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Morendil comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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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.