I've come across the so-called "problem of old evidence" in Bayesian statistics/epistemology.
First, let me summarize the problem as I see it so we're on the same page.
Suppose I have a theory H, and I suddenly realize that my theory predicts with certainty an experimental effect (E) that other theories have struggled to explain, that is p(E | H) = 1
I turn the Bayesian handle...
p(H | E) = p(E | H) * p(H) / p(E) = p(H) / p(E)
The reasoning now goes that since the experimental effect is old and long-known,
p(E) = 1, thus p(H | E) = p(H).
This seems surprising. To me, the mistake is that p(E) = 1, which is suggested because the data is old and well-established, we have faith in it. This begs the question; yes, p(E | E) = 1 trivially, but p(E) is our prior belief in seeing the phenomena E, not our belief that we saw the phenomena E given that we saw the phenomena E. We could have seen many different phenomena E', say.
P(E) should be expanded as normal,
p(E) = p(E | model 1) * p(model 1) + ... != 1
Solving the problem of old evidence. This is all rather trivial, so it strikes me that I'm missing something... What's wrong with my trivial resolution?
I'm happy with Bayes' theorem, Bayesian inference etc. I want to understand why people think that p(E) = 1, resulting in the problem of old evidence, or perhaps any mistakes I've made about the problem of old evidence. I honestly think the "problem" stems from a basic misunderstanding about p(E) - this is not our belief that we have obtained the evidence in E, it is our belief that we would obtain evidence E, a priori.