# Gaussian Process Models

Gaussian Process Model Definition
A Gaussian process model is a mathematical model for a nonlinear relationship, say f(z), which depends on some explanatory variable, z in this case. The model is commonly used to model all possible nonlinear relationships, such that in a random function model, any one particular nonlinear function is a single realisation of the random functions. Thus, the model is simply the class of realisations for the random functions.

The Gaussian process model places a probability distribution over the set of all possible relationships; hence, the joint probability distribution for [f(z1),…,f(zN)] for any finite set of values of [z1,…,zN] is specified. Since the chosen random function is Gaussian, its name is therefore given as Gaussian process model.

It follows that the model is simply defined by its mean function, and its covariance function.