**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(**z**_{1}),…,f(**z**_{N})] for any finite set of values of [**z**_{1},…,**z**_{N}] 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*.