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thomblake comments on Your Strength as a Rationalist - Less Wrong
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Sticking an adjective in front of the word evidence seems to work. "Evidence" includes things that give you 10^-15 bits of information; on the other hand "good evidence", "usable evidence" and "credible evidence" all imply that the strength of the evidence is at least not exponentially tiny.
I thought that "evidence", unmodified, would mean non-trivial evidence; otherwise, everything has to count as evidence because it will have some connection to the hypothesis, however weak. To specify a kind of evidence that includes the 1e-15 bit case, I think you would need to say "weak evidence" or "very weak evidence".
But I'm not the authority on this: How do others here interpret the term "evidence" (in the Bayesian sense) when it's unmodified?
I'm sympathetic to both views.
I have encountered a number of disputes that revolve around using these two different senses of the word, and am nonetheless blindsided by them consistently.
I try to always specify the strength of evidence in some sense when using the word. I think when I do use it unmodified I tend to use it in the technical sense (including even weak evidence).
It would be odd if 'evidence' excluded weak evidence, since then 'weak evidence' would be a contradiction in terms, or you could see people arguing things like "When I said 'weak evidence' I didn't mean the 1e-15 bit case, since that's not evidence at all!"