Theory Manual

Gaussian Processes

Contents:

• Gaussian Processes
  • Making Predictions
  • Posterior Predictive Distribution
  • Learning Hyperparameters
  • Computational Considerations
• Derivative-Enhanced Gaussian Processes
  • Hessian-Enhanced Gaussian Processes
    • Note on Simplified Test Predictions
• Derivative Screening
• Weighted Gradient Enchanced Gaussian Processes
  • Problem Formulation
  • Submodel Covariance Structure
  • Weighted Combination of Submodels
    • Computing Weight Functions
  • Hyperparameter Optimization
  • Computational Trade-offs
  • References
• Directional Derivative-Enhanced Gaussian Processes
  • Covariance Structure
  • Hyperparameter Optimization
  • Computational Advantages
  • References
• Cost-Aware Active Learning with Directional Derivatives
  • Acquisition Quantities
    • Function candidate
    • Directional-derivative candidates
    • Implementation note (Lanczos top-\(k\))
  • Unified Cost-Aware Score
    • Interpretation
  • Hyperparameter Optimisation
  • Algorithm Summary
  • Relationship to Classical Active Learning Criteria
  • Limitations and Future Extensions
  • References
• Multi-output Gaussian Processes
  • A Note on DEGPs
    • References
• Gradient Enhanced Multi-output Gaussian Processes
• Note on Observation Noise
  • Noisy Joint Distribution
  • Noisy Posterior
  • Noisy Marginal Log-Likelihood
    • Generalization
    • Heteroscedastic Noise