What are Gaussian processes? Explain the fundamental principles behind Gaussian processes, including their use for regression and probabilistic modeling. Discuss how Gaussian processes are defined by a mean function and a covariance function, and how they capture uncertainty in predictions. Additionally, elaborate on scenarios or applications where Gaussian processes are particularly advantageous compared to other regression models, and how their computational complexity might impact their practical usage in large-scale datasets.

Gaussian processes (GPs) are a flexible and powerful framework for modeling complex relationships between variables. At their core, GPs are a collection of random variables, any finite number of which have a joint Gaussian distribution. They are...