contents

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
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
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
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
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
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
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
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
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
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
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
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
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
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
Appendix A Data Sets 677
Appendix B Probability Distributions 685
Appendix C Properties of Matrices 695
Appendix D Calculus of Variations 703
Appendix E Lagrange Multipliers 707
References 711
Index 729