I'm reading a book about Bayesian data analysis (by Gelman et al.) and I bumped into the following text:
$x= \text{Football point spread}$
$y=\text{Game outcome}$
$d=y-x$
For the remainder of the discussion we take the distribution of $d$ to be independent of $x$ and normal with mean zero and standard deviation $14$ for each $x$; that is,
$$d|x\sim N(0,14^2).$$
My question is: What does $d|x \sim N(0,14^2)$ mean? Why didn't author just write $d\sim N(0,14^2)$?
Thank you!
P.S. In case your wondering where the $0$ and $14$ come from, the author uses empirical data of football games to compare the outcomes and point spreads of the games. The author calculates the mean of $d$ to be $0.07$ and sample standard deviation $13.86$.
Writing $D|X=x \sim N(0, 14^2)$ for all values of $x$ emphasize that the random variable $D$ is independent of $X$. Of course in this case we know that $D$ and $D|X = x$ has the same distribution as indicated by independence. Maybe, depending on the context, the distribution of interest is this conditional distribution so the author will need to present this at the very beginning, and this can be simplified afterward if the context does not confuse the readers.
In general when they are not independent, the conditional distribution of $D|X = x$ will depends on the value of $x$, acts like a parameter.