Lumifer comments on Open thread, Dec. 1 - Dec. 7, 2014 - Less Wrong Discussion
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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.