# What will happen if we condition on the parent of a confounder?

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!

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 $$Y.$$