Suppose that we have the following random variables $Y,X$ which are dependent on $Z, W$. However, $Z$, which is defined on a finite set $Z \in \{z_1, \ldots, z_n\}$ is further dependent on $W$. I would like to represent the joint distribution of $Y,X$. One approach (which is wrong) I thought of was to write:
$$ (Y,X) \mid Z=x \mid W \sim \mathcal{P}_{(Y,X) \mid Z \mid W} $$
Another approach is to have:
$$ (Y,X) \mid Z=z, W \sim \mathcal{P}_{(Y,X) \mid Z, W} $$
What is the correct way to represent the distribution? Thanks.
In these more complex cases we usually represent them using their joint distribution as $P_{X,Y,Z,W}(x,y,z,w)=P_{X|Z,W}(x|z,w)P_{Y|Z,W}(y|z,w)P_{Z|W}(z|w)P_{W}(w)$, this gives rise to what we call a factor graph representation of the joint distribution which gives nice intuition and have good properties. On these factor graph and their use in probability you may want to check "Probability on Graphs. Random processes on graph and lattices" by Geoffrey Grimmett, chapter 7.