Previously, we focused on general methods for approximating posterior distributions, such as MCMC and Variational Inference. Now, we turn our attention to Gaussian Processes (GPs), a powerful framework within Bayesian non-parametric modeling. Unlike models defined by a fixed number of parameters, GPs allow the model complexity to adapt to the data by placing priors directly over functions.

In this chapter, you will learn:

• The fundamentals of Gaussian Processes as distributions over functions.
• How covariance functions, or kernels, define the properties (like smoothness) of the functions drawn from a GP.
• The mathematical formulation for Gaussian Process regression, enabling predictions with associated uncertainty estimates, often represented as predictive mean $\mathbb{E}[f_* | \mathbf{y}]$ and variance $\text{Var}[f_* | \mathbf{y}]$.
• Methods for selecting appropriate kernels and optimizing hyperparameters by maximizing the marginal likelihood.
• Techniques for extending GPs to classification problems.
• Approximation methods, such as sparse GPs using inducing points, to apply GPs to larger datasets where exact computation becomes infeasible.

We will explore the theory behind these concepts and apply them through hands-on coding exercises.