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Yvain comments on Confidence levels inside and outside an argument - Less Wrong
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We have hypothesis H and evidence E, and we dutifully compute
P(H) * P(E | H) / P(E)
It sounds like your advice is: don't update yet! Especially if this number is very small. We might have made a mistake. But then how should we update? "Round up" seems problematic.
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.