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

We can see conditional independence for $D$ and $Y$ given $X$ means $E\left[D\mid Y,X\right]=E\left[D\mid X\right]$ or $E\left[Y\mid D,X\right]=E\left[Y\mid X\right]$ in the causal analysis. However, by Wiki the conditional independence is defined as
I cannot bridge these definitions. Is one of them stronger than the other one?
Thanks a lot.
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Alexzander Bowman
MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Use the property for sigma algebras ${\mathcal{G}}_{1}\subset {\mathcal{G}}_{\mathcal{2}}\subset \mathcal{F}$,

The $\sigma$-algebra generated by $X$ alone is a sub-$\sigma$-algebra of the one generated by $X,Y$, so the above property applies.

Assume that $E\left[D|XY\right]=E\left[D|X\right]$. Then,

The first equality follows by the property, the second equality by measurability of $Y$ with respect to $\sigma \left(X,Y\right)$, the third by the conditional independence hypothesis, and the fourth by measurability of $E\left[D|X\right]$ with respect to $\sigma \left(X\right).$
The reverse direction is similar.