The correct answer to this is that mathematically, E has to be certain. But mathematics (at least the mathematics which we currently use) does not correspond precisely to the reality of how beliefs work, but only approximately. In reality, it is possible for E (and everything that changed our mind about E or about H) to be somewhat uncertain. That simply says that reality is a bit more than math.
It seems like in order to go from P(H) to P(H|E) you have to become certain that E. Am I wrong about that?
Say you have the following joint distribution:
P(H&E) = a
P(~H&E) = b
P(H&~E) = c
P(~H&~E) = d
Where a,b,c, and d, are each larger than 0.
So P(H|E) = a/(a+b). It seems like what we're doing is going from assigning ~E some positive probability to assigning it a 0 probability. Is there another way to think about it? Is there something special about evidential statements that justifies changing their probabilities without having updated on something else?