gwern comments on Confidence levels inside and outside an argument - Less Wrong
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As I recall, there was a paper in 2008 or 2009 about the LHC problem which concluded effectively that the tiny errors that an analysis was incorrectly carried out cumulatively put a high floor on what small risk we could conclude the LHC posed.
Unfortunately, I can't seem to refind it to see whether it's a better version of this argument, so perhaps someone else remembers specifics.
Probing the Improbable
Looks like it, thanks: