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