinformatics. Extensive support is provided for course instructors, including more than 400 exercises, graded according to difficulty. Example solutions for a subset of the exercises are available from the book web site, while solutions for the remainder can be obtained by instructors from the publisher. The book is supported by a great deal of additional material, and the reader is encouraged to visit the book web site for the latest information.
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