My limited understanding of Bayes's theorem,
is that—even though one of it's terms is $P(E)$—it's applied when $E$ is definitely known to be true. (You went to the doctor and definitely got a positive test result.)
How can I apply the theorem when $E$ might be true with some known probability?
More detail: I have a machine-learning model that, evaluating $H$ and $E$ individually, estimates how probable each is. $H$ and $E$
may be are related, though, so I'd like to adjust these probabilities based on what the model said about the other. ($H$ could be true even if $E$ is not true—but $E$ being true makes it more likely $H$ is true.)
A concrete example adapted from Wikipedia's:
Suppose that a test for using a particular drug is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users. Suppose that 0.5% of people are users of the drug.
What is the probability that a randomly selected individual for whom it is 70% certain (somehow) that he got a positive result is a user of the drug? (This does not mean that's there's a 30% chance he got a negative result, but a 30% chance that he got no result.)