Your argument does all the things that are necessary to solve Monty Hall, but it doesn’t consider some things that could be necessary. (Now, maybe you would have realized that those things need to be considered, if they were necessary. I am just explaining how things can get tricky.)
Suppose instead of Monty Hall, we have Forgetful Monty Hall. Forgetful Monty Hall does not remember where the car is, so he opens a door (that you have not picked) at random, and luckily there is a goat behind it!
Here your line of reasoning still seems to apply: if you chose a goat, switching is 100% and staying 0%, while if you chose a car, switching is 0% and staying 100%. So shouldn’t switching still win with probability 2/3?
An extra thing happened, though. In the “if your prior was right” a.k.a. “if you chose a car” case, it’s not surprising that Forgetful Monty Hall opened a door with a goat. In the other case, if you chose a goat, then Forgetful Monty Hall had a 1 in 2 chance of opening the door with a car by mistake. He didn’t, so the probability you chose a goat should be penalized by that factor of 2. The 1:2 prior becomes 1:1, and then your argument (correctly) tells us that switching and staying are both 50%.
One final comment.
I don’t know if it is Bayesian what I am doing...
Here we are dealing with a problem that can be solved exactly. Any mathematician, Bayesian or otherwise, ought to agree with your answer. When solving harder problems, we might get something that cannot be solved exactly. For instance, I chose the 5 integers 4,3,2,4,3 and then chose the 5 integers 2,2,1,5,1. How likely is it that they came from the same distribution?
This is not a question we can answer and so we instead answer a different question or answer this question with simplifying assumptions. When we do this, if we end up talking about “conjugate priors” it is Bayesian; if we end up talking about “null hypothesis testing” it is not Bayesian.
(A clever trick of demagoguery is to take a question that can be solved exactly and point out that null hypothesis testing solves it incorrectly, and thus Bayesian methods are superior. This is silly! Obviously if you can solve a question exactly, you do so.)
Your argument does all the things that are necessary to solve Monty Hall, but it doesn’t consider some things that could be necessary. (Now, maybe you would have realized that those things need to be considered, if they were necessary. I am just explaining how things can get tricky.)
Suppose instead of Monty Hall, we have Forgetful Monty Hall. Forgetful Monty Hall does not remember where the car is, so he opens a door (that you have not picked) at random, and luckily there is a goat behind it!
Here your line of reasoning still seems to apply: if you chose a goat, switching is 100% and staying 0%, while if you chose a car, switching is 0% and staying 100%. So shouldn’t switching still win with probability 2/3?
An extra thing happened, though. In the “if your prior was right” a.k.a. “if you chose a car” case, it’s not surprising that Forgetful Monty Hall opened a door with a goat. In the other case, if you chose a goat, then Forgetful Monty Hall had a 1 in 2 chance of opening the door with a car by mistake. He didn’t, so the probability you chose a goat should be penalized by that factor of 2. The 1:2 prior becomes 1:1, and then your argument (correctly) tells us that switching and staying are both 50%.
One final comment.
Here we are dealing with a problem that can be solved exactly. Any mathematician, Bayesian or otherwise, ought to agree with your answer. When solving harder problems, we might get something that cannot be solved exactly. For instance, I chose the 5 integers 4,3,2,4,3 and then chose the 5 integers 2,2,1,5,1. How likely is it that they came from the same distribution?
This is not a question we can answer and so we instead answer a different question or answer this question with simplifying assumptions. When we do this, if we end up talking about “conjugate priors” it is Bayesian; if we end up talking about “null hypothesis testing” it is not Bayesian.
(A clever trick of demagoguery is to take a question that can be solved exactly and point out that null hypothesis testing solves it incorrectly, and thus Bayesian methods are superior. This is silly! Obviously if you can solve a question exactly, you do so.)