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Morendil comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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"ADDED: This is depressing. Here we have a collection of people who have studied probability problems and anthropic reasoning and all the relevant issues for years. And we have a question that is, on the scale of questions in the project of preparing for AGI, a small, simple one. It isn't a tricky semantic or philosophical issue; it actually has an answer. And the LW community is doing worse than random at it."
That's why I posted this to begin with. It is interesting that we can't come to an agreement on the solution to this problem, even though it involves very straightforward probability. Heck, I got heavily down voted after making statements that were correct. People are getting thrown off by doing the wrong kind of frequency counting.
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However, I should note that the event 'sleeping beauty is awake' is equivalent to 'sleeping beauty has been woken up at least once' because of the amnesia. The forgetfulness aspect of the problem is why the solution is 1/2.
Which of your down-voted statements were correct?
Well, I got -6 for this statement: "P(monday and heads)=1/2. P(monday and tails)=1/4. P(tuesday and tails)=1/4. Remember, these have to add to 1."
Initially there is a 50% chance for heads and 50% chance for tails. Given heads, it's monday with certainty. So, P(heads)=1/2, p(monday | heads)=1.
Do you dispute either of those?
Similarly, p(tails)=1/2, p(monday | tails)=1/2. p(tuesday | tails)=1/2.
Do you dispute either of those?
The above are all of the probabilities you need to know. From them, you can derive anything that is of interest here.
For example, on an awakening p(monday)=p(monday|tails)p(tails) + p(monday|heads) p(heads)=1/4+1/2=3/4
p(monday and heads)=p(heads)*p(monday|heads)=1/2
etc.
Re: "P(monday and heads)=1/2. P(monday and tails)=1/4. P(tuesday and tails)=1/4. Remember, these have to add to 1."
Yes, but those Ps are wrong - they should all be 1/3.
My assumptions and use of probability laws are clearly stated above. Tell me where I made a mistake, otherwise just saying "you're wrong" is not going to move things forward.
Well, the correct sum is this one:
"Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give is 1/3. This is the correct answer from Beauty's perspective."
That gives:
P(monday and heads)=500/1500. P(monday and tails)=500/1500. P(tuesday and tails)=500/1500.
You appear to have gone wrong by giving a different answer - based on a misinterpretation of the meaning of the interview question, it appears.
So you are not willing to tell me where I made a mistake?
P(heads)=1/2, p(monday | heads)=1. Which one of these is wrong?
You're using expected frequencies to estimate a probability, apparently. But you're counting the wrong thing. What you are calling P(monday and heads) is not that. There is a problem with your denominator. Think about it. Your numerator has a maximum value of 1000 (if the experiment was repeated 1000 times). Your denominator has a maximum value of 2000. If the maximum possible values of the numerator and denominator do not match, there is a problem. You have an outcome-dependent denominator. Try taking expectation of that. You won't get what you think you'll get.
Re: "If the maximum possible values of the numerator and denominator do not match, there is a problem.
The total possible number of awakenings is 2000.
That represents all tails - e.g.:
P(monday and heads) = 0/2000; P(monday and tails) = 1000/2000; P(tuesday and tails) = 1000/2000;
These values add up to 1 - i.e. the total numerators add up to the commonn denominator. That is the actual constraint. The maximum possible value of the numerator in each individual fraction is permitted to be smaller than the common denominator - that is not indicative of a problem.
Oh, it is a huge problem. It proves that your ratio isn't of the form # of events divided by # of trials. Your ratio is something else. The burden is on you to prove that it actually converges to a probability as the number of trials goes to infinity.
Using cell counts and taking a ratio leads to a probability as the number of trials goes to infinity if you have independent draws. You don't. You have a strange dependence in there that messes things up. Standard theory doesn't hold. Your thing there is estimating something, you just don't know what it is
The total number of events (statements by Beauty) adds up to the total number of trials (interviews).
You should not expect the number of statements by beauty on Monday to add up to the total number of interviews alltogether. It adds up to the number of interviews on Monday. This is not very complicated.
Or they all should be 1/2.
Impossible - if they are to add up to 1.
For Jack's bookie, I agree, you have to use 1/3 – but if you want to calculate a distribution on how much cash Beauty has after the experiment given different betting behavior, it no longer works to treat Monday and Tuesday as mutually exclusive.
I'd like to see a model of how a group of people is supposed to improve their initial distribution of beliefs in a problem with a true/false answer.
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:
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?
(?!?!!)
Yes, this is very alarming, considering this is a forum for aspiring rationalists.
I disagree; but I've already given my reasons.