# Gaussian Process (GP)

## What's that?

• a GP is a prior over functions
• a Gaussian process is a collection of random variables {X_i}, such that any new random variable vector consisting of a subset of some of these random variables is Gaussian distributed
• simple example #1: the {X_i} are Gaussian distributed, then any new random vector Y=(X_t1, …, X_tn), is Gaussian distributed as well
• simple example #2: {X_i=i*W}, where W is a normal distributed random variable is a GP as well

## What is it good for?

• you can define a family of functions using GPs indirectly by specifying a mean and a covariance function
• you can model a stochastic process using a GP and draw random samples from that model (e.g., Brownian motion)

## Videos

### Theorem: Existence of Gaussian Processes

Short video only presenting the theorem, but not the proof of the theorem:

Given some mean and covariance function, the theorem states, that there exists a GP such that it has this mean and for any two selected random variables the covariance is that specified by the covariance function

### Examples of GPs

Given different covariance functions (GP kernels), Matlab plots of samples of the resulting GP are shown.

Especially part II gives you a good intution about GP:

• GP allow you to draw random samples from multi-dimensional functions
• where you specify the functions indirectly
• by the mean function & the covariance function
• while the mean function is nearly always set to 0 for all points
• the covariance functions allows you to define the covariance between nearby (and far away) function values
• e.g., such that the function values change smoothly
• by taking the squared exponential kernel for the covariance function

Part I

Part II

### Which functions are valid covariance functions?

A function f is a covariance function (or: a positive semi-definite kernel) if for any collection of random variables of the GP the corresponding covariance matrix is positive semi-definite) 