The subjective interpretation: Uncertainty is in the mind, not the environment. If I flip a coin and slap it against my wrist, it’s already landed either heads or tails. The fact that I don’t know whether it landed heads or tails is a fact about me, not a fact about the coin. The claim “I think this coin is heads with probability 50%” is an expression of my own ignorance, and 50% probability means that I’d bet at 1 : 1 odds (or better) that the coin came up heads.
Hold on, you’re pulling a fast one here—you’ve substituted the question of “what is the probability that this coin which I have already flipped but haven’t looked at yet has already landed heads” for the question of “what is the probability that this coin which I am about to flip will land lands”!
It is obviously easy to see what the subjective interpretation means in the case of the former question—as you say, the coin is already heads or tails, no matter that I don’t know which it is. But it is not so easy to see how the subjective interpretation makes sense when applied to the latter question—and that is what people generally have difficulty with, when they have trouble accepting subjectivism.
Doesn’t it mean the same thing in either case? Either way, I don’t know which way the coin will land or has landed, and I have some odds at which I’ll be willing to make a bet. I don’t see the problem.
(Though my willingness to bet at all will generally go down over time in the “already flipped” case, due to the increasing possibility that whoever is offering the bet somehow looked at the coin in the intervening time.)
The difference is (to the naive view; I don’t necessarily endorse it) that in the case where the coin has landed, I do not know how it landed, but there’s a sense in which I could, in theory, know; there is, in any case, something to know; there is a fact of the matter about how the coin has landed, but I do not know that fact. So the “probability” of it having landed heads, or tails—the uncertainty—is, indeed, entirely in my mind.
But in the case where the coin has yet to be tossed, there is as yet no case of the matter about whether it’s heads or tails! I don’t know whether it’ll land heads or tails, but nor could I know; there’s nothing to know! (Or do you say the future is predetermined?—asks the naive interlocutor—Else how else may one talk about probability being merely “in the mind”, for something which has not happened yet?)
Whatever the answers to these questions may be, they are certainly not obvious or simple answers… and that is my objection to the OP: that it attempts to pass off a difficult and confusing conceptual question as a simple and obvious one, thereby failing to do justice to those who find it confusing or difficult.
the coin is already heads or tails, no matter that I don’t know which it is
it’s worse than that. All you know that the coin has landed. You need further observations to learn more. Maybe it will slip from your hand and fall on the ground. Maybe you will be distracted with reading LW and forget to check. Maybe you don’t remember which side to check, the wrist or the hand side. You can insist that the coin has already landed and therefore it has landed either heads or tails, but that is not a useful supposition until you actually look. Think just a little way back: the coin is about to land, but not quite yet. Is it the same as the coin has landed? Almost, but not quite. what about a little ways further back? The uncertainty about the outcome is even more. So, there is nothing special about the landed coin until you actually look, beyond a certain level of probabilities. A pragmatic approach (I refuse to wade into the ideological debate between militant frequentists and militant Bayesians) would be to use all available information to make the best prediction possible, depending on the question asked.
He never said “will land heads”, though. He just said “a flipped coin has a chance of landing heads”, which is not a timeful statement. EDIT: no longer confident that this is the case
Didn’t the post already counter your second paragraph? The subjective interpretation can be a superset of the propensity interpretation.
Actually, the assignment of probability 1 to an event that has happened is also subjective. You don’t know that it had to occur with complete inevitability, ie you don’t know that it had a conditional probability of 1 relative to the preceding state of the universe.
You are setting it to 1 because it is a given as far as you are concerned
The question is not “what is the probability that the coin would have landed heads”. The question is, “what is the probability that the coin has in fact landed heads”!
If you are interested in the objective probability of the coin flip,the it only has one value because it is only one event. In a deterministic universe the objective probability is 1, in a suitably indeterministic universe it is always 0.5.
If you think the questions “what will it be” and “what was it” are different, you are dealing with subjective probability, because the difference the passage of time makes is a difference in the information available to you, the subject.
Failing to distinguish objective and subjective probability leads to confusion. For instance, the sleeping beauty paradox is only a paradox if you expect all observers to calculate the same probability despite the different information available to them.
Hold on, you’re pulling a fast one here—you’ve substituted the question of “what is the probability that this coin which I have already flipped but haven’t looked at yet has already landed heads” for the question of “what is the probability that this coin which I am about to flip will land lands”!
It is obviously easy to see what the subjective interpretation means in the case of the former question—as you say, the coin is already heads or tails, no matter that I don’t know which it is. But it is not so easy to see how the subjective interpretation makes sense when applied to the latter question—and that is what people generally have difficulty with, when they have trouble accepting subjectivism.
Doesn’t it mean the same thing in either case? Either way, I don’t know which way the coin will land or has landed, and I have some odds at which I’ll be willing to make a bet. I don’t see the problem.
(Though my willingness to bet at all will generally go down over time in the “already flipped” case, due to the increasing possibility that whoever is offering the bet somehow looked at the coin in the intervening time.)
The difference is (to the naive view; I don’t necessarily endorse it) that in the case where the coin has landed, I do not know how it landed, but there’s a sense in which I could, in theory, know; there is, in any case, something to know; there is a fact of the matter about how the coin has landed, but I do not know that fact. So the “probability” of it having landed heads, or tails—the uncertainty—is, indeed, entirely in my mind.
But in the case where the coin has yet to be tossed, there is as yet no case of the matter about whether it’s heads or tails! I don’t know whether it’ll land heads or tails, but nor could I know; there’s nothing to know! (Or do you say the future is predetermined?—asks the naive interlocutor—Else how else may one talk about probability being merely “in the mind”, for something which has not happened yet?)
Whatever the answers to these questions may be, they are certainly not obvious or simple answers… and that is my objection to the OP: that it attempts to pass off a difficult and confusing conceptual question as a simple and obvious one, thereby failing to do justice to those who find it confusing or difficult.
it’s worse than that. All you know that the coin has landed. You need further observations to learn more. Maybe it will slip from your hand and fall on the ground. Maybe you will be distracted with reading LW and forget to check. Maybe you don’t remember which side to check, the wrist or the hand side. You can insist that the coin has already landed and therefore it has landed either heads or tails, but that is not a useful supposition until you actually look. Think just a little way back: the coin is about to land, but not quite yet. Is it the same as the coin has landed? Almost, but not quite. what about a little ways further back? The uncertainty about the outcome is even more. So, there is nothing special about the landed coin until you actually look, beyond a certain level of probabilities. A pragmatic approach (I refuse to wade into the ideological debate between militant frequentists and militant Bayesians) would be to use all available information to make the best prediction possible, depending on the question asked.
He never said “will land heads”, though. He just said “a flipped coin has a chance of landing heads”, which is not a timeful statement. EDIT: no longer confident that this is the case
Didn’t the post already counter your second paragraph? The subjective interpretation can be a superset of the propensity interpretation.
Actually, the assignment of probability 1 to an event that has happened is also subjective. You don’t know that it had to occur with complete inevitability, ie you don’t know that it had a conditional probability of 1 relative to the preceding state of the universe. You are setting it to 1 because it is a given as far as you are concerned
The question is not “what is the probability that the coin would have landed heads”. The question is, “what is the probability that the coin has in fact landed heads”!
If you are interested in the objective probability of the coin flip,the it only has one value because it is only one event. In a deterministic universe the objective probability is 1, in a suitably indeterministic universe it is always 0.5.
If you think the questions “what will it be” and “what was it” are different, you are dealing with subjective probability, because the difference the passage of time makes is a difference in the information available to you, the subject.
Failing to distinguish objective and subjective probability leads to confusion. For instance, the sleeping beauty paradox is only a paradox if you expect all observers to calculate the same probability despite the different information available to them.