Gaussian Processes for Machine Learning
Free

Gaussian Processes for Machine Learning

By Carl Edward Rasmussen and Christopher K. I. Williams
Free
Book Description
Table of Contents
  • 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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