Couldn’t we just say that 1 was heads and ~1 was tails? Then it would be the same, right?
The problem has enough information to be solved.
really? Could you explain further, what do i know besides that if infinite independent people give me this problem and then flip the coin, it will not land heads 50% of the time? Knowing that it is two sided doesn’t change anything as far as I can tell. And what problem are you solving exactly? Just to make sure we’re on the same page.
I always thought of the coin bayes frequentest thing as being a dispute over definitions. It didn’t seem like you really derive P(heads) ≠ 1⁄2 as a frequentest, it is kind of already in the premises of the situation given your interpretation of “probability”, and your model of bias. On the other hand, the bayesian using his/her interpretation of “probability”, makes a one step inference using the principle of indifference that P(heads) = 1⁄2.
Neither of these are deep theorems of the respective statistical disciplines, their proof is trivial in both traditions. They are the statistical consequences of interpreting probability in different ways, i.e., modeling different things with probability to deal with uncertainty. I didn’t think of this difference as showing some deep difference between the bayesian and the frequentist; the frequentist and the bayesian are different in terms of their most basic surface apparatus; they use their tool (probability) to model different things (freqeuncy and degree of belief), which then gets them to two statistical methods, but still only one probability theory.
Could you explain further, what do i know besides that if infinite independent people give me this problem and then flip the coin, it will not land heads 50% of the time?
My wording was unclear—I’ve edited to fix. If people flip the same coin over and over, it won’t land heads 50% of the time. “Independent” means different coins. The key idea is to imagine, rather than flipping the same coin over and over, being in the same state of information over and over.
I always thought of the coin bayes frequentist thing as being a dispute over definitions.
It is, in a sense. But if you want to make any decisions about the coin (say that “coin” is whether or not the next solar flare will knock out your satellite), and frequentist and bayesian estimators disagree, which should you use? If you have certain desiderata about your decisions (e.g. you won’t take bets that are guaranteed to lose), this is a math problem with a right answer.
And then of course the question is, if this “frequentist probability” stuff is almost always the same as this “bayesian probability” stuff, and when it’s different you shouldn’t ever base decisions on it, why keep it as an alternate definition? Words should be useful.
Well, let’s not keep frequentism as a statistical method, cause bayes almost always if not always does better. But it is a theoretically interesting fact that komolgorov models finite frequencies, and our intuitions about infinite frequencies, and a fact that it does.
Understanding exactly what degrees of belief are, becomes a lot easier (I suppose) if you know that for some reason they are isomorphic to frequencies, and that for some reason they are also isomorphic to spatial measures. This not only allows us to solve problems of degree of belief by solving problems of frequency and area. But also, if we understand what degree of belief has in common with frequency and area, then we understand what it has in common with bayes.
Understanding exactly what degrees of belief are, becomes a lot easier (I suppose) if you know that for some reason they are isomorphic to frequencies
If probabilities were systematically wrong about the frequency of success in independent trials, there would be some other method of reasoning from incomplete information that was better than probabilistic logic. But since the real world obeys all the requirements for probabilistic logic (basically, causality works), there is no such method, and so frequencies match probabilities.
for some reason they are also isomorphic to spatial measures
Read a introductory chapter on set theory that uses pictures to represent sets, and you will understand why.
It’s certainly an interesting fact that these things behave the same. But it’s not an unsolved problem. We don’t have to keep a definition around that’s useless in the real world because of any lurking mystery.
Couldn’t we just say that 1 was heads and ~1 was tails? Then it would be the same, right?
really? Could you explain further, what do i know besides that if infinite independent people give me this problem and then flip the coin, it will not land heads 50% of the time? Knowing that it is two sided doesn’t change anything as far as I can tell. And what problem are you solving exactly? Just to make sure we’re on the same page.
I always thought of the coin bayes frequentest thing as being a dispute over definitions. It didn’t seem like you really derive P(heads) ≠ 1⁄2 as a frequentest, it is kind of already in the premises of the situation given your interpretation of “probability”, and your model of bias. On the other hand, the bayesian using his/her interpretation of “probability”, makes a one step inference using the principle of indifference that P(heads) = 1⁄2.
Neither of these are deep theorems of the respective statistical disciplines, their proof is trivial in both traditions. They are the statistical consequences of interpreting probability in different ways, i.e., modeling different things with probability to deal with uncertainty. I didn’t think of this difference as showing some deep difference between the bayesian and the frequentist; the frequentist and the bayesian are different in terms of their most basic surface apparatus; they use their tool (probability) to model different things (freqeuncy and degree of belief), which then gets them to two statistical methods, but still only one probability theory.
My wording was unclear—I’ve edited to fix. If people flip the same coin over and over, it won’t land heads 50% of the time. “Independent” means different coins. The key idea is to imagine, rather than flipping the same coin over and over, being in the same state of information over and over.
It is, in a sense. But if you want to make any decisions about the coin (say that “coin” is whether or not the next solar flare will knock out your satellite), and frequentist and bayesian estimators disagree, which should you use? If you have certain desiderata about your decisions (e.g. you won’t take bets that are guaranteed to lose), this is a math problem with a right answer.
And then of course the question is, if this “frequentist probability” stuff is almost always the same as this “bayesian probability” stuff, and when it’s different you shouldn’t ever base decisions on it, why keep it as an alternate definition? Words should be useful.
Well, let’s not keep frequentism as a statistical method, cause bayes almost always if not always does better. But it is a theoretically interesting fact that komolgorov models finite frequencies, and our intuitions about infinite frequencies, and a fact that it does.
Understanding exactly what degrees of belief are, becomes a lot easier (I suppose) if you know that for some reason they are isomorphic to frequencies, and that for some reason they are also isomorphic to spatial measures. This not only allows us to solve problems of degree of belief by solving problems of frequency and area. But also, if we understand what degree of belief has in common with frequency and area, then we understand what it has in common with bayes.
If probabilities were systematically wrong about the frequency of success in independent trials, there would be some other method of reasoning from incomplete information that was better than probabilistic logic. But since the real world obeys all the requirements for probabilistic logic (basically, causality works), there is no such method, and so frequencies match probabilities.
Read a introductory chapter on set theory that uses pictures to represent sets, and you will understand why.
It’s certainly an interesting fact that these things behave the same. But it’s not an unsolved problem. We don’t have to keep a definition around that’s useless in the real world because of any lurking mystery.