...an easy solution to the monty-hall problem is to just apply the base theorem and calculate the probabilities.
It seems simple, but the claim (not mine) was that the theorum states that since you are only choosing "once" you can't use an estimation, as that is based on collected data... which somehow doesn't exist or apply to you.
The reason is that somehow there are no odds/percentages/probabilities if there is only one instance of an event, or at least they don't change. (ie, 2 door, always 1 in 2 odds, period)
Normally I wouldn't detract from a thread like this, but participation seems low, so I can take the down votes in hopes of getting a response that helps me with this conundrum.
Bayesian statistics is a way of updating your beliefs about the value of some unknown number (or value) after doing an experiment. You have to be uncertain about some variable, and then you work backwards to figure out its value.
In the case of the Monty Hall problem, we have a complete understanding of what's going on, so there are no beliefs to update. There are no hidden rules or magic numbers we haven't been told about and we need to work out. You literally can't use Bayesian statistics if you understand a system completely.
Also, Bayesian statistics is a tool. It's not a law of nature. It can get the wrong answers. There is literally no guarantee that you will get the right answer when using it. Garbage In Garbage Out. It's just that there are cases where, empirically, it can be useful.
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.
Thanks for this explanation, it certainly expels some of my confusion.
It seems then based on your reply that my friend is possibly wrong for invoking Bayesian theory regarding the Monty Hall problem for the exact reason he claimed it was the only answer? (note, my friend is a brilliant person and dedicated engineer who understands math far above my level. I only program with statistics and the PhD's tell me what to do.)
In other words, his argument is incorrect in his assessment because we actually _do_ have previous knowledge? (He pointed out that Bayes theorum was used for analysing insurance premiums for new kinds of insurance because there was no previous data to work with, which fits with your statement.)
This seems like too easy an answer for him to have missed, which makes me suspect of my understanding of your reply.
(yes, my initial question was serious, I appreciate you replying)
Not only do we have previous knowledge, we know basically everything about what's going on. You only use Bayesian inference if there is something you don't know which can't be computed from the things you do know. Then, after doing an experiment, you work backwards to figure out the value of the things you didn't know.
Don't confuse Bayesian inference with Bayes' theorem, which is a theorem that doesn't have any philosophy attached to it. Bayesian inference makes use of this theorem on top of some philosophical assumptions, which may be questionable depending on how you use them.
Bayesian inference also requires you to give your "prior" beliefs, which are the beliefs you have before you do the experiment. If these beliefs are strongly biased away from the right answer, you will get Garbage Out.
Ok, that is really neat. I need to read more about this. I just did some searching on bayesian interference solution for the monty hall problem. And I am guessing they are applying the math whilst pretending to not have foreknowledge of the outcome?
I certainly don't expect you to research this or explain every detail for me, I may have to take a class on this subject after I retire to satisfy my need to grasp this. Thank you for time, this was enlightening.
just to chime in. The monty-hall problem is a pure math (probability-theory) and not statistics-excersize, since we know all the probabilities beforehand and don't estimate.
It's very important to distinguish between bayesian inference and the bayesian theorem. The bayesian theorem can be proven and therefore must be true (if we accept the axioms). Bayesian inference has some real philosophical problems, since we often have no way to choose a justified prior. Frequentists approaches may also use the Bayesian theorem.
The important difference is that one is the probability of P(x == car), which is 1/3, and one is P(x == car | y == goat). Just apply the bayes theorem and calculate it yourself!
Comments
It seems simple, but the claim (not mine) was that the theorum states that since you are only choosing "once" you can't use an estimation, as that is based on collected data... which somehow doesn't exist or apply to you.
The reason is that somehow there are no odds/percentages/probabilities if there is only one instance of an event, or at least they don't change. (ie, 2 door, always 1 in 2 odds, period)
Normally I wouldn't detract from a thread like this, but participation seems low, so I can take the down votes in hopes of getting a response that helps me with this conundrum.
Bayesian statistics is a way of updating your beliefs about the value of some unknown number (or value) after doing an experiment. You have to be uncertain about some variable, and then you work backwards to figure out its value.
In the case of the Monty Hall problem, we have a complete understanding of what's going on, so there are no beliefs to update. There are no hidden rules or magic numbers we haven't been told about and we need to work out. You literally can't use Bayesian statistics if you understand a system completely.
Also, Bayesian statistics is a tool. It's not a law of nature. It can get the wrong answers. There is literally no guarantee that you will get the right answer when using it. Garbage In Garbage Out. It's just that there are cases where, empirically, it can be useful.
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.
Thanks for this explanation, it certainly expels some of my confusion.
It seems then based on your reply that my friend is possibly wrong for invoking Bayesian theory regarding the Monty Hall problem for the exact reason he claimed it was the only answer? (note, my friend is a brilliant person and dedicated engineer who understands math far above my level. I only program with statistics and the PhD's tell me what to do.)
In other words, his argument is incorrect in his assessment because we actually _do_ have previous knowledge? (He pointed out that Bayes theorum was used for analysing insurance premiums for new kinds of insurance because there was no previous data to work with, which fits with your statement.)
This seems like too easy an answer for him to have missed, which makes me suspect of my understanding of your reply.
(yes, my initial question was serious, I appreciate you replying)
Not only do we have previous knowledge, we know basically everything about what's going on. You only use Bayesian inference if there is something you don't know which can't be computed from the things you do know. Then, after doing an experiment, you work backwards to figure out the value of the things you didn't know.
Don't confuse Bayesian inference with Bayes' theorem, which is a theorem that doesn't have any philosophy attached to it. Bayesian inference makes use of this theorem on top of some philosophical assumptions, which may be questionable depending on how you use them.
Bayesian inference also requires you to give your "prior" beliefs, which are the beliefs you have before you do the experiment. If these beliefs are strongly biased away from the right answer, you will get Garbage Out.
Ok, that is really neat. I need to read more about this. I just did some searching on bayesian interference solution for the monty hall problem. And I am guessing they are applying the math whilst pretending to not have foreknowledge of the outcome?
https://sc5.io/posts/how-to-solve-the-monty-hall-problem-usi...
I certainly don't expect you to research this or explain every detail for me, I may have to take a class on this subject after I retire to satisfy my need to grasp this. Thank you for time, this was enlightening.
just to chime in. The monty-hall problem is a pure math (probability-theory) and not statistics-excersize, since we know all the probabilities beforehand and don't estimate.
It's very important to distinguish between bayesian inference and the bayesian theorem. The bayesian theorem can be proven and therefore must be true (if we accept the axioms). Bayesian inference has some real philosophical problems, since we often have no way to choose a justified prior. Frequentists approaches may also use the Bayesian theorem.
I always like the blog-posts of the angry statistician and he also has one about the monty-hall problem: http://angrystatistician.blogspot.com/2012/06/bayes-solution...
The important difference is that one is the probability of P(x == car), which is 1/3, and one is P(x == car | y == goat). Just apply the bayes theorem and calculate it yourself!