we have a complete understanding of what's going on, so there are no beliefs to update.
This is not true, unless by 'complete understanding' you imply that everything must be deterministic and we understand every single variable.
However, probabilities can be interpreted as statements of lack of knowledge at certain points of a process, even if the entire process itself is understood completely.
For example, if I have a process where randomly half the time I get a 55% heads biased coin, and half the time I get a fair coin, then I flip it 10 times, I can absolutely apply Bayes' Theorem here to figure out probability I had the fair or the biased coin after seeing the result of the flips.
This is a system that I understand completely, and part of that system is a single unknown bit, which we're trying to put bounds on.
Comments
This is not true, unless by 'complete understanding' you imply that everything must be deterministic and we understand every single variable.
However, probabilities can be interpreted as statements of lack of knowledge at certain points of a process, even if the entire process itself is understood completely.
For example, if I have a process where randomly half the time I get a 55% heads biased coin, and half the time I get a fair coin, then I flip it 10 times, I can absolutely apply Bayes' Theorem here to figure out probability I had the fair or the biased coin after seeing the result of the flips.
This is a system that I understand completely, and part of that system is a single unknown bit, which we're trying to put bounds on.
A correct Bayesian treatment of the Monty Hall problem is given in a cousin comment: https://news.ycombinator.com/item?id=17438740
No that's Bayes' theorem, not Bayesian inference. You haven't made any Bayesian assumptions.