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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.

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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.

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  • $\begingroup$ I asked a similar question about controlling for baseline outcomes in a longitudinal analysis: stats.stackexchange.com/questions/598107/… $\endgroup$
    – RobertF
    Commented Jan 15, 2023 at 13:50
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    $\begingroup$ I would encourage you to adopt a mindset of "Understand what conditioning on a collider does" as opposed to "Do not condition on a collider." $\endgroup$
    – Alexis
    Commented Jan 15, 2023 at 18:04
  • $\begingroup$ @Alexis the latter is not my mindset - the reason I asked is this question is because the author, a premier expert on the topic, is applying causal diagrams to what is a fairly trivial situation (compared to most real-world situations) - i.e. a game that has set rules. But in the model presented, the "spurious dependence" yields the correct answer and this is confusing to me. A spurious relationship, by definition, is when 2+ events/vars are associated but NOT causally related. Is there a better causal DAG that would be useful in modeling the impact of "switching" on winning more directly? $\endgroup$
    – Mark Z.
    Commented Jan 20, 2023 at 16:53
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    $\begingroup$ related: stats.stackexchange.com/questions/499455/… $\endgroup$
    – markowitz
    Commented Jan 20, 2023 at 17:39
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    $\begingroup$ This diagram imposes a particular (IMHO counterfactual) interpretation on the Monty Hall problem. It lacks information about Mr. Hall's choices. He was under no obligation to offer any alternative to the contestant and potentially his decision to offer the alternative was based on his knowledge of what was behind the doors. Obtaining the right answer to the wrong framing of the problem is at the heart of how controversy has arisen: different people frame it differently and each party knows they have the right answer! $\endgroup$
    – whuber
    Commented Jan 20, 2023 at 21:52

2 Answers 2

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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).

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    $\begingroup$ This explanation makes sense, thank you @Ben $\endgroup$
    – Mark Z.
    Commented Jan 21, 2023 at 5:56
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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.

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  • $\begingroup$ Appreciate your response.. Point 2 was actually clear to me already, as basically you can make other adjustments to compensate for that action (conditioning on a collider), but in this case there are no other adjustments one can make and yet we arrive at the correct solution - so how does one differentiate whether your results are still correct (or not) after doing so for Point 1? Just because there is no other choice than to condition on the collider does not make it the right thing to do - is this then just a situational decision point, or what are ways to validate? $\endgroup$
    – Mark Z.
    Commented Jan 15, 2023 at 13:27
  • $\begingroup$ I re-read your question and realized that I misunderstood it the first time I answered. In this case, you don't want the door you open to correctly estimate the number of cars behind doors; you want the door you open to be most likely to contain a car. $\endgroup$ Commented Jan 15, 2023 at 14:33

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