I’m just trying to understand your point a bit better. Hopefully you don’t mind the late reply (I’ve been on vacation for a while)
“In probability, “correlations” are always bidirectional.”
Can’t there be three separate, equally valid points which, if proven, would prove she was the murderer? Even if those three equally valid proofs of her guilt are contradictory? Once we know she is guilty, they can’t all three be true, can they?
I’m not sure how one would accurately express this, given what you’re saying. The probability that A implies Guilt, B implies Guilt, and C implies Guilt can all be 100%, yes? Obviously, the probability that guilt implies all of A+B+C is 0%, since they are contradictory. Therefor, how can it be correct to assume the opposite correlation, that Guilt implies A at 100% certainty?
In general it is not true that P(A|B) = P(B|A). P(A|Guilt) depends on the prior probabilities of A and Guilt, as well as P(Guilt|A). For example, say we have four possible proofs A, B, C, D, and P(Guilt|A or B or C) = 1, and P(Guilt|D) = 0. Our prior is all four are equally likely: P(A) = P(B) = P(C) = P(D) = 0.25. P(Guilt) is then 0.75 = P(Guilt|A)P(A) + P(Guilt|B)P(B)...
P(A|Guilt) isn’t 1. But it’s 33%, which is still higher than the prior %25: that is, Guilt is evidence for A.
By the way I think it might help if you avoid talking in proofs and implication and 100% certainty. In hypothetical examples it’s useful to set things to P(X) = 1, but in the real world evidence is always probabilistic; nothing’s ever 100%.
Ahhh, that helps clear things up. For some reason I’d been understanding you as saying that, given P(Guilt|A) = 1, P(A|Guilt) was also 1. It looks like what you meant was just that Guilt is evidence for, but not necessarily 100% proof of, A. Am I getting that all correct?
I’m just trying to understand your point a bit better. Hopefully you don’t mind the late reply (I’ve been on vacation for a while)
“In probability, “correlations” are always bidirectional.”
Can’t there be three separate, equally valid points which, if proven, would prove she was the murderer? Even if those three equally valid proofs of her guilt are contradictory? Once we know she is guilty, they can’t all three be true, can they?
I’m not sure how one would accurately express this, given what you’re saying. The probability that A implies Guilt, B implies Guilt, and C implies Guilt can all be 100%, yes? Obviously, the probability that guilt implies all of A+B+C is 0%, since they are contradictory. Therefor, how can it be correct to assume the opposite correlation, that Guilt implies A at 100% certainty?
It isn’t!
In general it is not true that P(A|B) = P(B|A). P(A|Guilt) depends on the prior probabilities of A and Guilt, as well as P(Guilt|A). For example, say we have four possible proofs A, B, C, D, and P(Guilt|A or B or C) = 1, and P(Guilt|D) = 0. Our prior is all four are equally likely: P(A) = P(B) = P(C) = P(D) = 0.25. P(Guilt) is then 0.75 = P(Guilt|A)P(A) + P(Guilt|B)P(B)...
Given this, we have:
P(A|Guilt) isn’t 1. But it’s 33%, which is still higher than the prior %25: that is, Guilt is evidence for A.
By the way I think it might help if you avoid talking in proofs and implication and 100% certainty. In hypothetical examples it’s useful to set things to P(X) = 1, but in the real world evidence is always probabilistic; nothing’s ever 100%.
Ahhh, that helps clear things up. For some reason I’d been understanding you as saying that, given P(Guilt|A) = 1, P(A|Guilt) was also 1. It looks like what you meant was just that Guilt is evidence for, but not necessarily 100% proof of, A. Am I getting that all correct?
Yes.
P(Guilt|A) = P(A|Guilt) only when P(A) = P(Guilt). In which case it would be 100% proof. But that is a rare situation.
Nitpick: the two conditionals also be equal if A and Guilt were mutually exclusive. (in that case, of course, the two conditionals would be both zero)