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Consider four random vectors $X, Z, C$ and $W$ in which $Z_i = W_i+N(0,\sigma)$: iid Gaussian noise for each element of $W$ If $X$ is conditionally independent of $Z$ given $C,$ will X be conditionally independent of $W$ given $C$?

Thank you very much.

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Not necessarily, given the conditions as stated. We can work in one dimension. Let $\eta, \xi$ be two iid $N(0,1)$ random variables. Set $X = \eta - \xi$, $W = \eta$, $Z = \eta + \xi$, and $C=0$ (so conditional independence given $C$ is just independence). Then $X$ and $Z$ are independent (they are jointly Gaussian with zero covariance) but $X$ and $W$ are not (their covariance is 1).

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