Updating by Bayesian conditionalization does assume that you are treating E as if its probability is now 1. If you want an update rule that is consistent with maintaining uncertainty about E, one proposal is Jeffrey conditionalization. If P1 is your initial (pre-evidential) distribution, and P2 is the updated distribution, then Jeffrey conditionalization says:
P2(H) = P1(H | E) P2(E) + P1(H | ~E) P2(~E).
Obviously, this reduces to Bayesian conditionalization when P2(E) = 1.
Yeah, the problem i have with that though is that I'm left asking: why did I change my probability in that? Is it because i updated on something else? Was I certain of that something else? If not, then why did I change my probability of that something else, and on we go down the rabbit hole of an infinite regress.
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?