In Bayesian statistics, what does this notation formally mean? I've seen Bayesian models specified as
\begin{align*}
Y_i|v_i &\overset{ind}{\sim} f_i(y_i|v_i),\\
v_i & \overset{ind}{\sim} g_i(v_i).
\end{align*}
My question is about the top line $Y_i|v_i\overset{ind}{\sim}f_i(y_i;v_i)$. It seems like, strictly speaking, this isn't mathematically precise because $Y_i$ and $Y_j$ are being conditioned on two separate variables. I can think of two ways to interpret this notation: $Y_i\perp Y_j|v_i,v_j$, for all $i,j$, or $\{Y_i\}_i$ is an independent family of random variables given the vector $\{v_i\}_i$. I can't tell whether or not they're the same. So are these two interpretations equivalent, and if not, which is correct? 
Edit: To be clear, I'm confused about the "ind" above the "distributed as" sign $\sim$ on the first line, given that the conditioning variable has the subscript $i$. 
Another possibility also occurred to me, which is that the "ind" is redundant and that the joint distribution of all $Y_i$ and $v_i$ would be determined by just writing
\begin{align*}
Y_i|v_i & \sim f_i(y_i;v_i),\\
v_i & \overset{ind}{\sim} g_i(v_i),
\end{align*}
but I'm not sure how to show that either. Edit: I realized this can't be true, because
\begin{align*}
v_1, v_2 &\text{ iid } N(0, 1),\\
Y_i | v_i & \sim N(v_i, 2),\quad \text{for } i =1, 2,
\end{align*}
is true for both $Y_i|v_1,v_2 \text{ iid } N(v_1+v_2,1)$, and $Y_i|v_1,v_2 \overset{ind}{\sim} N(v_i,2)$.
 A: Taking the simpler case
$$
v_i \overset{ind}{\sim} g_i(v_i)
$$
presumably means that "$v_i$'s are independently distributed according to $g_i$ distributions each". The notation is a shortcut for describing each $i$-th variable as separate case. There's often trade-off between simplicity and formality.
But the notation you quote is strange, e.g. it puts $v_i$ on both sides of $\sim$, while usually you'd see variable on left hand side and distribution with parameters on right hand side, so it is redundant if used like this.
A: You are correct to say that the statement is not mathematically precise; it is a shorthand that is sometimes used to set out a hierarchical model.  I'm not a fan of this notation personally, since it is an unnecessary abuse of notation.  (I prefer to specify independence separately.)
Presumably the intention of such statements is to set up mutually independent random variables with specified conditional distributions.  Implicitly, the argument variable is taken to be independent of any other iteration of the conditioning variable not mentioned in the conditioning statement.  So I would think that  the shorthand statement,
$$Y_i|v_i \overset{ind}{\sim} f_i(y_i|v_i),$$
would formally mean that $\{ Y_k \}$ are mutually independent conditional on  $\{ v_k \}$, and each element of the former has conditional distribution $Y_i | \{ v_k \} \sim f_i(v_i)$.  So your second interpretation is the one I would use.  The reason I would think that this is the interpretation is simply that you cannot proceed with your analysis with weaker assumptions.  If what was intended by the notation was certain kinds of pairwise independence, or independence conditional only on subsets of $\{ v_k \}$ then there is not enough specification to set up a hierarchical model with a fixed likelihood function.
