Hmmm… I’m afraid I don’t really understand your problem. I was hoping that looking at one of the proofs would give me a clue as to what you were missing, but it didn’t.
The symbol p(A|B) is normally defined as p(AB)/p(B). What we need to check is that this matches up with our intuitive notion of conditional probability. Different people don’t always have the same intuitive notions of probability, and the line that wikipedia takes is that probabilities conditional on B should be the probabilities you get when you set the chance of elementary events inconsistent with B to zero, and then renormalise everything else. They prove from there that this gives p(AB)/p(B).
Doesn’t this limit the use of this equivalency to cases where you are, in fact, conditioning on B—that is, you can’t use this to make inferences about B’s conditional probability given A? Or am I misunderstanding the proof?
This is the part of your question I don’t understand. The symbol p(A|B) refers to some particular number. The proof shows that this is, in fact, the probability that you should ascribe to A, given that you know B. The symbol p(B|A) refers to some other number. We have p(A|B)=p(AB)/p(B) and p(B|A)=p(AB)/p(A). Smushing these equations together gives p(B|A)=p(A|B)p(B)/p(A), a formula for p(B|A) involving p(A|B).
The issue I have is whether or not it is valid to smush the equations together; whether the equation for p(A|B) is valid in the context of the equation for p(B|A). It may be an issue of intuition mismatch, but it seems analogous to simplifying the equation (1-X)*X^2/(1-X) - the value of 1 is still supposed to be undefined, even after you simplify. Here, we have two “versions” of the same set with disagreeing assigned probabilities.
But your description suggests the issue is that I’m trying to think of the set from the proof p(A|B) as still being there, instead of considering p(A|B) as a specific number; that is, I’m trying to interpret it as a variable whose value remains unresolved. If I consider it in the latter terms, the issue goes away.
Hmmm… I’m afraid I don’t really understand your problem. I was hoping that looking at one of the proofs would give me a clue as to what you were missing, but it didn’t.
The symbol p(A|B) is normally defined as p(AB)/p(B). What we need to check is that this matches up with our intuitive notion of conditional probability. Different people don’t always have the same intuitive notions of probability, and the line that wikipedia takes is that probabilities conditional on B should be the probabilities you get when you set the chance of elementary events inconsistent with B to zero, and then renormalise everything else. They prove from there that this gives p(AB)/p(B).
This is the part of your question I don’t understand. The symbol p(A|B) refers to some particular number. The proof shows that this is, in fact, the probability that you should ascribe to A, given that you know B. The symbol p(B|A) refers to some other number. We have p(A|B)=p(AB)/p(B) and p(B|A)=p(AB)/p(A). Smushing these equations together gives p(B|A)=p(A|B)p(B)/p(A), a formula for p(B|A) involving p(A|B).
The issue I have is whether or not it is valid to smush the equations together; whether the equation for p(A|B) is valid in the context of the equation for p(B|A). It may be an issue of intuition mismatch, but it seems analogous to simplifying the equation (1-X)*X^2/(1-X) - the value of 1 is still supposed to be undefined, even after you simplify. Here, we have two “versions” of the same set with disagreeing assigned probabilities.
But your description suggests the issue is that I’m trying to think of the set from the proof p(A|B) as still being there, instead of considering p(A|B) as a specific number; that is, I’m trying to interpret it as a variable whose value remains unresolved. If I consider it in the latter terms, the issue goes away.