komponisto comments on Take heed, for it is a trap - Less Wrong

47 Post author: Zed 14 August 2011 10:23AM

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Comment author: komponisto 15 August 2011 04:35:08PM *  3 points [-]

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.

Comment author: Tyrrell_McAllister 16 August 2011 07:24:49PM 0 points [-]

EDIT: This is logically delicate, but perhaps can be clarified via the following dialogue:

I believe that this dialogue is logically confused, as I argue in this comment.

Comment author: benelliott 15 August 2011 05:32:30PM 0 points [-]

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.

Comment author: komponisto 15 August 2011 06:26:12PM 0 points [-]

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.

Did you mean "can't"? Because "can" is my position (as illustrated in the dialogue!).

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.

This exemplifies the point in my original comment:

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.

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