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