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jimrandomh comments on Conditioning on Observers - Less Wrong
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No, P(H)=P(H|W)P(W) is incorrect because the W in P(H|W) is different than the W in P(W): the former is a probability distribution over a set of three events, while the latter is a boolean. Using the former definition, as a probability distribution, P(W) isn't meaningful at all, it's just a type error.