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AlephNeil comments on Conditioning on Observers - Less Wrong
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Intuition Pump
Suppose 50% of people in a population have an asymptomatic form of cancer. None of them know if they have it. One of them is randomly selected and a diagnostic test is carried out (the result is not disclosed to them). If they don't have cancer, they are woken up once. If they do have it, they are woken up 9 times (with amnesia-inducing drug administered each time, blah blah blah). Each time they are woken up, they are asked their credence (subjective probability) for cancer.
Imagine we do this repeatedly, randomly selecting people from a population that has 50% cancer prevalence.
World A: Everyone uses thirder logic
Someone without cancer will say: "I'm 90% sure I have cancer"
Someone with cancer will say: "I'm 90% sure I have cancer." "I'm 90% sure I have cancer." "I'm 90% sure I have cancer." "I'm 90% sure I have cancer." "I'm 90% sure I have cancer." "I'm 90% sure I have cancer." "I'm 90% sure I have cancer." "I'm 90% sure I have cancer." "I'm 90% sure I have cancer."
Notice, everyone says they are 90% sure they have cancer, even though only 50% of them actually do.
Sure, the people who have cancer say it more often, but does that matter? At an awakening (you can pick one), people with cancer and people without are saying the same thing.
World B: Everyone uses halfer logic
Someone without cancer will say: "I'm 50% sure I have cancer"
Someone with cancer will say: "I'm 50% sure I have cancer." "I'm 50% sure I have cancer." "I'm 50% sure I have cancer." "I'm 50% sure I have cancer." "I'm 50% sure I have cancer." "I'm 50% sure I have cancer." "I'm 50% sure I have cancer." "I'm 50% sure I have cancer." "I'm 50% sure I have cancer."
Here, half of the people have cancer, and all of them say they are 50% sure they have cancer.
My question: which world contains the more rational people?
I do like this intuition pump - it gets across how weird the situation is - but here's one thing you may not have realised:
Once the experiment is over, then even based on thirder logic, people ought to cease saying "I'm 90% sure..." and start saying "I'm 50% sure..."
(Because they have forgotten the 'extra information' that led them to give 90% rather than 50% as their probability of cancer.)