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potato comments on What is Evidence? - Less Wrong
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Why not just say e is evidence for X if P(X) is not equal to P(X|e)?
Incidentally, I don't really see the difference between probabilistic dependence (as above) and entanglement. Entanglement is dependence in the quantum setting.
"This should not be confused with the technical sense of "entanglement" used in physics - here I'm just talking about "entanglement" in the sense of two things that end up in correlated states because of the links of cause and effect between them."
That's literally in the third paragraph.
I think you mean, if P(x)<P(x|e) then e is evidence for x. That is a good definition for evidence, but it doesn't function on the same level as Yudkowsky's above. Yudkowsky is explaining not just what function evidence has in truth finding, he is also explaining how evidence is built into a physical system, e.g., camera, human, or other entanglement device. The Bayesian def of evidence you gave tells us what evidence is, but it doesn't tell us how evidence works, which Yudkowsky's does.
X : precence of flower A in certain area e : there are bees on that area then you would possibly have that P(X) < P(X|e), given that bees help doing pollinization. Then should we phrase "probability of having flower A in an area is greater if we have bees, therefore e is evidence for X (bees are evidence for flower A)" and what if X is "having presents brought by santa claus", and e is "we are in USA instead of cambodia" (which increases the probability of having presents because that date is more commonly celebrated with presents in USA).