Fair enough, although I think my point stands, it would be fairly silly if you could deduce P(A|B) = 1 simply from the fact that you know nothing about A and B.
This is the same confusion I was originally having with Zed. Both you and he appear to consider knowing the explicit form of a statement to be knowing something about the truth value of that statement, whereas I think you can know nothing about a statement even if you know what it is, so you can update on finding out that C is a conjunction.
Given that we aren’t often asked to evaluate the truth of statements without knowing what they are, I think my sense is more useful.
Of course, we almost never reach this level of ignorance in practice, which makes this the type of abstract academic point that people all-too-characteristically have trouble with. The step of calculating the complexity of a hypothesis seems “automatic”, so much so that it’s easy to forget that there is a step there.
Fair enough, although I think my point stands, it would be fairly silly if you could deduce P(A|B) = 1 simply from the fact that you know nothing about A and B.
Well, you can’t—you would have to know nothing about B and A&B, a very peculiar situation indeed!
EDIT: This is logically delicate, but perhaps can be clarified via the following dialogue:
-- What is P(A)?
-- I don’t know anything about A, so 0.5
-- What is P(B)?
-- Likewise, 0.5
-- What is P(C)?
-- 0.5 again.
-- Now compute P(C)/P(B)
-- 0.5/0.5 = 1
-- Ha! Gotcha! C is really A&B; you just said that P(A|B) is 1!
-- Oh; well in that case, P(C) isn’t 0.5 any more: P(C|C=A&B) = 0.25.
As per my point above, we should think of Bayesian updating as the function P varying, rather than its input.
I believe that this dialogue is logically confused, as I argue in this comment.
This is the same confusion I was originally having with Zed. Both you and he appear to consider knowing the explicit form of a statement to be knowing something about the truth value of that statement, whereas I think you can know nothing about a statement even if you know what it is, so you can update on finding out that C is a conjunction.
Given that we aren’t often asked to evaluate the truth of statements without knowing what they are, I think my sense is more useful.
Did you mean “can’t”? Because “can” is my position (as illustrated in the dialogue!).
This exemplifies the point in my original comment: