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beriukay comments on Open Thread: July 2010 - Less Wrong
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I know this thread is a bit bloated already without me adding to the din, but I was hoping to get some assistance on page 11 of Pearl's Causality (I'm reading 2nd edition).
I've been following along and trying to work out the examples, and I'm hitting a road block when it comes to deriving the property of Decomposition using the given definition (X || Y | Z) iff P( x | y,z ) = P( x | z ), and the basic axioms of probability theory. Part of my problem comes because I haven't been able to meaningfully define the 'YW' in (X || YW | Z), and how that translates into P( ). My best guess was that it is a union operation, but then if they aren't disjoint we wouldn't be using the axioms defined earlier in the book. I doubt someone as smart as Pearl would be sloppy in that way, so it has to be something I am overlooking.
I've been googling variations of the terms on the page, as well as trying to get derivations from Dawid, Spohn, and all the other sources in the footnote, but they all pretty much say the same thing, which is slightly unhelpful. Help would be appreciated.
Edit: It appears I failed at approximating the symbol used in the book. Hopefully that isn't distracting. It should look like the symbol used for orthogonality/perpendicularity, except with a double bar in the vertical.
Do not worry about that. Pearl's Causality is part of the canon of this place.
You are right that YW means "Y and W". (The fact that they might be disjoint doesn't matter. It looks like the property you are referring to follows from the definition of conditional independence, but I'm not good at these kinds of proofs.)
And welcome to LW, don't feel bad about adding a question to the open thread.
You're probably right, Silas, that "YW" means "Y and W" (or "y and w" or what have you), but you confuse the matter by stating falsely that the original poster (beriukay) was right in his guess: if it was a union operation, Pearl would write it "Y cup W" or "y or w" or some such.
I do not have the book in front of me, beriukay, so that is the only guidance I can give you given what you have written so far.
Added. I now recall the page you refer to: there are about a dozen "laws" having to do with conditional independence. Now that I remember, I am almost certain that "YW" means "Y intersection W".
Sorry, I'm bad about that terminology. Thanks for the correction.
First, thanks for taking an interest in my question. I just realized that instead of typing my question into a different substrate, google likely had a scan of the page in question. I was correct. And unless I am mistaken, when he introduces his probability axioms he explicitly stated that he would use a comma to indicate intersection.
I am afraid I cannot agree with you.
Have you succeeded in your stated intention of "deriving the property of Decomposition using the given definition (X || Y | Z) iff P( x | y,z ) = P( x | z ), and the basic axioms of probability theory"?
If you wish to continue discussing this problem with me, I humbly suggest that the best way forward is for you to show me your proof of that. And we might take the discussion to email if you like.
It is great that you are studying Pearl.