jimrandomh 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?
Hmm. Maybe the strength of the evidence isn't the right thing to use, but rather the confidence with which we know the sign of the correlation.