# Confusion about derivation of regression function

I know that there are already some questions with the same title and my question is kind of similar but requires different derivation (I think).

I am reading "The Elements of Statistical Learning" and I don't understand how they got to the expression in equation $$(2.27)$$: $$EPE(x_0) = E_{y_0 | x_0}E_\mathcal{T}(y_0 -\hat{y}_0)^2$$

and how does it relate to the definition of the expected (squared) prediction error (EPE) in equation $$(2.9)$$?

$$EPE(f) = E(Y-f(X))^2$$

$EPE(f) = E_{x_0}EPE(x_0)$ where $f(x) = x_0^T\beta$ The expected predicted error for the linear regression is the expectation of $EPE(x)$

I don't understand how they got to the expression in equation (2.27):

The conditioning on $$y_0$$ stems from Expected prediction error - derivation you should read it as:

$$\underbrace{E_{y_0|x_0} \underbrace{\left( E_{\tau} (y_0-\hat{y}_0)^2 | y_0,x_0 \right)}_{\substack{\text{expected error due to:} \\ \text{ error in estimate \hat{y}_0} \\ \text{with fixed y_0}}}}_{\substack{\text{expected error due to:} \\ \text{ error in estimate \hat{y}_0} \\ \text{and error in sample y_0} \\ \text{ with variable y_0}}}$$

Eventually you end up taking the sum of the error in the estimate $$\hat y_0$$ (which can be decomposed in variance and bias, where the bias is zero in this case) and the error in the sampled variable $$y_0$$. For the expression of those two, $$y_0$$ and $$\hat{y}_0$$, see also the question $$\hat{y} \sim\mathcal{N}(X\beta, \sigma^{2}X(X^{T}X)^{-1}X^{T}) = y \sim\mathcal{N}(X\beta, \sigma^{2}I_n)$$

how does it relate to the definition of the expected (squared) prediction error (EPE) in equation (2.9)?

You should read that $$EPE(x_0)$$ as the predicted error of $$f = x\beta$$ in the point $$x_0$$. It is shorthanded for $$EPE(f(x_0)) = E(\hat y_0 - y_0)^2$$. It is the expectation value for the squared difference between a prediction $$\hat y_0$$ in the point $$x_0$$ (based on the training sample $$\tau$$) and a new sample $$y_0$$ in the point $$x_0$$