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neq1 comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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Variation Alpha:
10 people. If heads, one of the ten is randomly selected to be revived. If tails, all ten are revived. (If you like, suppose that the ten are revived one at a time on consecutive days - but it doesn't make any difference.)
Variation Beta:
Same as Alpha except the 10 people are clones of yours, with mental state identical to your own.
Variation Gamma:
Same as Beta except the cloning is done after you fall asleep.
Variation Delta:
Same as Gamma except that the way the clones are not created all at once. Rather, successive clones are created on subsequent days by erasing one days' worth of memory of the previous clone.
It seems clear to me that in variation Alpha, 1/11 is the answer and not 1/2. And clearly variation Delta is isomorphic to the Sleeping Beauty problem (except with 10 days rather than 2). And clearly each step from Alpha to Delta doesn't change anything essential.
Right?
Variation Alpha is unclear, as worded. Let's say one of the 10 people is Sleeping Beauty, and the other people have different names. Sleeping Beauty was identified ahead of time, and she knows it. If she is not selected, then no one is interviewed. Then, if she is revived, she should think it was heads with probability 10/11.
But... if we will interview everyone who is revived, and no one was labeled as special ahead of time, then all each person that was interviewed knows is that at least one person was revived, which was a probability 1 event under heads and tails.
This is just the self-indication assumption situation.
Consider an example. Suppose we want to know if it's common for people to get struck by lightening. We could choose one person ahead of time to monitor. If they get struck by lightening in the next, say, year, then it's likely that getting struck by lightening is common. But... if instead everyone is monitored, but we are only told about one person who was struck by lightening (there could be others, we don't know), then we have no information about whether getting struck by lightening is common or not.
Variation Alpha is intended in such a way that, from the perspective of the experimenters, none of the ten subjects is 'special'.
See <a href="http://lesswrong.com/lw/286/beauty_quips_id_shut_up_and_multiply/1zzh">here</a> for why 1/11 is the correct posterior probability for heads.