(EDIT: Disregard this, the source of my confusion was me mixing up marked and unmarked arrows. I was thinking "marked arrow = actual causal arrow _or_ latent common cause in the underlying reality, unmarked arrow = actual causal arrow in the underlying reality", but I mixed up the definitions and it's actually the other way around. Actually, I think I was being stupid in multiple ways, but the overall answer to this post is that I was being stupid and mixed up multiple things, including to what extent steps 1 and 2 could be trusted to find all v-structures, given the possibility of unknown latent variables. (EDIT2: Wait, if there truly was an adb v-structure, shouldn't step 2 have at least put an unmarked arrowhead from a to b?))
Am having another go at trying to go through the book (second edition, first printing), and find myself rather confused about rule 2 in step 3 of the IC* algorithm.
Suppose a, b, c, d are all observed variables, a and d are adjacent by an initially undirected edge, a*->b*->c*->d (where *-> represents a marked arrow), and c is not adjacent to d.
Rule 2 would have us now turn the a-d edge into an unmarked arrow from a to d
But... shouldn't an unmarked arrow from a to d be the one thing that is absolutely forbidden? ie, one of the possible underlying reality states allowed by the above is that c has causal influence over d. If you also have a -> d, then you have a v structure with a,d,c. But assuming stable distribution relative to the true latent structure and having the actual distribution of the observable variables rather than just a sample, and all those other "it's all well behaved" assumptions, then steps 1 and 2 should have already found all of the v structures among the observed variables.
So how is rule 2 of step 3 allowed to introduce potential new (possible) v structures among observed variables? Illumination would be appreciated, thank you. :)
(EDIT: Disregard this, the source of my confusion was me mixing up marked and unmarked arrows. I was thinking "marked arrow = actual causal arrow _or_ latent common cause in the underlying reality, unmarked arrow = actual causal arrow in the underlying reality", but I mixed up the definitions and it's actually the other way around. Actually, I think I was being stupid in multiple ways, but the overall answer to this post is that I was being stupid and mixed up multiple things, including to what extent steps 1 and 2 could be trusted to find all v-structures, given the possibility of unknown latent variables. (EDIT2: Wait, if there truly was an adb v-structure, shouldn't step 2 have at least put an unmarked arrowhead from a to b?))
Am having another go at trying to go through the book (second edition, first printing), and find myself rather confused about rule 2 in step 3 of the IC* algorithm.
Suppose a, b, c, d are all observed variables, a and d are adjacent by an initially undirected edge, a*->b*->c*->d (where *-> represents a marked arrow), and c is not adjacent to d.
Rule 2 would have us now turn the a-d edge into an unmarked arrow from a to d
But... shouldn't an unmarked arrow from a to d be the one thing that is absolutely forbidden? ie, one of the possible underlying reality states allowed by the above is that c has causal influence over d. If you also have a -> d, then you have a v structure with a,d,c. But assuming stable distribution relative to the true latent structure and having the actual distribution of the observable variables rather than just a sample, and all those other "it's all well behaved" assumptions, then steps 1 and 2 should have already found all of the v structures among the observed variables.
So how is rule 2 of step 3 allowed to introduce potential new (possible) v structures among observed variables? Illumination would be appreciated, thank you. :)