Bayes' Theorem Assumption

[Kriminal99]

I have been debating in a many page thread in an actuary forum debating the common explanations for the Monty Hall problem. In any case here is the conclusion I came to:

Bayes' Theorem is dependent on there not being any correlation between the frequency with which given information is given, and the outcome. Whether or not you are aware of such a correlation.

An example is a slight variation of the Monty Hall problem. Suppose you didn't know that Monty always shows a goat in response to your picking the car or either of the two goats. Here you would use Bayes' Theorem to calculate the probability that you chose a goat given that you did not choose the one he showed you. You would get a 1/2 chance to get the car by switching doors. But empyrically the chance to get the car by switching is 2/3.

The formula:

Prob(A|B) = (Prob(B|A)*Prob(A)*Prob(B was given|A))/(Prob(B)) / ((Prob(B|A)*Prob(A)*Prob(B was given|A))/(Prob(B)) + (Prob(B|A)*Prob(A)*Prob(B was given|A compliment))/(Prob(B))

Seems to work for adjusting when there is such a correlation and you know what it is.

[Qfwfq]

After having heard the MH problem a few years ago, I had worked out how to resolve the paradox. I perfectly agree the probability will be 1/2 if MH actually didn't know himself which door to open and therefore could have stumbled on the winning door. As this simply didn't happen you count the probability conditioned to the event that the opened door wasn't the winning one.

If instead MH knew wich door was winning and you're sure of this, you can apply the usual reasoning correctly and you are thus partly exploiting his information. If you're not sure he knew, you should consider the probability somewhere in between, according to how sure you are that he knew. This shows that your estimate of probability depends on information you have, equiprobability is an assumption when lacking better information. Compare with a knowledge quiz with multiple choice answers. You might be totally incompetent on the topic of the quiz, or highly expert and certain of the answer, or anything in between. If it's a question you know the answer to, Bayes Schmayes.

Try enumerating the cases including those where MH stumbles on the winning door and reasoning on conditional probability.

[Kriminal99]

Originally Posted by Qfwfq

After having heard the MH problem a few years ago, I had worked out how to resolve the paradox. I perfectly agree the probability will be 1/2 if MH actually didn't know himself which door to open and therefore could have stumbled on the winning door. As this simply didn't happen you count the probability conditioned to the event that the opened door wasn't the winning one.

If instead MH knew wich door was winning and you're sure of this, you can apply the usual reasoning correctly and you are thus partly exploiting his information. If you're not sure he knew, you should consider the probability somewhere in between, according to how sure you are that he knew. This shows that your estimate of probability depends on information you have, equiprobability is an assumption when lacking better information. Compare with a knowledge quiz with multiple choice answers. You might be totally incompetent on the topic of the quiz, or highly expert and certain of the answer, or anything in between. If it's a question you know the answer to, Bayes Schmayes.

Try enumerating the cases including those where MH stumbles on the winning door and reasoning on conditional probability.

Yes I agree... If you don't know anything about what monty is doing or why (or the door just magically opens and you don't know what caused it to do so), and he just opens a door with a goat then based on what only information you have the conditional probability would be 1/2, and yet because of this unkown correlation between how frequently you would have been given the information and the outcome, that answer would show to be wrong.

The extra term in the above formula only adjusts based on whatever that correlation is.

So my next question is, how can you ever be sure when using Bayes' theorem that there is not such a correlation that you simply are not aware of?