Distressingly few people have publicly changed their mind on this thread. Various people show great persistence in believing the wrong answer—even when the problem has been explained. Perhaps overconfidence is involved.
I changed my mind from “1/3 is the right answer” to “The answer is obviously 1⁄2 or 1⁄3 once you’ve gotten clear on what question is being asked”. I’m not sure if I did so publicly. It seems to me that other folks have changed their minds similarly. I think I see an isomorphism to POAT here, as well as any classic Internet debate amongst intelligent people.
I’m also not sure if you’re serious, but if you assign a 50% probability to the relevant question being the one with the correct answer of ‘1/2’ and a 50% probability to the relevant question being the one with the correct answer of ‘1/3’ then ‘5/12’ should maximize your payoff over multiple such cases if you’re well-calibrated.
Phil and I seem to think the problem is sufficiently clearly specified to give an answer to. If you think 1⁄2 is a defensible answer, how would you reply to Robin Hanson’s comment?
FWIW, on POAT I am inclined towards “Whoever asked this question is an idiot”.
there are some problems similar to this one for which the answer is 1⁄2
there are some problems similar to this one for which the answer is 1⁄3
people seem to be disagreeing which sort of problem this is
all debate has devolved to debate over the meanings of words (in the problem statement and elsewhere)
Given this, I think it’s obvious that the problem is ambiguous, and arguing whether the problem is ambiguous is counterproductive as compared to just sorting out which sort of problem you’re responding to and what the right answer is.
IMHO, different people giving different answers to problems does not mean it is ambiguous. Nor does people disagreeing over the meanings of words. Words do have commonly-accepted meanings—that is how people communicate.
I’m coming around to the 1⁄2 point of view, from an initial intuition that 1⁄3 made most sense, but that it mostly depended on what you took “credence” to mean.
My main new insight is that the description of the set-up deliberately introduces confusion, it makes it seem as if there are two very different situations of “background knowledge”, X being “a coin flip” and X’ being “a coin flip plus drugs and amnesia”. So that P(heads|X) may not equal P(heads|X’).
This comment makes the strongest case I’ve seen that the difference is one that makes no difference. Yes, the setup description strongly steers us in the direction of taking “credence” to refer to the number of times my guess about the event is right. If Beauty got a candy bar each time she guessed right she’d want to guess tails. But on reflection what seems to matter in terms of being well-calibrated on the original question is how many distinct events I’m right about.
Take away the drug and amnesia, and suppose instead that Beauty is just absent-minded. On Tuesday when you ask her, she says: “Oh crap, you asked me that yesterday, and I said 1⁄2. But I totally forget if you were going to ask me twice on tails or on heads. You’d think with all they wrote about this setup I’d remember it. I’ve no idea really, I’ll have to go with 1⁄2 again. Should be 1 for one or the other, but what can I say, I just forget.”
I’m less than impressed with the signal-to-noise ratio in the recent discussion, in particular the back-and-forth between neq1 and timtyler. As a general observation backed by experience in other fora, the more people are responding in real time to a controversial topic, the less likely they are to be contributing useful insights.
I’ve been thinking 1⁄2 as well (though I’m also definitely in the “problem is underdefined” camp).
Here is how describe the appropriate payoff scheme. Prior to the experiment (but after learning the details) Beauty makes a wager with the Prince. If the coin comes up heads the Prince will pay Beauty $10. If it comes up tails Beauty will pay $10. Even odds. This wager represents Beauty’s prior belief that the coin is fair and head/tails have equal probability: her credence that heads will or did come up. At any point before Beauty learns what day of the week it is she is free alter the bet such that she takes tails but must pay $10 more dollars to do so (making the odds 2:1).
Beauty should at no point (before learning what day of the week it is) alter the wager. Which means when she is asked what her credence is that the coin came up heads she should continue to say 1⁄2.
This seems at least as good an payoff interpretation as a new bet every time Beauty is asked about her credence.
You don’t measure an agent’s subjective probability like that, though—not least because in many cases it would be bad experimental methodology. Bets made which are intended to represent the subject’s probability at a particular moment should pay out—and not be totally ignored. Otherwise there may not be any motivation for the subject making the bet to give an answer that represents what they really think. If the subject knows that they won’t get paid on a particular bet, that can easily defeat the purpose of offering them a bet in the first place.
If Beauty forgets what is going on—or can’t add up—her subjective probability could potentially be all over the shop.
However, the problem description states explicitly that: “During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.”
This seems to me to weigh pretty heavily against the hypothesis that she may have forgotten the details of the experiment.
In the case where she remembers what’s going on, when you ask her on Tuesday what her credence is in Heads, she says “Well, since you asked me yesterday, the coin must have come up Tails; therefore I’m updating my credence in Heads to 0.”
The setup makes her absent-minded (in a different way than I suggest above). It erases information she would normally have. If you told her “It’s Monday”, she’d say 1⁄2. If you told her “It’s Tuesday”, she’d say 0. The amnesia prevents Beauty from conditioning on what day it is when she’s asked.
Prior to the experiment, Beauty has credence 1⁄2 in either Heads or Tails. To argue that she updates that credence to 1⁄3, she must be be taking into account some new information, but we’ve established that it can’t be the day, as that gets erased. So what it is?
Jonathan_Lee’s post suggests that Beauty is “conditioning on observers”. I don’t really understand what that means. The first analogy he makes is to an identical-copy experiment, but we’ve been over that already, and I’ve come to the conclusion that the answer in that case is “it depends”.
Yes. I noted then that the description of the setup could make a difference, in that it represents different background knowledge.
It does not follow that it does make a a difference.
When I say “prior to the experiment”, I mean chronologically, i.e. if you ask Beauty on Sunday, what her credence is then in the proposition “the coin will come up heads”, she will answer 1⁄2.
Once Beauty wakes up and is asked the question, she conditions on the fact that the experiment is now ongoing. But what information does that bring, exactly?
When Beauty knows she will be the subject of the experiment (and its design), she will know she is more likely to be observing tails. Since the experiment involves administering Beauty drugs, it seems fairly likely that she knew she would be the subject of the experiment before it started—and so she is likely to have updated her expectations of observing heads back then.
If she is asked: “if you wake up with amnesia in this experiment, what odds of the coin being heads will you give”, then yes. She doesn’t learn anything to make her change her mind about the odds she will give after the experiment has started.
That isn’t a symmetrical question. We’re not asking for her belief about what odds she will give. We’re asking what her odds are for a particular event (namely a coin flip at time t1 being heads).
The question “What is your credence now for the proposition that our coin landed heads?” doesn’t appear to make very much sense before the coin is flipped. Remember that we are told in the description that the coin is only flipped once—and that it happens after Beauty is given a drug that sends her to sleep.
Beauty should probably clarify with the experimenters which previous coin is being discussed, and then, based on what she is told about the circumstances surrounding that coin flip, she should use her priors to answer.
The English language doesn’t have a timeless tense. So we can’t actually phrase the question without putting the speaker into some time relative to the event we’re speaking of. But that doesn’t mean we can’t recognize that the question being asked is a timeless one. We have a coordinate system that lets us refer to objects and events throughout space and time… it doesn’t matter when the agent is: the probability of the event occurring can be estimated before, after and during just as easily (easy mathematically, not practically). That is why I used the phrasing “the coin flip at time t1 being heads”. The coin flip at t1 can be heads or tails. Since we know it is a fair coin toss we start with P=1/2 for heads. If you want the final answer to be something other than 1⁄2 you need to show when and how Beauty gets additional information about the coin toss.
The question asked in the actual problem has the word “now” in it. You said I didn’t answer a “symmetrical” question—but it seems as though the question you wanted me to answer is not very “symmetrical” either.
If Beauty is asked before the experiment the probabality she expects the coin to show heads at the end of the experiment, she will answer 1⁄2. However, in the actual problem she is not asked that.
We’re supposed to be Bayesians. It doesn’t matter whether the question asks “now” “in 500 B.C.E.” or “at the heat death of the universe” unless our information has changed, the time the prediction is made is irrelevant.
(ETA: Okay, I guess at the heat death of the universe the information would have changed. But you get my point :-)
But here you’ve put the time-indexical “now” into your description of the event. You’re asking for P(it is night, now). In the Beauty case question asked is what is P(heads), now. In the first case every moment that goes by we’re talking about a temporally distinct event. You’re actually asking about a different event every moment- so it isn’t surprising that the answer changes from moment to moment. The Sleeping Beauty problem is always about the same event.
The coin flip doesn’t change—but Beauty does. She goes in one end of the expertiment and comes out the other side, and she knows roughly where she is on that timeline. Probabalities are subjective—and in this example we are asked for Beauty’s “credence”—i.e. her subjective probability at a particular point in time. That’s a function of the observer, not just the observed.
Yes. But subjective probability is a function of the information someone has not where they are on the time-line. Which is why people keep asking what information Beauty is updating on. We’re covering 101 stuff at this point.
...and going round in circles, I might note. We did already discuss the issue of exactly when Beauty updates close by—here.
Also, we already know where we differ. We consider “subjective probability” to refer to different things. Given your notion of “subjective probability”, your position makes perfect sense, IMO. I just don’t think that is how scientists generally use the term.
Well you tried to answer the question. I suggested your answer was ridiculous and explained why and I have been rebutting your responses since then. So no, we’re not going in circles. I’m objecting to your answer to the updating question and rebutting your responses to my objection.
Here is what happened in this thread.
You suggested that Beauty would have estimated heads at 1⁄3 prior to the experiment.
I said ‘Wha?!?’
You tried to make Beauty’s pre-experiment estimation about what she was going to say when she woke up.
I pointed out that that question was about a different event (the saying) than the question “What is your credence now for the proposition that our coin landed heads?” is about (the coin flip)
You claimed that it didn’t make sense to ask that question (about the coin having landed heads) before the coin flip happens.
I showed how even though English requires us to use tense we can make the question time symmetrical by inventing temporal coordinates (t1) and speaking of subjective probability of heads at t1 at any time Beauty exists.
You claimed that the probability of heads at the end of the experiment was somehow different from the probability of heads at some other time (presumably when she is asked).
I pointed out that time is irrelevant and what matters is her information- an elementary point which I shouldn’t have to make to someone who was last night trashing the OP for supposedly not knowing anything about probability (and I’m a philosopher not a math guy!).
In conclusion: My claim is that for Beauty to answer 1⁄3 for the probability of the time invariant event “coin toss by experimenter at time t1 being heads” she needs to get new information since the prior for that even is obviously 1⁄2. No one has ever pointed to what new information she gets. You tried to claim that Beauty updates as soon as she gets the details of the experiment: but that can’t be right. The details of the experiment can’t alter the outcome of a fair coin toss. So where is the updating?!
It’s hard to tell but I’m not sure your notion of “subjective probability” is coherent- specifically because you keep talking about different events depending on what time you’re in. That sounds like a recipe for disaster. But alright.
I just don’t think that is how scientists generally use the term.
Does this mean we can just agree to specify payouts in our probability problems from now on? Or must we now investigate which one of us is using the term the way scientists do? Unfortunately this disagreement suggest to me that scientists may not know exactly what they mean by subjective probability.
Subjective probability is a basic concept in decision theory. Scientists have certainly tried hard to say exactly what they mean by the term. E.g. see this one, from 1963:
“A Definition of Subjective Probability”—F. J. Anscombe; R. J. Aumann
Sure. I don’t see anything in there to suggest that subjective probability isn’t time symmetrical (by which I mean that a subjective probability regarding an event can be held at any time and there is not reason for the probability to change unless the person’s evidence changes). Can you do a better job formalizing what your alternative is?
Distressingly few people have publicly changed their mind on this thread. Various people show great persistence in believing the wrong answer—even when the problem has been explained. Perhaps overconfidence is involved.
I changed my mind from “1/3 is the right answer” to “The answer is obviously 1⁄2 or 1⁄3 once you’ve gotten clear on what question is being asked”. I’m not sure if I did so publicly. It seems to me that other folks have changed their minds similarly. I think I see an isomorphism to POAT here, as well as any classic Internet debate amongst intelligent people.
I’m not sure whether this is legitimate or a joke, but if the question is unclear about whether 1⁄2 or 1⁄3 is better, maybe 5⁄12 is a good answer.
I’m also not sure if you’re serious, but if you assign a 50% probability to the relevant question being the one with the correct answer of ‘1/2’ and a 50% probability to the relevant question being the one with the correct answer of ‘1/3’ then ‘5/12’ should maximize your payoff over multiple such cases if you’re well-calibrated.
Phil and I seem to think the problem is sufficiently clearly specified to give an answer to. If you think 1⁄2 is a defensible answer, how would you reply to Robin Hanson’s comment?
FWIW, on POAT I am inclined towards “Whoever asked this question is an idiot”.
Actually I think it would make more sense to reply to my own comment in response to this. link
I am not sure that is going anywhere.
Personally, I think I pretty-much nailed what was wrong with the claim that the problem was ambiguous here.
I think that we’ve established the following:
there are some problems similar to this one for which the answer is 1⁄2
there are some problems similar to this one for which the answer is 1⁄3
people seem to be disagreeing which sort of problem this is
all debate has devolved to debate over the meanings of words (in the problem statement and elsewhere)
Given this, I think it’s obvious that the problem is ambiguous, and arguing whether the problem is ambiguous is counterproductive as compared to just sorting out which sort of problem you’re responding to and what the right answer is.
IMHO, different people giving different answers to problems does not mean it is ambiguous. Nor does people disagreeing over the meanings of words. Words do have commonly-accepted meanings—that is how people communicate.
I’m coming around to the 1⁄2 point of view, from an initial intuition that 1⁄3 made most sense, but that it mostly depended on what you took “credence” to mean.
My main new insight is that the description of the set-up deliberately introduces confusion, it makes it seem as if there are two very different situations of “background knowledge”, X being “a coin flip” and X’ being “a coin flip plus drugs and amnesia”. So that P(heads|X) may not equal P(heads|X’).
This comment makes the strongest case I’ve seen that the difference is one that makes no difference. Yes, the setup description strongly steers us in the direction of taking “credence” to refer to the number of times my guess about the event is right. If Beauty got a candy bar each time she guessed right she’d want to guess tails. But on reflection what seems to matter in terms of being well-calibrated on the original question is how many distinct events I’m right about.
Take away the drug and amnesia, and suppose instead that Beauty is just absent-minded. On Tuesday when you ask her, she says: “Oh crap, you asked me that yesterday, and I said 1⁄2. But I totally forget if you were going to ask me twice on tails or on heads. You’d think with all they wrote about this setup I’d remember it. I’ve no idea really, I’ll have to go with 1⁄2 again. Should be 1 for one or the other, but what can I say, I just forget.”
I’m less than impressed with the signal-to-noise ratio in the recent discussion, in particular the back-and-forth between neq1 and timtyler. As a general observation backed by experience in other fora, the more people are responding in real time to a controversial topic, the less likely they are to be contributing useful insights.
I’m not ruling out changing my mind again. :)
I’ve been thinking 1⁄2 as well (though I’m also definitely in the “problem is underdefined” camp).
Here is how describe the appropriate payoff scheme. Prior to the experiment (but after learning the details) Beauty makes a wager with the Prince. If the coin comes up heads the Prince will pay Beauty $10. If it comes up tails Beauty will pay $10. Even odds. This wager represents Beauty’s prior belief that the coin is fair and head/tails have equal probability: her credence that heads will or did come up. At any point before Beauty learns what day of the week it is she is free alter the bet such that she takes tails but must pay $10 more dollars to do so (making the odds 2:1).
Beauty should at no point (before learning what day of the week it is) alter the wager. Which means when she is asked what her credence is that the coin came up heads she should continue to say 1⁄2.
This seems at least as good an payoff interpretation as a new bet every time Beauty is asked about her credence.
You don’t measure an agent’s subjective probability like that, though—not least because in many cases it would be bad experimental methodology. Bets made which are intended to represent the subject’s probability at a particular moment should pay out—and not be totally ignored. Otherwise there may not be any motivation for the subject making the bet to give an answer that represents what they really think. If the subject knows that they won’t get paid on a particular bet, that can easily defeat the purpose of offering them a bet in the first place.
This doesn’t make any sense to me. Or at least the sense it does make doesn’t sound like sufficient reason to reject the interpretation.
If Beauty forgets what is going on—or can’t add up—her subjective probability could potentially be all over the shop.
However, the problem description states explicitly that: “During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.”
This seems to me to weigh pretty heavily against the hypothesis that she may have forgotten the details of the experiment.
In the case where she remembers what’s going on, when you ask her on Tuesday what her credence is in Heads, she says “Well, since you asked me yesterday, the coin must have come up Tails; therefore I’m updating my credence in Heads to 0.”
The setup makes her absent-minded (in a different way than I suggest above). It erases information she would normally have. If you told her “It’s Monday”, she’d say 1⁄2. If you told her “It’s Tuesday”, she’d say 0. The amnesia prevents Beauty from conditioning on what day it is when she’s asked.
Prior to the experiment, Beauty has credence 1⁄2 in either Heads or Tails. To argue that she updates that credence to 1⁄3, she must be be taking into account some new information, but we’ve established that it can’t be the day, as that gets erased. So what it is?
Jonathan_Lee’s post suggests that Beauty is “conditioning on observers”. I don’t really understand what that means. The first analogy he makes is to an identical-copy experiment, but we’ve been over that already, and I’ve come to the conclusion that the answer in that case is “it depends”.
Re: “Prior to the experiment, Beauty has credence 1⁄2 in either Heads or Tails.”
IMO, we’ve been over that adequately here. Your comment there seemed to indicate that you understood exactly when Beauty updates.
Yes. I noted then that the description of the setup could make a difference, in that it represents different background knowledge.
It does not follow that it does make a a difference.
When I say “prior to the experiment”, I mean chronologically, i.e. if you ask Beauty on Sunday, what her credence is then in the proposition “the coin will come up heads”, she will answer 1⁄2.
Once Beauty wakes up and is asked the question, she conditions on the fact that the experiment is now ongoing. But what information does that bring, exactly?
When Beauty knows she will be the subject of the experiment (and its design), she will know she is more likely to be observing tails. Since the experiment involves administering Beauty drugs, it seems fairly likely that she knew she would be the subject of the experiment before it started—and so she is likely to have updated her expectations of observing heads back then.
The question is
Your claim is that Beauty answers “1/3” before the experiment even begins?
(?!?!!)
If she is asked: “if you wake up with amnesia in this experiment, what odds of the coin being heads will you give”, then yes. She doesn’t learn anything to make her change her mind about the odds she will give after the experiment has started.
That isn’t a symmetrical question. We’re not asking for her belief about what odds she will give. We’re asking what her odds are for a particular event (namely a coin flip at time t1 being heads).
The question “What is your credence now for the proposition that our coin landed heads?” doesn’t appear to make very much sense before the coin is flipped. Remember that we are told in the description that the coin is only flipped once—and that it happens after Beauty is given a drug that sends her to sleep.
Beauty should probably clarify with the experimenters which previous coin is being discussed, and then, based on what she is told about the circumstances surrounding that coin flip, she should use her priors to answer.
The English language doesn’t have a timeless tense. So we can’t actually phrase the question without putting the speaker into some time relative to the event we’re speaking of. But that doesn’t mean we can’t recognize that the question being asked is a timeless one. We have a coordinate system that lets us refer to objects and events throughout space and time… it doesn’t matter when the agent is: the probability of the event occurring can be estimated before, after and during just as easily (easy mathematically, not practically). That is why I used the phrasing “the coin flip at time t1 being heads”. The coin flip at t1 can be heads or tails. Since we know it is a fair coin toss we start with P=1/2 for heads. If you want the final answer to be something other than 1⁄2 you need to show when and how Beauty gets additional information about the coin toss.
The question asked in the actual problem has the word “now” in it. You said I didn’t answer a “symmetrical” question—but it seems as though the question you wanted me to answer is not very “symmetrical” either.
If Beauty is asked before the experiment the probabality she expects the coin to show heads at the end of the experiment, she will answer 1⁄2. However, in the actual problem she is not asked that.
We’re supposed to be Bayesians. It doesn’t matter whether the question asks “now” “in 500 B.C.E.” or “at the heat death of the universe” unless our information has changed, the time the prediction is made is irrelevant.
(ETA: Okay, I guess at the heat death of the universe the information would have changed. But you get my point :-)
If you are locked in a lead-lined box, the answer to question “is it night time outside now” varies over time—even though you learn nothing new.
Similarly with Beauty, as she moves through the experimental procedure.
But here you’ve put the time-indexical “now” into your description of the event. You’re asking for P(it is night, now). In the Beauty case question asked is what is P(heads), now. In the first case every moment that goes by we’re talking about a temporally distinct event. You’re actually asking about a different event every moment- so it isn’t surprising that the answer changes from moment to moment. The Sleeping Beauty problem is always about the same event.
The coin flip doesn’t change—but Beauty does. She goes in one end of the expertiment and comes out the other side, and she knows roughly where she is on that timeline. Probabalities are subjective—and in this example we are asked for Beauty’s “credence”—i.e. her subjective probability at a particular point in time. That’s a function of the observer, not just the observed.
Yes. But subjective probability is a function of the information someone has not where they are on the time-line. Which is why people keep asking what information Beauty is updating on. We’re covering 101 stuff at this point.
...and going round in circles, I might note. We did already discuss the issue of exactly when Beauty updates close by—here.
Also, we already know where we differ. We consider “subjective probability” to refer to different things. Given your notion of “subjective probability”, your position makes perfect sense, IMO. I just don’t think that is how scientists generally use the term.
Well you tried to answer the question. I suggested your answer was ridiculous and explained why and I have been rebutting your responses since then. So no, we’re not going in circles. I’m objecting to your answer to the updating question and rebutting your responses to my objection.
Here is what happened in this thread.
You suggested that Beauty would have estimated heads at 1⁄3 prior to the experiment.
I said ‘Wha?!?’
You tried to make Beauty’s pre-experiment estimation about what she was going to say when she woke up.
I pointed out that that question was about a different event (the saying) than the question “What is your credence now for the proposition that our coin landed heads?” is about (the coin flip)
You claimed that it didn’t make sense to ask that question (about the coin having landed heads) before the coin flip happens.
I showed how even though English requires us to use tense we can make the question time symmetrical by inventing temporal coordinates (t1) and speaking of subjective probability of heads at t1 at any time Beauty exists.
You claimed that the probability of heads at the end of the experiment was somehow different from the probability of heads at some other time (presumably when she is asked).
I pointed out that time is irrelevant and what matters is her information- an elementary point which I shouldn’t have to make to someone who was last night trashing the OP for supposedly not knowing anything about probability (and I’m a philosopher not a math guy!).
In conclusion: My claim is that for Beauty to answer 1⁄3 for the probability of the time invariant event “coin toss by experimenter at time t1 being heads” she needs to get new information since the prior for that even is obviously 1⁄2. No one has ever pointed to what new information she gets. You tried to claim that Beauty updates as soon as she gets the details of the experiment: but that can’t be right. The details of the experiment can’t alter the outcome of a fair coin toss. So where is the updating?!
It’s hard to tell but I’m not sure your notion of “subjective probability” is coherent- specifically because you keep talking about different events depending on what time you’re in. That sounds like a recipe for disaster. But alright.
Does this mean we can just agree to specify payouts in our probability problems from now on? Or must we now investigate which one of us is using the term the way scientists do? Unfortunately this disagreement suggest to me that scientists may not know exactly what they mean by subjective probability.
Subjective probability is a basic concept in decision theory. Scientists have certainly tried hard to say exactly what they mean by the term. E.g. see this one, from 1963:
“A Definition of Subjective Probability”—F. J. Anscombe; R. J. Aumann
http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/anscombeaumann.pdf
Sure. I don’t see anything in there to suggest that subjective probability isn’t time symmetrical (by which I mean that a subjective probability regarding an event can be held at any time and there is not reason for the probability to change unless the person’s evidence changes). Can you do a better job formalizing what your alternative is?
Except she doesn’t. She’ll give the same answer on Monday as she will on Tuesday, because she doesn’t learn anything by waking up.
Yes, this is very alarming, considering this is a forum for aspiring rationalists.