We can see conditional independence for $D$ and $Y$ given $X$ means $E[D\mid Y,X]=E[D\mid X]$ or $E[Y\mid D,X]=E[Y\mid X]$ in the causal analysis. However, by Wiki the conditional independence is defined as $E[YD\mid X]=E[Y\mid X]E[D\mid X]\ a.e.$.
I cannot bridge these definitions. Is one of them stronger than the other one?
Thanks a lot.
Use the property that for sigma algebras $\mathcal{G}_1 \subset \mathcal {G_2} \subset \mathcal{F}$, $$ E[E[X | \mathcal{G}_2 | \mathcal{G}_1 ]] = E[X | \mathcal{G}_1].$$ 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 $E[D | XY] = E[D | X]$. Then, \begin{align*} E[YD | X] &= E[E[YD | XY] | X] \\ &= E[Y E[D | XY] | X] \\ &= E[Y E[D | X] | X] \\ &= E[D | X] E[Y|X]. \end{align*} The first equality follows by the property, the second equality by measurability of $Y$ with respect to $\sigma(X,Y)$, the third by the conditional independence hypothesis, and the fourth by measurability of $E[D|X]$ with respect to $\sigma(X).$
The reverse direction is similar.