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Sleeping Beauty is kidnapped by a mad philosopher.
The philosopher explains to her the following:
Sleeping Beauty, my dear, I've brought you here for a special purpose. You are honored to the subject of...
A PHILOSOPHICAL EXPERIMENT
Today is Saturday. Tomorrow will be Sunday, the Day After That will be Monday, and the Following Day will be Tuesday. Just so we're clear.
Tomorrow I will give you a magic potion that will put you to sleep. After that, I will flip a Coin.
The Coin is fair, scout's honor.
If the COIN IS HEADS:
I will wake you up on Monday, and we will have tea. Then I will give you a second potion which will put you to sleep and erase all memory of Monday. Then I will wake you up on Tuesday and we will have tea.
If the COIN IS TAILS:
I will not wake you up on Monday. You will sleep through Monday, looking beautiful. Then I will wake you up on Tuesday and we will have tea.
OK SO:
Sleeping Beauty drinks the potion on Sunday. She's asleep, and then the mad philosopher wakes her up. They have tea. Then the philosopher asks:
Sleeping Beauty, my ravishing somnolent darling, what credence do you ascribe to the proposition that "the Coin was tails"
 
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So its supposed to be a 1/3 chance the coin was tails?
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that's one possible answer
but there's a new thought experiment you can propose that makes that seem like a bad belief. can you find it?
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I'm assuming it's perfectly valid that at the moment of the question, it is Monday, since Sleeping Beauty doesn't forget all that shit until after Monday is over with?
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yeah it could be monday, no way to tell, they seem the same
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Eh then it would seem to be 1/2. Really you gain no useful information from waking up and having tea because it was bound to happen anyways. And we have no reason to believe the random choice of heads or tails was biased.
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From Sleeping Beauty's subjective perspective, Pr(tails) should be 1/2. As far as she knows, the priors are equal (fair coin), and the conditionals are equal (being woken up at some point occurs regardless of the flip, and she doesn't know on what day she's awoken), so the posteriors should also be equal. What am I missing here?
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And actually this instantly reminded me of an analogue with entangled quantum bits. Say we have (1/sqrt(2))|01> and (1/sqrt(2))|10>, and then one dude measures 0. Well, obviously the other dude's is 1, but fuck if we know which guy measured first, since we didn't actually gain any information. So it's 50/50 who measured first, until we go and ask the other guy when they measured.
Same thing seems to apply to this, cause knowing that I'm awake and drinking tea didn't give me any useful information.
The only thing I can imagine that might be a possible source of information is the fact that if it were heads, we have 2 days on which to wake up, so the day we are being asked could be either Monday or Tuesday, whereas for tails it is Tuesday. But still, either way we only remember it once, so to Sleeping Beauty they are one and the same.
And if we consider the Pr(tea) = 1, and Pr(tea | tails)=1 as well, then we are left only with Pr(tails), so yea definitely looks smells and reads like a case of absolutely 0 information gain... so 1/2.
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Just looked at the wikipedia article, nothing new there, but it says it would make sense that because being woken up gives sleeping beauty no new information, the probability may be one half, as well as giving an explanation for the one third answer. I still think the one third answer makes more sense, because from sleeping beauties perspective, she will be right twice if she chooses heads every time and it is heads, but right only once if she chooses tails and it is tails.
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On January 04 2013 15:27 32 wrote:
Just looked at the wikipedia article, nothing new there, but it says it would make sense that because being woken up gives sleeping beauty no new information, the probability may be one half, as well as giving an explanation for the one third answer. I still think the one third answer makes more sense, because from sleeping beauties perspective, she will be right twice if she chooses heads every time and it is heads, but right only once if she chooses tails and it is tails.
Lol I just wiki'd it. I think I'll go with the religious argument and appeal to divine solution.
+ Show Spoiler +
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If she says 1/2 and then makes book on it she'll have terrible EV
edit: I'm not going to say what I think, I wrote a paper about it in college and argued about it for hours in seminar so that'll take the fun out of it. but people should argue because it's a fun problem and it's an issue for bayesians
edit: 32 is on a right track, if it's heads you get sampled twice and that's a big part of the issue. But how do express a credence in the proposition then?
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On January 04 2013 15:34 EtherealDeath wrote:Show nested quote +On January 04 2013 15:27 32 wrote:
Just looked at the wikipedia article, nothing new there, but it says it would make sense that because being woken up gives sleeping beauty no new information, the probability may be one half, as well as giving an explanation for the one third answer. I still think the one third answer makes more sense, because from sleeping beauties perspective, she will be right twice if she chooses heads every time and it is heads, but right only once if she chooses tails and it is tails. Lol I just wiki'd it. I think I'll go with the religious argument and appeal to divine solution. + Show Spoiler +
What if we applied a platonic solution, that really the tea and day were imperfect and that somewhere there is a perfect tea and day and therefore none of this really happened. I might be putting Descarte before the horse here though HUEHUEHUE.
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On January 04 2013 15:39 sam!zdat wrote:
If she says 1/2 and then makes book on it she'll have terrible EV
edit: I'm not going to say what I think, I wrote a paper about it in college and argued about it for hours in seminar so that'll take the fun out of it. but people should argue because it's a fun problem and it's an issue for bayesians
Um, I demand you upload paper and argue.
Also yea, at first I was going to say 1/3, which is why I asked about Monday and memory, but then I looked at it in a Bayesian sense and was like nahhhhh 1/2.
But then, it's godawful EV. Meh.
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I don't have the paper anymore.
But what happens if instead of 2 days on heads, it is arbitrarily large?
The point is we were talking about bayesianism in another thread and I wonder how bayesian solves the EV problem
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On January 04 2013 15:42 docvoc wrote:Show nested quote +On January 04 2013 15:34 EtherealDeath wrote:On January 04 2013 15:27 32 wrote:
Just looked at the wikipedia article, nothing new there, but it says it would make sense that because being woken up gives sleeping beauty no new information, the probability may be one half, as well as giving an explanation for the one third answer. I still think the one third answer makes more sense, because from sleeping beauties perspective, she will be right twice if she chooses heads every time and it is heads, but right only once if she chooses tails and it is tails. Lol I just wiki'd it. I think I'll go with the religious argument and appeal to divine solution. + Show Spoiler + What if we applied a platonic solution, that really the tea and day were imperfect and that somewhere there is a perfect tea and day and therefore none of this really happened. I might be putting Descarte before the horse here though HUEHUEHUE.
Actually here is a simple, simple solution.
Monday didn't exist. Scummy bastard could scam us out of our betting money anyways if we bet on it hue hue hue hue hue (not like we would ever remember being scammed zzzzzz).
But then since we are apparently a she in this case, good and potent use of puppy eyes could counter that. Mmmm the possibilities. You'd never remember all the awfully loose stuff you did the previous day to guarantee you weren't scammed! I mean, even pregnancy would have to reverse itself, else we'd realize we were up to something on Monday, and thus the assumptions of the problem broken!
Yes, it is it!
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Man, this is pretty dang messed up.
Moral implications aside, I unfortunately can't see any argument that the chance is not 1/2. So, to answer the question, moderate credence.
I made a probability tree to show my thinking: + Show Spoiler +
Edit: Does EV mean expected value?
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I think you are forced to accept that one can have two separate modalities of credence, or something.
because "objectively" it's definitely 1/2. but "subjectively" it feels like it must be1/3. of course you get sampled twice in one instance, so the payoff is like leveraged or something? god idk
@above yes EV=expected value
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i end up thinking it's 75%. :<
edit: no i don't. i read question wrong.
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On January 04 2013 15:46 sam!zdat wrote:
I think you are forced to accept that one can have two separate modalities of credence, or something.
because "objectively" it's definitely 1/2. but "subjectively" it feels like it must be1/3. of course you get sampled twice in one instance, so the payoff is like leveraged or something? god idk
@above yes EV=expected value
Man, but when one mode ends up losing you money, it's hard to view it as valid (even if you were using it before).
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you can also ask the question, which I originally had and edited out because I'm dumb, "what is the probability that it is Monday?"
but that involves an indexical so it's harder
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EPT![[image loading]](http://i.imgur.com/grLQs.png?1)
