# Randomness Assumption (IID assumption)

The randomness assumption (also known as the IID assumption) is that the observations in a sequence are generated independently from the same probability distribution `⚠ $Q$`

on the space of possible observations `⚠ $\mathbf{Z}$`

(often `⚠ $\mathbf{Z}=\mathbf{X}\times\mathbf{Y}$`

). A weaker (for a wide class of `⚠ $\mathbf{Z}$`

, according to de Finetti's theorem) assumption is that of exchangeability.

The randomness assumption is used in stochastic prediction and conformal prediction. It is a standard assumption in machine learning. In applications, algorithms developed under this assumption (such as SVM) are often applied when the assumption is violated. However, if the observations `⚠ $z_1,z_2,\ldots$`

, `⚠ $z_i=(x_i,y_i)$`

, are coming from a stationary measure on `⚠ $\mathbf{Z}^{\infty}$`

, the IID assumption can be often made "almost satisfied" by extending the objects `⚠ $x_i$`

. For example, in the case of time series we may add the pre-history of `⚠ $y_i$`

to `⚠ $x_i$`

(and this will work very well if the time series is Markov).