I was also confused by this, and think that it does work out with the usual ‘given that’ (I’ll write P(A) instead of A as I get confused with the other notation):
The statement becomes
If A is evidence of B, then P(A|B)>P(A)
where I would have intuitively phrased this as B being evidence of A. But this turns out to be the same thing: If knowing A makes B more likely, finding out that B is true also makes A more likely.
If we already know Bayes theorem, this becomes clear:
P(A|B)=P(B|A)P(B)P(A)>P(A)⇔P(A|B)P(A)P(B)=P(B|A)>P(B)
where the fractions being >1 is equivalent to the two things being evidence for each other.
Hmm, ok, I see that that’s true provided that we assume A necessarily makes B more likely, which certainly seems like the intended reading. Seems like kind of a weird point for them to make in that context (partly because B may often only be trivial evidence of A, as in the raven paradox), so I wonder if it may have been a typo on their part. But as you point out it does work either way. Thanks!
(minor note: I realized I had an error in my comment—unless my thought process at the time was pretty different than I now imagine it to be—so I edited it slightly. Doesn’t really affect your point)
I was also confused by this, and think that it does work out with the usual ‘given that’ (I’ll write P(A) instead of A as I get confused with the other notation):
The statement becomes
where I would have intuitively phrased this as B being evidence of A. But this turns out to be the same thing: If knowing A makes B more likely, finding out that B is true also makes A more likely.
If we already know Bayes theorem, this becomes clear: P(A|B)=P(B|A)P(B)P(A)>P(A)⇔P(A|B)P(A)P(B)=P(B|A)>P(B)
where the fractions being >1 is equivalent to the two things being evidence for each other.
Hmm, ok, I see that that’s true provided that we assume A necessarily makes B more likely, which certainly seems like the intended reading. Seems like kind of a weird point for them to make in that context (partly because B may often only be trivial evidence of A, as in the raven paradox), so I wonder if it may have been a typo on their part. But as you point out it does work either way. Thanks!
(minor note: I realized I had an error in my comment—unless my thought process at the time was pretty different than I now imagine it to be—so I edited it slightly. Doesn’t really affect your point)