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Lumifer comments on Open thread, Dec. 1 - Dec. 7, 2014 - Less Wrong Discussion
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An interesting question. Let me offer a different angle.
You don't have weak evidence. You have data. The difference is that "evidence" implies a particular hypothesis that the data is evidence for or against.
One problem with being in love with Bayes is that the very important step of generating hypotheses is underappreciated. Notably, if you don't have the right hypothesis in the set of hypotheses that you are considering, all the data and/or evidence in the world is not going to help you.
To give a medical example, if you are trying to figure out what causes ulcers and you are looking at whether evidence points at diet, stress, or genetic predisposition, well, you are likely to find lots of weak evidence (and people actually did). Unfortunately, ulcers turned out to be an bacterial disease and all that evidence, actually, meant nothing.
Another problem with weak evidence is that "weak" can be defined as evidence that doesn't move you away from your prior. And if you don't move away from your prior, well, nothing much changed, has it?
"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.