# Definitions of conditional independence

How are conditional independence defined for various cases?

For one case, given two random variables $X$ and $Y$, and a subsigma algebra $\mathcal G$ of the underlying sigma algebra $\mathcal F$, or another random variable $Z$.

What is the definition of the conditional independence between $X$ and $Y$ given $\mathcal G$ or $Z$?

Is it related to the independence between $E(X\mid \mathcal G)$ and $E(Y \mid \mathcal G)$, and independence between $P(X\in A\mid \mathcal G)$ and $P(Y\in A\mid \mathcal G)$, for any $A \in \mathcal F$? I am trying to see if there is some relation between conditional independence and independence?

Thanks and regards!

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The random variables $X$ and $Y$ are independent conditionally on $\mathcal G$ if and only if, for every measurable $A$ and $B$, one has $\mathbb P((X,Y)\in A\times B\mid\mathcal G)=\mathbb P(X\in A\mid\mathcal G)\cdot\mathbb P(Y\in B\mid\mathcal G)$ almost surely.
Some random variables are independent conditionally on the random variable $Z$ if and only if they are independent conditionally on $\sigma(Z)$.
Thanks! Is it related to the independence between $E(X\mid \mathcal G)$ and $E(Y \mid \mathcal G)$, and the independence between $P(X\in A\mid \mathcal G)$ and $P(Y\in B\mid \mathcal G)$, for any $A, B \in \mathcal F$? –  Tim Mar 3 '13 at 18:13
By "almost surely" do you mean relative to $\mathcal{G}$ or to the underlying $\sigma$-algebra? –  Evan Aad Mar 10 '13 at 20:31
Almost surely refers to $\mathbb P$, hence to the underlying sigma-algebra $\mathcal F$. In the present case however, since the LHS and the RHS are both $\mathcal G$-measurable, they coincide except on a null event in $\mathcal F$ if and only if they coincide except on a null event in $\mathcal G$. –  Did Mar 10 '13 at 21:28