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nerzhin 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 read it to mean "update again" based on the probability that E is flawed. This well tend to adjust back toward your prior.
While you do that, the probability for the estimate being dynamically unstable should go up and then down again. Otherwise, you might make some strange decisions in-between, where the tradeoff between waiting for new information and deciding right now will be as for the honest estimate and not an intermediate step in a multi-step updating procedure with knowably incorrect intermediate results.