Causal Inference - when Conditioning on a Collider is correct I have been reading Judea Pearl's Book of Why and in it, he tackles the famous Monty Hall problem through a causal lens. Although it may still grind away at our initial instincts, hopefully nobody here will disagree that the right answer, when addressing the problem as framed in the game show, would be to switch your choice of door as your chances of success in doing so would be 2/3 vs. 1/3 if you stick with your original choice.

What is throwing me for a loop now, however, is that in the causal world, the general advice is to NOT condition on what are called "colliders", which in this case, would be the "Door Opened". This flies in the face of the causal explanation for the RIGHT answer in this problem:

Once   we
obtain  information  on this     variable,   all     our     probabilities   become  conditional
on  this    information.    But when    we  condition   on  a   collider,   we  create  a   spurious
dependence   between     its     parents.    The     dependence  is  borne   out     in  the
probabilities:   if  you     chose   Door    1,  the     car     location    is  twice   as  likely  to  be
behind  Door     2   as Door     1; if   you     chose  Door     2, the  car    location    is  twice    as
likely  to  be  behind  Door    1.

Meaning, by conditioning on the collider Door Opened (which in this case, happens via an action that occurs in reality, e.g. say during the game show) you create a non-causal dependence (or "flow of information", aligning to his metaphor of opening the pipe when you condition on a collider) between "Your Door" and "Location of Car" which Pearl still labels spurious and yet that is how you obtain the right answer to the problem, again despite the fact that collider-conditioning-induced-bias is what you are taught (in general) to avoid unless other adjustments are made to compensate for when you have no other choice.
How does one reconcile this? Am I missing something? Please do tell. Any insight would be much appreciated.

In this situation, the "true" probability $\theta$ of choosing a door with a car is 1/3. If your goal was to estimate $\theta$ , you would want to not condition on the collider, which would result in you choosing a car with a door 1/3 of the time.
However, if your goal is to get a car (and not to accurately estimate $\theta$) your priorities change. So, you will make decisions to introduce upward bias into the number of cars you will observe receiving.
In an analysis, you want to condition on a collider when:

*

*The only way to collect data to answer your question is by sampling based on the collider. (For example, the collider may be admission to a certain hospital, and the only way to sample enough cases of a rare disease is to recruit based on hospital admission).


*Conditioning on the collider is the only way to block an open backdoor path, and the path opened by conditioning on the collider may be blocked by conditioning on another variable.
