Say we have a confounder $U$ between $X$ and $Y$, i.e. graphically, $U\rightarrow X$ and $U \rightarrow Y$, $X$ and $Y$ are not directly connected.
According to the d-seperation rule, I know conditioning on $U$ will break the correlation between $X$ and $Y$, i.e. $X\perp Y|U$.
However, what I want to ask is that, if we have a parent of $U$, say $P \rightarrow U \rightarrow X$, and $P \rightarrow U \rightarrow Y$, $X$ and $Y$ are not directly connected. Will conditioning on $P$ d-seperate the correlation between $X$ and $Y$, i.e. $X \perp Y|P$?
I think of this problem using the structure causal model (SCM). I think conditioning on $P$ will not entirely break the correlation between $X$ and $Y$, because $u=f(p)+\epsilon$, and the exogenous variable $\epsilon$ is assumed to be independent with $P$.
Don't know whether my thinking is correct. Hopeful for some help. Thanks in advance!
Best Answer
It's possible that conditioning on $P$ will weaken the correlation between $X$ and $Y.$ However, it will not $d$-separate $X$ and $Y,$ because causal information can still flow along the backdoor path $X\leftarrow U\to Y.$
What you might be thinking of is if you condition on a child of a collider, that also opens up the collider to allow causal information to pass through. That is, suppose you have the graph $X\to Z,\;Y\to Z,\;Z\to T.$ There is a collider at $Z,$ but if it is unconditioned, causal information cannot flow from $X$ to $Y.$ If you condition on $Z,$ causal information can flow. But it is also the case that if you condition on $T,$ causal information can flow from $X$ to $Y.$
Conditioning on parents of colliders, chains, or forks does not inherently change the nature of those connections, though it might change correlations somewhat.