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I am trying to read the Elements of Statistical Learning Tibshirani, Hastie and Friedman, however I have a problem with understanding the expected (squared) prediction error ($EPE$) formula that they provide on page $26$:

The start they assume that the relationship between $X$ and $Y$ is linear so:

$Y = X^TB+\epsilon$, where $\epsilon$~$N(0,\sigma^2)$, the task is to feed the model to the training data. Now

$EPE(x_0) = E_{x_0|y_0}[E_T(y_0-\hat y_0)^2]$

What is the $E_T$? What is the reason to compute the $EPE$ of $x_0$ insted of $\hat y_0$?

On page $23$ there is written that $T$ is the training set, so my understanding is that it consists of some $X$'s. Is it right?

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$E_T$ is the expectation taken over the training set. $\hat y_0$ is a deterministic function of $x_0,$ i.e., $\hat y_0 = \hat \beta^Tx_0.$ The training set consists of rows of covariate values, that is correct.

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  • $\begingroup$ So, the training set consists of pairs (x,y)'s, input and output pairs. It was defined in the beginning of the book. Can you explain to me why only $\hat y$ is dependent on $E_T$ and not also y ? $\endgroup$
    – user13
    Oct 18 '18 at 14:03

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