calibration according to Kuleshov i was reading this paper  https://arxiv.org/pdf/1807.00263.pdf in which Kuleshov gives a definition of a well calibrated neural network in a regression task


I would like to ask one thing:
why does he makes a special cases for when $x_t,y_t$ are i.i.d. realizations of random variables $X, Y \sim \mathbb{P}$? when we do regression don't we always assume that  both the independent and dependent variables are jointly distributed? also why is a sufficient condition?

You can have realizations $(x_t, y_t)$ from their joint distribution $\mathbb P$ that are not independent. E.g., you could decide to choose $x_{t+1}$ as a function of $y_t$.
As to why Equation (4) is sufficient for Equation (3):
I think he just wants to say that if he uses the quantile function of the $Y$-marginal of $\mathbb P$ as the forcaster $H(X)$, i.e. $H(X)$ would be the same for all $X$, then it would be calibrated. But that is by definition: provided you sample i.i.d. from $\mathbb P$, you will for $t\to\infty$ get relative frequencies that converge to the marginal $Y$-distribution.
