Advanced Mathematical Statistics

Introduction to Gaussian processes and their probabilistic foundations. Finite-dimensional distributions, mean and covariance functions, Gaussian processes, and fundamental existence results. Positive definite kernels, Mercer and Bochner theorems, spectral representations, and an introduction to reproducing kernel Hilbert spaces. Bayesian inference with Gaussian processes: Gaussian conditioning, regression, posterior and predictive distributions, marginal likelihood, and hyperparameter estimation. Models for multidimensional and spatio-temporal data. Applications to machine learning, including kernel methods, Gaussian process classification, Bayesian optimization, and approximation techniques. Introduction to Deep Gaussian Processes, connections with neural networks, and probabilistic approaches to artificial intelligence. Computational laboratories in Python and PyTorch focused on the implementation and analysis of Gaussian process models.