Application of Gaussian Processes in Regression and Classification Problems

Gaussian Processes serve as a widely adopted probabilistic model in addressing both regression and classification problems. This Bayesian non-parametric approach models data distributions by defining a prior over functions and updating it based on observed data through kernel-based covariance functions. Key implementations typically involve: 1) Kernel selection (e.g., RBF, Matern) for capturing data patterns, 2) Hyperparameter optimization via marginal likelihood maximization, and 3) Predictive distribution computation using Gaussian conditioning. For classification, common adaptations include Laplace approximation or expectation propagation to handle non-Gaussian likelihoods. The scikit-learn library provides essential functions like GaussianProcessRegressor and GaussianProcessClassifier, where critical parameters include kernel configurations and optimization algorithms. Various Gaussian Process variants exhibit distinct characteristics - sparse approximations (e.g., SVGP) enhance scalability for large datasets, while multi-task GPs handle correlated outputs. Selection requires careful evaluation of computational complexity, data characteristics, and problem constraints, often necessitating kernel customization or inference method modifications to optimize performance for specific applications.