Irreducible error

In any form of modelling, inherent irreducible error arises due to limitations in our knowledge, unobservable hidden variables, and measurement constraints. For a function $F: A \rightarrow B$, irreducible error is typically represented as an random variable $\epsilon$ in the target space $B$, with $\mathbb{E}[\epsilon] = 0$ —expressing that, on average, the irreducible error lacks preference in $B$.

For example suppose we have some perfectly random sample that is either 0 or 1 with a half chance. No further data about the world will help us decide if it is either 0 or 1, it is all irreducible error.