what the probability of heads is, as a property of the coin and the flip
I meant that the outcome of the toss was not known yet by me
Map/territory distinction. As a property of the actual coin and flip, the probability of heads is 0 or 1 (modulo some nonzero but utterly negligible quantum uncertainty); as a property of your state of knowledge, it can be 0.5.
This comment helped things come into better focus for me.
A frequentist believes that there is a probability of flipping heads, as a property of the coin and (yes, certainly) the conditions of the flipping. To a frequentist, this probability is independent of whether the outcome is determined or not and is even independent of what the outcome is. Consider the following sequence of flips: H T T
A frequentist believes that the probability of flipping heads was .5 all along right? The first ‘H’ and the second ‘T’ and the third ‘T’ were just discrete realizations of this probability.
The reasons why I’ve been calling this a Platonic perspective is because I think the critical difference in philosophy is the frequentist idea of this non-empirical “probability’ existing independent of realizations. The probability of flipping heads for a set of conditions is .5 whether you actually flip the coins or not. However, frequentists agree you must flip the coin to know that the probability was .5.
You might think this perspective is wrong-headed, and from a strict empirical view where you allow no Platonic entities/concepts, it kind of is. But the question I am really interested in is the following: to what extent is this point of view a choice we can be wrong or right about, or a perspective that some (or most?) people have hard-wired in their physical brain? Further, how can you argue that it isn’t useful when it demonstrably has been so useful? Perhaps it facilitates or is necessary for some categories of abstract thought.
But the question I am really interested in is the following: to what extent is this point of view a choice we can be wrong or right about, or a perspective that most people have hard-wired in their physical brain algorithms?
It could be hard-wired and still be right or wrong.
Correct, generally. But how could a perspective be wrong?
I can think of two ways a perspective can be wrong: either because it (a) asserts a fact about external reality that is not true or (b) yields false conclusions about the external world.
(a) Frequentists don’t assert anything extra about the empirical world, they assert the use of (and obstensibly, the “existence” of) something symbolic. From the empiricist perspective, it’s not really there. Like a little icon floating above or around the actual thing that your cursor doesn’t interact with, so it can’t be false in the empirical sense.
(b) It would be fascinating if the frequentist perspective yielded false conclusions,and in such a case, is there any doubt that people would develop and embrace new mathematics that avoided such errors? In fact, we already see this happening where physics at extreme scales seems to defy intuition. If someone wanted to propose a new theory of everything I don’t think anyone would ever criticize it on the grounds of not being frequentist. I guess the point here is just that it’s useful or not.
Later edit: Ok, I finally get it. Maybe the reason we don’t understand physics at the extreme scales is because the frequentist approach was evolved (hard-wired) for understanding intermediate physical scales and it’s (apparently) beginning to fail. You guys are using empirical philosophy to try and develop a new brand mathematics that won’t have these inborn errors of intuition. So while I argue that frequentism has definitely been productive so far, you argue that it is intrinsically limited based on philosophical principles.
A perspective can be wrong if it arbitrarily assigns a probability of 1 to an event that has a symmetrical alternative. Read the intro to My Bayesian Enlightenment for Eliezer’s description of a frequentist going wrong in this way with respect to the problem of the mathematician with two children, at least one of which is a boy.
No, Bayesian probability and orthodox statistics give exactly the same answers if the context of the problem is the same. The two schools may tend to have different ideas about what is a “natural” context, but any good textbook will always define exactly what the context is so that there is no guessing and no disagreement.
Nevertheless, which event with a symmetrical alternative were you referring to? (You are given that the women said she has at least 1 boy, so it would be correct to assign that probability 1 in the context of a given assumption, obviously when applying the orthodox method.) Both approaches work differently, but they both work.
Nevertheless, which event with a symmetrical alternative were you referring to?
Given that the women does have a boy and a girl, what is the probability that she would state that at least one of them is a boy? By symmetry, you would expect a priori, not knowing anything about this person’s preferences, that in the same conditions, she is equally likely to state that at least one of her children is a girl, so to assign the conditional probability higher than .5 does not make sense, so it is definitely not right for the frequentist Eliezer was talking with to act as though the conditional probability were 1. (The case could be made that the statement is also evidence that the woman has a tendency to say at least once child is a boy rather than that at least one child is a girl. But this is a small effect, and still does not justify assigning a conditional probability of 1.)
I think the frequentist approach could handle this problem if applied correctly, but it seems that frequentist in practice get it wrong because they do not even consider the conditional probability that they would observe a piece of evidence if a theory they are considering is true.
any good textbook will always define exactly what the context is so that there is no guessing and no disagreement.
If you read the article I cited, Eliezer did explain that this was a mangling of the original problem, in which the mathematician made the statement in response to a direct question, so one could reasonably approximate that she would make the statement exactly when it is true.
However, life does not always present us with neat textbook problems. Sometimes, the conditional probabilities are hard to figure out. I prefer the approach that says figure them out anyways to the one that glosses over their importance.
so to assign the conditional probability higher than .5 does not make sense, so it is definitely not right for the frequentist Eliezer was talking with to act as though the conditional probability were 1
In the “correct” formulation of the problem (the one in which the correct answer is 1⁄3), the frequentist tells us what the mother said as a given assumption; considering the prior <1 probability of this is rendered irrelevant because we are now working in the subset of probability space where she said that.
it seems that frequentist in practice get it wrong because they do not even consider the conditional probability that they would observe a piece of evidence if a theory they are considering is true.
Considering whether a theory is true is science—I completely agree science has important, necessary Bayesian elements.
Giving “probably” of actual outcome for the coin flip as ~1 looks like a type error, although it’s clear what you are saying. It’s more like P(coin is heads|coin is heads), tautologically 1, not really a probability.
As a property of the actual coin and flip, the probability of heads is 0 or 1 (modulo some nonzero but utterly negligible quantum uncertainty)
This mixes together two different kinds of probability, confusing the situation. There is nothing fuzzy about the events defining the possible outcomes, the fact that there is also indexical uncertainty imposed on your mind while it observes the outcome is from a different problem.
Map/territory distinction. As a property of the actual coin and flip, the probability of heads is 0 or 1 (modulo some nonzero but utterly negligible quantum uncertainty); as a property of your state of knowledge, it can be 0.5.
This comment helped things come into better focus for me.
A frequentist believes that there is a probability of flipping heads, as a property of the coin and (yes, certainly) the conditions of the flipping. To a frequentist, this probability is independent of whether the outcome is determined or not and is even independent of what the outcome is. Consider the following sequence of flips: H T T
A frequentist believes that the probability of flipping heads was .5 all along right? The first ‘H’ and the second ‘T’ and the third ‘T’ were just discrete realizations of this probability.
The reasons why I’ve been calling this a Platonic perspective is because I think the critical difference in philosophy is the frequentist idea of this non-empirical “probability’ existing independent of realizations. The probability of flipping heads for a set of conditions is .5 whether you actually flip the coins or not. However, frequentists agree you must flip the coin to know that the probability was .5.
You might think this perspective is wrong-headed, and from a strict empirical view where you allow no Platonic entities/concepts, it kind of is. But the question I am really interested in is the following: to what extent is this point of view a choice we can be wrong or right about, or a perspective that some (or most?) people have hard-wired in their physical brain? Further, how can you argue that it isn’t useful when it demonstrably has been so useful? Perhaps it facilitates or is necessary for some categories of abstract thought.
It could be hard-wired and still be right or wrong.
Correct, generally. But how could a perspective be wrong?
I can think of two ways a perspective can be wrong: either because it (a) asserts a fact about external reality that is not true or (b) yields false conclusions about the external world.
(a) Frequentists don’t assert anything extra about the empirical world, they assert the use of (and obstensibly, the “existence” of) something symbolic. From the empiricist perspective, it’s not really there. Like a little icon floating above or around the actual thing that your cursor doesn’t interact with, so it can’t be false in the empirical sense.
(b) It would be fascinating if the frequentist perspective yielded false conclusions,and in such a case, is there any doubt that people would develop and embrace new mathematics that avoided such errors? In fact, we already see this happening where physics at extreme scales seems to defy intuition. If someone wanted to propose a new theory of everything I don’t think anyone would ever criticize it on the grounds of not being frequentist. I guess the point here is just that it’s useful or not.
Later edit: Ok, I finally get it. Maybe the reason we don’t understand physics at the extreme scales is because the frequentist approach was evolved (hard-wired) for understanding intermediate physical scales and it’s (apparently) beginning to fail. You guys are using empirical philosophy to try and develop a new brand mathematics that won’t have these inborn errors of intuition. So while I argue that frequentism has definitely been productive so far, you argue that it is intrinsically limited based on philosophical principles.
A perspective can be wrong if it arbitrarily assigns a probability of 1 to an event that has a symmetrical alternative. Read the intro to My Bayesian Enlightenment for Eliezer’s description of a frequentist going wrong in this way with respect to the problem of the mathematician with two children, at least one of which is a boy.
No, Bayesian probability and orthodox statistics give exactly the same answers if the context of the problem is the same. The two schools may tend to have different ideas about what is a “natural” context, but any good textbook will always define exactly what the context is so that there is no guessing and no disagreement.
Nevertheless, which event with a symmetrical alternative were you referring to? (You are given that the women said she has at least 1 boy, so it would be correct to assign that probability 1 in the context of a given assumption, obviously when applying the orthodox method.) Both approaches work differently, but they both work.
Given that the women does have a boy and a girl, what is the probability that she would state that at least one of them is a boy? By symmetry, you would expect a priori, not knowing anything about this person’s preferences, that in the same conditions, she is equally likely to state that at least one of her children is a girl, so to assign the conditional probability higher than .5 does not make sense, so it is definitely not right for the frequentist Eliezer was talking with to act as though the conditional probability were 1. (The case could be made that the statement is also evidence that the woman has a tendency to say at least once child is a boy rather than that at least one child is a girl. But this is a small effect, and still does not justify assigning a conditional probability of 1.)
I think the frequentist approach could handle this problem if applied correctly, but it seems that frequentist in practice get it wrong because they do not even consider the conditional probability that they would observe a piece of evidence if a theory they are considering is true.
If you read the article I cited, Eliezer did explain that this was a mangling of the original problem, in which the mathematician made the statement in response to a direct question, so one could reasonably approximate that she would make the statement exactly when it is true.
However, life does not always present us with neat textbook problems. Sometimes, the conditional probabilities are hard to figure out. I prefer the approach that says figure them out anyways to the one that glosses over their importance.
In the “correct” formulation of the problem (the one in which the correct answer is 1⁄3), the frequentist tells us what the mother said as a given assumption; considering the prior <1 probability of this is rendered irrelevant because we are now working in the subset of probability space where she said that.
Considering whether a theory is true is science—I completely agree science has important, necessary Bayesian elements.
Considering whether a theory is true is not science, althought the two are certainly useful to each other.
Giving “probably” of actual outcome for the coin flip as ~1 looks like a type error, although it’s clear what you are saying. It’s more like P(coin is heads|coin is heads), tautologically 1, not really a probability.
Edited to clarify.
This mixes together two different kinds of probability, confusing the situation. There is nothing fuzzy about the events defining the possible outcomes, the fact that there is also indexical uncertainty imposed on your mind while it observes the outcome is from a different problem.
Yeah, it just felt like too much work to add ”...randomly sampling from future Everett branches according to the Born probabilities” or the like.
My point is that most of the time decision-theoretic problems are best handled in a deterministic world.