The error introduced when estimating a function $f$ with $\hat{f}$ is called reducible error because it can be reduced by using more suitable statistical models. Even if we could reduce the reducible error to zero, meaning $\hat{Y}=f(X)$, we would still suffer from irreducible error because $Y=f(X)+\epsilon$, which implies $Y=\hat{Y}+\epsilon$. Since the error term $\epsilon$ is independent of $X$, the irreducible error cannot be reduced regardless of the statistical techniques used. The expected squared error of predicting $Y$ using $\hat{Y}$ is decomposed as: $E(Y-\hat{Y})^2 = E(f(X)+\epsilon-\hat{f}(X))^2 = \underbrace{[f(X)-\hat{f}(X)]^2}_{\text{Reducible}} + \underbrace{Var(\epsilon)}_{\text{Irreducible}}$ Here, $E(Y-\hat{Y})^2$ is the expected value of the squared error, and $Var(\epsilon)$ is the variance of the error term $\epsilon$.