We can see conditional independence for D and Y given X means E [ D &#x2223;<!-

cooloicons62 2022-07-08 Answered
We can see conditional independence for D and Y given X means E [ D Y , X ] = E [ D X ] or E [ Y D , X ] = E [ Y X ] in the causal analysis. However, by Wiki the conditional independence is defined as E [ Y D X ] = E [ Y X ] E [ D X ]   a . e .
I cannot bridge these definitions. Is one of them stronger than the other one?
Thanks a lot.
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Answers (1)

Alexzander Bowman
Answered 2022-07-09 Author has 19 answers
MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Use the property for sigma algebras G 1 G 2 F ,
E [ E [ X | G 2 | G 1 ] ] = E [ X | G 1 ] .
The σ-algebra generated by X alone is a sub- σ-algebra of the one generated by X , Y, so the above property applies.

Assume that E [ D | X Y ] = E [ D | X ]. Then,
E [ Y D | X ] = E [ E [ Y D | X Y ] | X ] = E [ Y E [ D | X Y ] | X ] = E [ Y E [ D | X ] | X ] = E [ D | X ] E [ Y | X ] .
The first equality follows by the property, the second equality by measurability of Y with respect to σ ( X , Y ), the third by the conditional independence hypothesis, and the fourth by measurability of E [ D | X ] with respect to σ ( X ) .
The reverse direction is similar.
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