What is the difference between "probability of H given E" as calculated by Bayes theorem and the "positive predictive value" of E?
AFAIK, those two are pretty unrelated.
Probability of H given E is simply the probability that the event H will happen when we already know that the event E happened.
On the other hand, when we do a statistical inference (that is, draw conclusions about population using a sample of it), we often want to know if something like the mean of a variable is zero.
We might do a test and come to the conclusion that it's indeed zero. However, we are never 100% sure, and we might have told that it's zero, but it's not, whenever this happens we have a false alarm.
1-(proportion of false alarms) gives you the positive predictive value, that is the proportion of times we don't have a false alarm in our test.