I have in mind very simple examples. Suppose that first I roll a die. If it doesn’t land on a 6, I then flip a biased coin that lands on heads 3⁄5 of the time. If it does land on a 6 I just record the result as ‘tails’. What is the probability that I get heads?
This is contrived so that the probability of heads is
5⁄6 x 3⁄5 = 1⁄2.
But do you think that that in saying this I mean something like “I don’t know the exact initial conditions… well enough to have any meaningful knowledge of how it’s going to land, and I can’t distinguish between the two options.” ?
Another example: Have you heard of the puzzle about the people randomly taking seats on the airplane? It’s a well-known probability brainteaser to which the answer is 1⁄2 but I don’t think many people would agree that saying the answer is 1⁄2 actually means something like “I don’t know the exact initial conditions… well enough to have any meaningful knowledge of how it’s going to land, and I can’t distinguish between the two options.”
There needn’t be any ‘indistinguishability of outcomes’ or ‘lack of information’ for something to have probability 0.5, it can just..well… be the actual result of calculating two distinguishable complementary outcomes.
I don’t understand how either of those are supposed to be a counterexample. If I don’t know what seat is going to be chosen randomly each time, then I don’t have enough information to distinguish between the outcomes. All other information about the problem (like the fact that this is happening on a plane rather than a bus) is irrelevant to the outcome I care about.
This does strike me as somewhat tautological, since I’m effectively defining “irrelevant information” as “information that doesn’t change the probability of the outcome I care about”. I’m not sure how to resolve this; it certainly seems like I should be able to identify that the type of vehicle is irrelevant to the question posed and discard that information.
OK I think this will be my last message in this exchange but I’m still confused. I’ll try one more time to explain what I’m getting at.
I’m interested in what your precise definition of subjective probability is.
One relevant thing I saw was the following sentence:
If I say that a coin is 50% likely to come up heads, that’s me saying that I don’t know the exact initial conditions of the coin well enough to have any meaningful knowledge of how it’s going to land, and I can’t distinguish between the two options.
It seems to give something like a definition of what it means to say something has a 50% chance. i.e. I interpret your sentence as claiming that a statement like ‘The probability of A is 1⁄2’ means or is somehow the same as a statement a bit like
[*] ‘I don’t know the exact conditions and don’t have enough meaningful/relevant knowledge to distinguish between the possible occurrence of (A) and (not A)‘
My reaction was: This can’t possibly be a good definition.
The airplane puzzle was supposed to be a situation where there is a clear ‘difference’ in the outcomes—either the last person is in the 1 seat that matches their ticket number or they’re not. - they’re in one of the other 99 seats. It’s not as if it’s a clearly symmetric situation from the point of view of the outcomes. So it was supposed to be an example where statement [*] does not hold, but where the probability is 1⁄2. It seems you don’t accept that; it seems to me like you think that statement [*] does in fact hold in this case.
But tbh it feels sorta like you’re saying you can’t distinguish between the outcomes because you already know the answer is 1/2! i.e. Even if I accept that the outcomes are somehow indistinguishable, the example is sufficiently complicated on a first reading that there’s no way you’d just look at it and go “hmm I guess I can’t distinguish so it’s 1/2”, i.e. if your definition were OK it could be used to justify the answer to the puzzle, but that doesn’t seem right to me either.
No, I think what I said was correct? What’s an example that you think conflicts with that interpretation?
I have in mind very simple examples. Suppose that first I roll a die. If it doesn’t land on a 6, I then flip a biased coin that lands on heads 3⁄5 of the time. If it does land on a 6 I just record the result as ‘tails’. What is the probability that I get heads?
This is contrived so that the probability of heads is
5⁄6 x 3⁄5 = 1⁄2.
But do you think that that in saying this I mean something like “I don’t know the exact initial conditions… well enough to have any meaningful knowledge of how it’s going to land, and I can’t distinguish between the two options.” ?
Another example: Have you heard of the puzzle about the people randomly taking seats on the airplane? It’s a well-known probability brainteaser to which the answer is 1⁄2 but I don’t think many people would agree that saying the answer is 1⁄2 actually means something like “I don’t know the exact initial conditions… well enough to have any meaningful knowledge of how it’s going to land, and I can’t distinguish between the two options.”
There needn’t be any ‘indistinguishability of outcomes’ or ‘lack of information’ for something to have probability 0.5, it can just..well… be the actual result of calculating two distinguishable complementary outcomes.
I don’t understand how either of those are supposed to be a counterexample. If I don’t know what seat is going to be chosen randomly each time, then I don’t have enough information to distinguish between the outcomes. All other information about the problem (like the fact that this is happening on a plane rather than a bus) is irrelevant to the outcome I care about.
This does strike me as somewhat tautological, since I’m effectively defining “irrelevant information” as “information that doesn’t change the probability of the outcome I care about”. I’m not sure how to resolve this; it certainly seems like I should be able to identify that the type of vehicle is irrelevant to the question posed and discard that information.
OK I think this will be my last message in this exchange but I’m still confused. I’ll try one more time to explain what I’m getting at.
I’m interested in what your precise definition of subjective probability is.
One relevant thing I saw was the following sentence:
It seems to give something like a definition of what it means to say something has a 50% chance. i.e. I interpret your sentence as claiming that a statement like ‘The probability of A is 1⁄2’ means or is somehow the same as a statement a bit like
[*] ‘I don’t know the exact conditions and don’t have enough meaningful/relevant knowledge to distinguish between the possible occurrence of (A) and (not A)‘
My reaction was: This can’t possibly be a good definition.
The airplane puzzle was supposed to be a situation where there is a clear ‘difference’ in the outcomes—either the last person is in the 1 seat that matches their ticket number or they’re not. - they’re in one of the other 99 seats. It’s not as if it’s a clearly symmetric situation from the point of view of the outcomes. So it was supposed to be an example where statement [*] does not hold, but where the probability is 1⁄2. It seems you don’t accept that; it seems to me like you think that statement [*] does in fact hold in this case.
But tbh it feels sorta like you’re saying you can’t distinguish between the outcomes because you already know the answer is 1/2! i.e. Even if I accept that the outcomes are somehow indistinguishable, the example is sufficiently complicated on a first reading that there’s no way you’d just look at it and go “hmm I guess I can’t distinguish so it’s 1/2”, i.e. if your definition were OK it could be used to justify the answer to the puzzle, but that doesn’t seem right to me either.