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Given random variables $Y, Z, X_1, X_2$

Is there some relation between

  • $Y $ and $Z$ are conditionally independent given $(X_1, X_2)$
  • $Y $ and $Z$ are conditionally independent given $X_1$, and given $X_2$

My guess is that the former implies the latter, but not vice versa, since the sigma algebras generated by the random variables $\sigma(X_1),\sigma(X_2) \subseteq \sigma(X_1, X_2)$?

When will they become equivalent?

Is there some reference which state properties of conditional independence? Thanks!

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Let $Y$, $X_1$ and $X_2$ be independent random variables with uniform distribution on $[0,1]$.

Now let $$Z=X_1+X_2+Y\mod 1.$$

Then the random variables $Z$ and $Y$ are independent conditionally on $X_1$ and conditionally on $X_2$ but clearly not conditional on $X_1$ and $X_2$. So the second condition does not imply the first one.

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  • $\begingroup$ Thanks! Do you think the first imply the second? $\endgroup$ – Tim Mar 4 '14 at 21:31
  • $\begingroup$ @Tim I have o think some more about it. $\endgroup$ – Michael Greinecker Mar 5 '14 at 6:49
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The first condition does not imply the second one. Take a constant random variable $X_1$. Then the conditional independence between $Y$ and $Z$ given $(X_1,X_2)$ is the conditional independence between $Y$ and $Z$ given $X_2$, and the conditional independence between $Y$ and $Z$ given $X_1$ is the independence between $Y$ and $Z$. It is possible that $Y$ and $Z$ are conditionally independent given $X_2$, but not independent.

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