Why are $y^*$ and $\hat{y}^*$ independent? A common assertion that is made in several contexts, and used to prove some pretty important results, is that, because a given observation wasn't explicitly used in the creation of a certain statistical model, that the prediction given by that model is independent of that observation.
For example, assume $y$ can be modeled (the true model) on $x$ linearly by
\begin{equation*} 
y = f(x) + \varepsilon
\end{equation*}
for some linear function $f$ and random noise $\varepsilon$.  Let $\hat{f}$ be the least-squares estimate of $f$ one gets by linear regression on some collection of simultaneous observations of $y$ and $x$.  Let $(x^*,y^*)$ be an observation not part of the training data used to construct $\hat{f}$, and let $\hat{y}^* = \hat{f}(x^*)$, the value of $y$ predicted by your least-squares model.  In proving that, for example, that the distribution of $y^* - \hat{y}^*$ has mean $0$ and variance = (formula not important here), one encounters the assertion
"since $y^*$ was not used in the computation of $\hat{y}^*$, it follows that $y^*$ and $\hat{y}^*$ are independent, and hence $\text{Cov}(y^*,\hat{y}^*) = 0$."
You can find analogous assertions made in other contexts, for instance in the bias-variance trade-off theorem for statistical learning methods.
This assertion makes no sense to me.  Yes, I understand that $\hat{f}$ is not directly dependent on the value $y^*$ since it wasn't included in the training data.  But saying that they are independent is to say that knowing one should give you no better idea of the other.  This is obviously nonsense; what is the point of constructing and $\hat{f}$ to begin with if its value at $x^*$ gives you no better guess as to the value of $y^*$?
Is there an implicit conditional independence that's being implied here?  Because, as stated, this seems patently false.
Thank you in advance for any clarification.

If x were random you'd need to talk about conditional independence.  But x is traditionally regarded as constant unless explicitly stated otherwise. So the independence assertion is correct.
And, yes, it IS confusing.
