# gaussian process

Let be a random variable indexed by . Think of as being time. Stochastic processes are sets of random variables of the form

For actual data we can only ever have one observation of each , which is why more assumptions are helpful.

### Special Cases

- If is vector valued then this is called a
**random field**. - If a linear combination of samples from a stochastic process is jointly
normal then this is called a
**Gaussian process**, or**Gaussian random field**for the multivariate case. - If is discrete and the value of only depends on then
the process is a discrete time
**Markov chain**.