# Expected optimism equals sum of covariances

I am reading section for 7.4 in elements of statistical learning, and I have some questions about some definitions and results in this section, and I would like to check if my understanding is correct.

1. In the definition for generalization error $$Err_T=\mathbb{E}_{X^0,Y^0}[L(Y^0,\hat{f}(X^0)|T]$$, why do we condition on training set $$T$$ if $$X^0,Y^0$$ are simply drawn from the true joint distribution? Is it just because the fit model $$\hat{f}$$ is dependent on the training set $$T$$?

2. They define in-sample error as $$Err_{in}=\frac{1}{N}\sum_{i=1}^N E_{Y^0}[L(Y_i^0,\hat{f}(x_i))|T]$$. Here, I assume $$Y_i^0~p_{Y|X}(\cdot|x_i)$$ What is the difference between $$\mathbb{E}_\mathbf{y}[Err_{in}]$$ and $$Err_{in}$$, as the latter already has an expectation over $$Y$$?

3. Finally, in the derived result $$\omega=\frac{2}{N}\sum_{i=1}^N Cov(\hat{y}_i,y_i)$$, how does this covariance even make sense as I am assuming $$y_i$$ is the training label for data point $$i$$ in the train set $$T$$, so it should just be deterministic, no? Unless we are treating the train set as a random variable here?

1. Yes, $$\hat f$$ is random because $$\hat f$$ is a function of the training set. The generalisation error is also a function of $$T$$: a 'good' training set might give you lower generalisation error
2. The book defines $$E_{\bf y}$$ as the expectation over the outcome variable in the training set, fixing the predictors in the training set. If you're using version 2 of the book, it's after equation 7.20. If you're using version 1 you should look for version 2, which is free now. This section has a footnote Indeed, in the first edition of our book, this section wasn’t sufficiently clear [about what is fixed and what is random]
• Thanks for the answer. I am still a little confused about point 2. Namely, $Err_{in}$ does not depend at all on the true outcome variable $y_i$'s, only the true (fixed) predictors $x_i$'s, and new variable $Y_i^0$'s, which are independent of all other randomness in the setting. So why is it not true that $\mathbb{E}_{\mathbf{y}}[Err_{in}]=Err_{in}$? Or am I misunderstanding something?