COVER
Preface
Mathematical notation
Contents
1. Introduction
	1.1. Example: Polynomial Curve Fitting
	1.2. Probability Theory
		1.2.1 Probability densities
		1.2.2 Expectations and covariances
		1.2.3 Bayesian probabilities
		1.2.4 The Gaussian distribution
		1.2.5 Curve fitting re-visited
		1.2.6 Bayesian curve fitting
	1.3. Model Selection
	1.4. The Curse of Dimensionality
	1.5. Decision Theory
		1.5.1 Minimizing the misclassification rate
		1.5.2 Minimizing the expected loss
		1.5.3 The reject option
		1.5.4 Inference and decision
		1.5.5 Loss functions for regression
	1.6. Information Theory
		1.6.1 Relative entropy and mutual information
	Exercises
2. Probability Distributions
	2.1. Binary Variables
		2.1.1 The beta distribution
	2.2. Multinomial Variables
		2.2.1 The Dirichlet distribution
	2.3. The Gaussian Distribution
		2.3.1 Conditional Gaussian distributions
		2.3.2 Marginal Gaussian distributions
		2.3.3 Bayes’ theorem for Gaussian variables
		2.3.4 Maximum likelihood for the Gaussian
		2.3.5 Sequential estimation
		2.3.6 Bayesian inference for the Gaussian
		2.3.7 Student’s t-distribution
		2.3.8 Periodic variables
		2.3.9 Mixtures of Gaussians
	2.4. The Exponential Family
		2.4.1 Maximum likelihood and sufficient statistics
		2.4.2 Conjugate priors
		2.4.3 Noninformative priors
	2.5. Nonparametric Methods
		2.5.1 Kernel density estimators
		2.5.2 Nearest-neighbour methods
	Exercises
3. Linear Models for Regression
	3.1. Linear Basis Function Models
		3.1.1 Maximum likelihood and least squares
		3.1.2 Geometry of least squares
		3.1.3 Sequential learning
		3.1.4 Regularized least squares
		3.1.5 Multiple outputs
	3.2. The Bias-Variance Decomposition
	3.3. Bayesian Linear Regression
		3.3.1 Parameter distribution
		3.3.2 Predictive distribution
		3.3.3 Equivalent kernel
	3.4. Bayesian Model Comparison
	3.5. The Evidence Approximation
		3.5.1 Evaluation of the evidence function
		3.5.2 Maximizing the evidence function
		3.5.3 Effective number of parameters
	3.6. Limitations of Fixed Basis Functions
	Exercises
4. Linear Models for Classification
	4.1. Discriminant Functions
		4.1.1 Two classes
		4.1.2 Multiple classes
		4.1.3 Least squares for classification
		4.1.4 Fisher’s linear discriminant
		4.1.5 Relation to least squares
		4.1.6 Fisher’s discriminant for multiple classes
		4.1.7 The perceptron algorithm
	4.2. Probabilistic Generative Models
		4.2.1 Continuous inputs
		4.2.2 Maximum likelihood solution
		4.2.3 Discrete features
		4.2.4 Exponential family
	4.3. Probabilistic Discriminative Models
		4.3.1 Fixed basis functions
		4.3.2 Logistic regression
		4.3.3 Iterative reweighted least squares
		4.3.4 Multiclass logistic regression
		4.3.5 Probit regression
		4.3.6 Canonical link functions
	4.4. The Laplace Approximation
		4.4.1 Model comparison and BIC
	4.5. Bayesian Logistic Regression
		4.5.1 Laplace approximation
		4.5.2 Predictive distribution
	Exercises
5. Neural Networks
	5.1. Feed-forward Network Functions
		5.1.1 Weight-space symmetries
	5.2. Network Training
		5.2.1 Parameter optimization
		5.2.2 Local quadratic approximation
		5.2.3 Use of gradient information
		5.2.4 Gradient descent optimization
	5.3. Error Backpropagation
		5.3.1 Evaluation of error-function derivatives
		5.3.2 A simple example
		5.3.3 Efficiency of backpropagation
		5.3.4 The Jacobian matrix
	5.4. The Hessian Matrix
		5.4.1 Diagonal approximation
		5.4.2 Outer product approximation
		5.4.3 Inverse Hessian
		5.4.4 Finite differences
		5.4.5 Exact evaluation of the Hessian
		5.4.6 Fast multiplication by the Hessian
	5.5. Regularization in Neural Networks
		5.5.1 Consistent Gaussian priors
		5.5.2 Early stopping
		5.5.3 Invariances
		5.5.4 Tangent propagation
		5.5.5 Training with transformed data
		5.5.6 Convolutional networks
		5.5.7 Soft weight sharing
	5.6. Mixture Density Networks
	5.7. Bayesian Neural Networks
		5.7.1 Posterior parameter distribution
		5.7.2 Hyperparameter optimization
		5.7.3 Bayesian neural networks for classification
	Exercises
6. Kernel Methods
	6.1. Dual Representations
	6.2. Constructing Kernels
	6.3. Radial Basis Function Networks
		6.3.1 Nadaraya-Watson model
	6.4. Gaussian Processes
		6.4.1 Linear regression revisited
		6.4.2 Gaussian processes for regression
		6.4.3 Learning the hyperparameters
		6.4.4 Automatic relevance determination
		6.4.5 Gaussian processes for classification
		6.4.6 Laplace approximation
		6.4.7 Connection to neural networks
	Exercises
7. Sparse Kernel Machines
	7.1. Maximum Margin Classifiers
		7.1.1 Overlapping class distributions
		7.1.2 Relation to logistic regression
		7.1.3 Multiclass SVMs
		7.1.4 SVMs for regression
		7.1.5 Computational learning theory
	7.2. Relevance Vector Machines
		7.2.1 RVM for regression
		7.2.2 Analysis of sparsity
		7.2.3 RVM for classification
8. Graphical Models
	8.1. Bayesian Networks
		8.1.1 Example: Polynomial regression
		8.1.2 Generative models
		8.1.3 Discrete variables
		8.1.4 Linear-Gaussian models
	8.2. Conditional Independence
		8.2.1 Three example graphs
		8.2.2 D-separation
	8.3. Markov Random Fields
		8.3.1 Conditional independence properties
		8.3.2 Factorization properties
		8.3.3 Illustration: Image de-noising
		8.3.4 Relation to directed graphs
	8.4. Inference in Graphical Models
		8.4.1 Inference on a chain
		8.4.2 Trees
		8.4.3 Factor graphs
		8.4.4 The sum-product algorithm
		8.4.5 The max-sum algorithm
		8.4.6 Exact inference in general graphs
		8.4.7 Loopy belief propagation
		8.4.8 Learning the graph structure
	Exercises
9. Mixture Models and EM
	9.1. K-means Clustering
		9.1.1 Image segmentation and compression
	9.2. Mixtures of Gaussians
		9.2.1 Maximum likelihood
		9.2.2 EM for Gaussian mixtures
	9.3. An Alternative View of EM
		9.3.1 Gaussian mixtures revisited
		9.3.2 Relation to K-means
		9.3.3 Mixtures of Bernoulli distributions
		9.3.4 EM for Bayesian linear regression
	9.4. The EM Algorithm in General
	Exercises
10. Approximate Inference
	10.1. Variational Inference
		10.1.1 Factorized distributions
		10.1.2 Properties of factorized approximations
		10.1.3 Example: The univariate Gaussian
		10.1.4 Model comparison
	10.2. Illustration: Variational Mixture of Gaussians
		10.2.1 Variational distribution
		10.2.2 Variational lower bound
		10.2.3 Predictive density
		10.2.4 Determining the number of components
		10.2.5 Induced factorizations
	10.3. Variational Linear Regression
		10.3.1 Variational distribution
		10.3.2 Predictive distribution
		10.3.3 Lower bound
	10.4. Exponential Family Distributions
		10.4.1 Variational message passing
	10.5. Local Variational Methods
	10.6. Variational Logistic Regression
		10.6.1 Variational posterior distribution
		10.6.2 Optimizing the variational parameters
		10.6.3 Inference of hyperparameters
	10.7. Expectation Propagation
		10.7.1 Example: The clutter problem
		10.7.2 Expectation propagation on graphs
	Exercises
11. Sampling Methods
	11.1. Basic Sampling Algorithms
		11.1.1 Standard distributions
		11.1.2 Rejection sampling
		11.1.3 Adaptive rejection sampling
		11.1.4 Importance sampling
		11.1.5 Sampling-importance-resampling
		11.1.6 Sampling and the EM algorithm
	11.2. Markov Chain Monte Carlo
		11.2.1 Markov chains
		11.2.2 The Metropolis-Hastings algorithm
	11.3. Gibbs Sampling
	11.4. Slice Sampling
	11.5. The Hybrid Monte Carlo Algorithm
		11.5.1 Dynamical systems
		11.5.2 Hybrid Monte Carlo
	11.6. Estimating the Partition Function
	Exercises
12. Continuous Latent Variables
	12.1. Principal Component Analysis
		12.1.1 Maximum variance formulation
		12.1.2 Minimum-error formulation
		12.1.3 Applications of PCA
		12.1.4 PCA for high-dimensional data
	12.2. Probabilistic PCA
		12.2.1 Maximum likelihood PCA
		12.2.2 EM algorithm for PCA
		12.2.3 Bayesian PCA
		12.2.4 Factor analysis
	12.3. Kernel PCA
	12.4. Nonlinear Latent Variable Models
		12.4.1 Independent component analysis
		12.4.2 Autoassociative neural networks
		12.4.3 Modelling nonlinear manifolds
	Exercises
13. Sequential Data
	13.1. Markov Models
	13.2. Hidden Markov Models
		13.2.1 Maximum likelihood for the HMM
		13.2.2 The forward-backward algorithm
		13.2.3 The sum-product algorithm for the HMM
		13.2.4 Scaling factors
		13.2.5 The Viterbi algorithm
		13.2.6 Extensions of the hidden Markov model
	13.3. Linear Dynamical Systems
		13.3.1 Inference in LDS
		13.3.2 Learning in LDS
		13.3.3 Extensions of LDS
		13.3.4 Particle filters
	Exercises
14. Combining Models
	14.1. Bayesian Model Averaging
	14.2. Committees
	14.3. Boosting
		14.3.1 Minimizing exponential error
		14.3.2 Error functions for boosting
	14.4. Tree-based Models
	14.5. Conditional Mixture Models
		14.5.1 Mixtures of linear regression models
		14.5.2 Mixtures of logistic models
		14.5.3 Mixtures of experts
	Exercises
Appendix A. Data Sets
Appendix B. Probability Distributions
Appendix C. Properties of Matrices
Appendix D. Calculus of Variations
Appendix E. Lagrange Multipliers
References
Index