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.
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.)
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.
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.
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.)
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.