Gaussian Processes for Machine Learning
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Description
Contents
Reviews
Language
English
ISBN
026218253X
Series Foreword
Preface
Symbols and Notation
Introduction
A Pictorial Introduction to Bayesian Modelling
Roadmap
Regression
Weight-space View
The Standard Linear Model
Projections of Inputs into Feature Space
Function-space View
Varying the Hyperparameters
Decision Theory for Regression
An Example Application
Smoothing, Weight Functions and Equivalent Kernels
* Incorporating Explicit Basis Functions
Marginal Likelihood
History and Related Work
Exercises
Classification
Classification Problems
Decision Theory for Classification
Linear Models for Classification
Gaussian Process Classification
The Laplace Approximation for the Binary GP Classifier
Posterior
Predictions
Implementation
Marginal Likelihood
* Multi-class Laplace Approximation
Implementation
Expectation Propagation
Predictions
Marginal Likelihood
Implementation
Experiments
A Toy Problem
One-dimensional Example
Binary Handwritten Digit Classification Example
10-class Handwritten Digit Classification Example
Discussion
* Appendix: Moment Derivations
Exercises
Covariance Functions
Preliminaries
* Mean Square Continuity and Differentiability
Examples of Covariance Functions
Stationary Covariance Functions
Dot Product Covariance Functions
Other Non-stationary Covariance Functions
Making New Kernels from Old
Eigenfunction Analysis of Kernels
* An Analytic Example
Numerical Approximation of Eigenfunctions
Kernels for Non-vectorial Inputs
String Kernels
Fisher Kernels
Exercises
Model Selection and Adaptation of Hyperparameters
The Model Selection Problem
Bayesian Model Selection
Cross-validation
Model Selection for GP Regression
Marginal Likelihood
Cross-validation
Examples and Discussion
Model Selection for GP Classification
* Derivatives of the Marginal Likelihood for Laplace's Approximation
* Derivatives of the Marginal Likelihood for EP
Cross-validation
Example
Exercises
Relationships between GPs and Other Models
Reproducing Kernel Hilbert Spaces
Regularization
* Regularization Defined by Differential Operators
Obtaining the Regularized Solution
The Relationship of the Regularization View to Gaussian Process Prediction
Spline Models
* A 1-d Gaussian Process Spline Construction
* Support Vector Machines
Support Vector Classification
Support Vector Regression
* Least-squares Classification
Probabilistic Least-squares Classification
* Relevance Vector Machines
Exercises
Theoretical Perspectives
The Equivalent Kernel
Some Specific Examples of Equivalent Kernels
* Asymptotic Analysis
Consistency
Equivalence and Orthogonality
* Average-case Learning Curves
* PAC-Bayesian Analysis
The PAC Framework
PAC-Bayesian Analysis
PAC-Bayesian Analysis of GP Classification
Comparison with Other Supervised Learning Methods
* Appendix: Learning Curve for the Ornstein-Uhlenbeck Process
Exercises
Approximation Methods for Large Datasets
Reduced-rank Approximations of the Gram Matrix
Greedy Approximation
Approximations for GPR with Fixed Hyperparameters
Subset of Regressors
The Nyström Method
Subset of Datapoints
Projected Process Approximation
Bayesian Committee Machine
Iterative Solution of Linear Systems
Comparison of Approximate GPR Methods
Approximations for GPC with Fixed Hyperparameters
* Approximating the Marginal Likelihood and its Derivatives
* Appendix: Equivalence of SR and GPR Using the Nyström Approximate Kernel
Exercises
Further Issues and Conclusions
Multiple Outputs
Noise Models with Dependencies
Non-Gaussian Likelihoods
Derivative Observations
Prediction with Uncertain Inputs
Mixtures of Gaussian Processes
Global Optimization
Evaluation of Integrals
Student's t Process
Invariances
Latent Variable Models
Conclusions and Future Directions
Appendix Mathematical Background
Joint, Marginal and Conditional Probability
Gaussian Identities
Matrix Identities
Matrix Derivatives
Matrix Norms
Cholesky Decomposition
Entropy and Kullback-Leibler Divergence
Limits
Measure and Integration
Lp Spaces
Fourier Transforms
Convexity
Appendix Gaussian Markov Processes
Fourier Analysis
Sampling and Periodization
Continuous-time Gaussian Markov Processes
Continuous-time GMPs on R
The Solution of the Corresponding SDE on the Circle
Discrete-time Gaussian Markov Processes
Discrete-time GMPs on Z
The Solution of the Corresponding Difference Equation on PN
The Relationship Between Discrete-time and Sampled Continuous-time GMPs
Markov Processes in Higher Dimensions
Appendix Datasets and Code
Bibliography
Author Index
Subject Index
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