For linear regression
$$y_i=x_i^T \beta+\epsilon_i$$
what is been sampled? Sometimes I see that our sample is $\{X_1,X_2,\ldots,X_n\}$ while other times I see that our sample is $\{Y_1,Y_2,\ldots,Y_n\}$ where each $Y_i$ and $X_i$ are random variables.
Few times I saw $X_1,\ldots,X_n \, \stackrel{i.i.d.}{\sim} f$ and sometimes $X_1,\ldots,X_n \,|\, p \stackrel{i.i.d.}{\sim} p,\quad p\in \Pi$.
Are we sampling $y_i$ or $x_i^T$ or both?
I am specifically focusing on posterior consistency in https://www.dianacai.com/blog/2021/02/14/schwartz-theorem-posterior-consistency/#:~:text=Consistency%20serves%20a%20check%20on,around%20the%20true%20generating%20value
where we study the convergence of the sequence of posteriors $\{ Π(.|X1,...,X_n) \}$. Does $X_i$ here refer to both $y_i$ and $x^T_i$ or just $x^T_i$?
