Therefore, for every expectation of evidence, there is an equal and opposite expectation of counter-evidence.
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.
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).
The changes are equal and opposite:
[ P(H|E) - P(H) ]*P(E) + [ P(H|~E) - P(H) ]*P(~E) = 0
See Nick Hay’s much earlier comment.
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.