Firstly, you can't have a two-side arrow in DAG (Direct Acyclic Graph), since it creates a cycle of length 2. Also, I'm not sure if "Error" nodes U1 and U2 are necessary: possible errors can be expressed as noise in CPD (in node "W" and "Y" CPD, to be specific). I will assume next, that U1 and U2 nodes were removed, so we don't have the two-side arrow now.
As for d-separability: you have a direct edge from S to Y; there is no way they can be d-separated.
In general case, the separation set Z is not unique: consider graph A -> B -> C -> D, first set is {A}, second is {D}, and there are multiple sets by which they can be separated: {B}, {C}, {B, C} If you want to find one of them, you can enumerate all the undirected paths from first set of nodes X to the second set of nodes Y, and then separate them, until no non-separated paths are left.
To find out whether two sets of nodes X and Y are d-separated by a set of nodes Z, you can either use the definition of d-separability (i.e. check all paths from X to Y), or the Bayes Ball algorithm.
Note on terminology: this object is called a Directed Graphical Model (DGM) or a Bayesian network, or a belief network, or a causal network. Sure, DGMs are DAGs, but not all DAGs are DGMs; DAG is a far more general term.
P. S. Why Lesswrong, and not, say, StackExchange?
Am not on SE; will register there. Thank you. Deleting this now. (Although I think it might be useful, generally, to have a 'models' thread on LW for small discussions.)
Why Lesswrong, and not, say, StackExchange?
Because it is about a bayesian network? It tries to apply the methodology taught here.
It would have been better suited as a comment in the open or stupid questions thread but he commented about that. No need to downvote a newbie for that.
@Romashka: You can embed images in comments as explained here.
Well, every instrument has its purpose. Why post a question here and have to include apologizing disclaimer and be downvoted and wait a few hours for a single answer if you can go to a specialized questions/answers site where you'll be welcome and will have your answer in minutes?
P. S. I have neither downvoted nor upvoted this post.
(Sorry, I did not find an easy way to post this as a comment in Open Thread or a private message, so putting it here. When there are some definite responses I promise to delete it.)
I tried to make a DAG to represent a process of recognizing a season of the year from an old, damaged photo of a plant. The object was to see how 'Colour of the plant' becomes d-separated from 'Season of the year' and what variables screen 'colour' off.
Here's my attempt (first one ever, so there might be mistakes):
X is season, Y is colour on the photo, and Z is whatever can be used for analysis. U1 is error introduced when the photo was taken, and the two-sided arrow between it and 'Species' means that some plants can be grown indoors as well as outdoors and even though they are capable of photosynthesis they don't tell us useful things about what season it is.
Could you help me?