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kmccarty comments on Conditioning on Observers - Less Wrong
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Consider the case of Sleeping Beauty with an absent-minded experimenter.
If the coin comes up Heads, there is a tiny but non-zero chance that the experimenter mixes up Monday and Tuesday.
If the coin comes up Tails, there is a tiny but non-zero chance that the experimenter mixes up Tails and Heads.
The resulting scenario is represented in a new sheet, Fuzzy two-day, of my spreadsheet document.
Under these assumptions, Beauty may no longer rule out Tuesday & Heads. She has no justification to assign all of the Heads probability mass to Monday & Heads. She is therefore constrained to conditioning on being woken in the way that the usual two-day variant suggests she should, and ends up with a credence arbitrarily close to 1/3 if we make the "absent-minded" probability tiny enough.
Why should we get a discontinuous jump to 1/2 as this becomes zero?
This sounds like the continuity argument, but I'm not quite clear on how the embedding is supposed to work, can you clarify? Instead of telling me what the experimenter rightly or wrongly believes to be the case, spell out for me how he behaves.
What does this mean operationally? Is there a nonzero chance, let's call it epsilon or e, that the experimenter will incorrectly behave as if it's Tuesday when it's Monday? I.e., with probability e, Beauty is not awoken on Monday, the experiment ends, or is awoken and sent home, and we go on to next Sunday evening without any awakenings that week? Then Heads&Tuesday still with certainty does not occur. So maybe you meant that on Monday, he doesn't awaken Beauty at all, but awakens her on Tuesday instead? Is this confusion persistent across days, or is it a random confusion that happens each time he needs to examine the state of the coin to know what he should do?
And on Tuesday
So when the coin comes up Tails, there is a nonzero probability, let's call it delta or d, that the experimenter will incorrectly behave as if it's Heads? I.e., on Tuesday morning, he will not awaken Beauty or will wake her and send her home until next Sunday? Then Tails&Tuesday is a possible nonoccurrence.
On reflection, my verbal description doesn't rmatch the reply I wanted to give, which was: the experimenter behaves such that the probability mass is allocated as in the spreadsheet.
Make it "on any day when Beauty is scheduled to remain asleep, the experimenter has some probability of mistakenly waking her, and vice-versa".