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Kindly comments on Conservation of Expected Evidence - Less Wrong
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Eliezer, isn't the "equal" part untrue? I like the parallel with Newton's 3rd law, but the two terms P(H|E)*P(E) and P(H|~E)*P(~E) aren't numerically equal - we only know that they sum to P(H).
P(H) is the belief where you start, and P(H|E) and P(H|~E) are the possible beliefs where you end. You could go to one with probability P(E) and to the other with probability P(~E), but due to the identity you quote, in expectation you do not move at all.