Judea Pearl on the Meaning of the Monty Hall problem: Judea Pearl: (2018): The Book of Why: The New Science of Cause and Effect (New York: Basic Books: 046509760X): "Even today, many people seeing the puzzle for the first time find the result hard to believe. Why? What intuitive nerve is jangled?...
...There are probably 10,000 different reasons, one for each reader, but I think the most compelling argument is this: vos Savant’s solution seems to force us to believe in mental telepathy. If I should switch no matter what door I originally chose, then it means that the producers somehow read my mind. How else could they position the car so that it is more likely to be behind the door I did not choose?...
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. It is a bizarre dependence for sure, one of a type that most of us are unaccustomed to. It is a dependence that has no cause. It does not involve physical communication between the producers and us. It does not involve mental telepathy. It is purely an artifact of Bayesian conditioning: a magical transfer of information without causality. Our minds rebel at this possibility because from earliest infancy, we have learned to associate correlation with causation. If a car behind us takes all the same turns that we do, we first think it is following us (causation!). We next think that we are going to the same place (i.e., there is a common cause behind each of our turns). But causeless correlation violates our common sense. Thus, the Monty Hall paradox is just like an optical illusion or a magic trick: it uses our own cognitive machinery to deceive us....
Notice that I have really given two explanations of the Monty Hall paradox. The first one uses causal reasoning to explain why we observe a spurious dependence between Your Door and Location of Car; the second uses Bayesian reasoning to explain why the probability of Door 2 goes up in Let’s Make a Deal. Both explanations are valuable. The Bayesian one accounts for the phenomenon but does not really explain why we perceive it as so paradoxical. In my opinion, a true resolution of a paradox should explain why we see it as a paradox in the first place. Why did the people who read her column believe so strongly that vos Savant was wrong? It wasn’t just the know-it-alls. Paul Erdos, one of the most brilliant mathematicians of modern times, likewise could not believe the solution until a computer simulation showed him that switching is advantageous. What deep flaw in our intuitive view of the world does this reveal?
“Our brains are just not wired to do probability problems very well, so I’m not surprised there were mistakes,” said Persi Diaconis, a statistician at Stanford University, in a 1991 interview with the New York Times. True, but there’s more to it. Our brains are not wired to do probability problems, but they are wired to do causal problems. And this causal wiring produces systematic probabilistic mistakes, like optical illusions. Because there is no causal connection between My Door and Location of Car, either directly or through a common cause, we find it utterly incomprehensible that there is a probabilistic association. Our brains are not prepared to accept causeless correlations, and we need special training—through examples like the Monty Hall paradox or the ones discussed in Chapter 3—to identify situations where they can arise. Once we have “rewired our brains” to recognize colliders, the paradox ceases to be confusing...
#shouldread
#cognition
