Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question

1 Answer 1

up vote 4 down vote accepted

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 $\sigma(Z)$.

share|improve this answer
    
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
    
You already asked it in the question and I answered by omission. Since you insist: no. (What makes you imagine that it is?) –  Did Mar 3 '13 at 18:16
    
Thanks! I am trying to see if there is some relation between conditional independence and independence? –  Tim Mar 3 '13 at 18:17
    
By "almost surely" do you mean relative to $\mathcal{G}$ or to the underlying $\sigma$-algebra? –  Evan Aad Mar 10 '13 at 20:31
1  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.