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Yvain comments on Confidence levels inside and outside an argument - Less Wrong
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I'm not saying not to use Bayes' theorem, I'm saying to consider very carefully what to plug into "E". In the election example, your evidence is "A guy on a website said that there was a 999,999,999 in a billion chance that the incumbent would win." You need to compute the probability of the incumbent winning given this actual evidence (the evidence that a guy on a website said something), not given the evidence that there really is a 999,999,999/billion chance. In the cosmic ray example, your evidence would be "There's an argument that looks like it should make a less than 10^20 chance of apocalypse", which may have different evidence value depending on how well your brain judges the way arguments look.
EDIT: Or what nerzhin said.
I think this amounts to saying: real-world considerations force an upper bound on abs(log(P(E | H) / P(E))). I'm on board with that, but can we think about how to compute and increase this bound?
Yes.