Causal inference, stratification to mitigate confounders in continuous variables? Handling confounders in continuous variables
In Statistical Rethinking, the author shows that in different situations, a confounder (fork, pipe, collider, descendent) will induce spurious correlations. For the collider, he shows that if Z is Bernoulli variable dependent on both X and Y, X and Y could appear uncorrelated. However, if you condition on Z=0, the correlation looks much stronger and likewise for Z=1.
How does this stratification work for continuous variables? Say that Z was normally distributed (but still dependent on X and Y), conditioning on 0, 1, or any specific value doesn't make sense.
Source: https://www.youtube.com/watch?v=UpP-_mBvECI&list=PLDcUM9US4XdMROZ57-OIRtIK0aOynbgZN&index=7 (45:28)
 A: First:
I have not watched the video, so I don't know what definitions  are used there, but in the books and papers I have read:

*

*A confounder is only a fork, not a collider.

*An unobserved collider is completely blocking the path between $X$ and $Y$, it is not just making it "appear" to be uncorrelated. This is d-separation. So if there is no other path between $X$ and $Y$, those two are independent.

Your question:
IIUC, your point is that, for a given dataset, "conditioning a continuous variable $Z$ on e.g. 1" would refer to reducing the dataset to those observations which contain $1$ in the column for $Z$ (stratification), and since the probability of a continuous variable to have a single value is zero, this will not return any observations given sufficient resolution (or it would return only one observation if you chose to condition on a value that appears in your data). That is correct and in this case you cannot really do conditioning in this way.
What you can do, is to partially condition on the collider $Z$, e.g. by conditioning on a discrete child $D$ of $Z$. This $D$ could be e.g. a binary variable describing whether $Z$ is inside some interval or not. This partial conditioning would result in "partial opening" of the connection between $X$ and $Y$. The larger this interval gets, the less conditioning you have.
