Causal Inference - when Conditioning on a Collider is correct

Question

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.

Answer 1

This is an excellent question giving a really incisive inquiry into causal reasoning in a "simple" problem. The issue here is that when you are playing the Monty Hall game, you are making a predictive inference only --- you want to use the information you have to predict where the car is placed. You are not attempting to make a causal inference about the effect of you choosing a door --- i.e., you are not seeking to determine whether or not your choice of a door causes the car to move.

A causal inference in the Monty Hall game would be asking whether or not you choosing a door caused the location of the car to move (e.g., an assistant quickly and quietly drove it to another door based on hearing your choice). If you want to know this then you would not condition on the collider. You would compute the probability that the car is behind a particular door, conditional only on you having chosen that door and without Monty opening another door. Suppose you conclude that this conditional probability is $\tfrac{1}{3}$, which matches the prior probability that the car is behind that door. Since these probabilities match, you would infer that there is no causal effect --- i.e., your initial choice of a door had no effect on the location of the car.

A predictive inference in the Monty Hall game is the standard question we are asking --- given all the information we have, what is the conditional probability for the position of the car? Since we are making a predictive inference we condition on all available information including the collider variable, and this leads to the famous solution you have outlined.

As you can see from the above, the Monty Hall problem actually is consistent with the general advice given by Pearl --- when making a causal inference we do not condition on the collider variable. Another way of looking at this is that if we are making a predictive inference we don't care if a statistical relationship is "spurious" or not (see my related answer on "spurious" correlation).

Answer 2

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:

1. 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).

2. 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.