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.