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Alsadius comments on Open thread, Dec. 1 - Dec. 7, 2014 - Less Wrong
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"Weak" means that it doesn't change your beliefs very much - if the prior probability is 50%, and the posterior probability is 51%, calling it weak evidence seems pretty natural. But it still helps improve your estimates.
Only if it's actually good evidence and you interpret it correctly. Another plausible interpretation of "weak" is "uncertain".
Consider a situation where you unknowingly decided to treat some noise as evidence. It's weak and it only changed your 50% prior to a 51% posterior, but it did not improve your estimate.