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I would like to know if the following is true:

$E\left[\ E[X|Y] \ | \ Y,Z \ \right] = E[X |Y,Z]$

Can someone provide a proof to show that the equation above is either true or not?

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I believe the equation is not true in general. Note that $E[X \mid Y]$ is a function of $Y$, since $X$ has been integrated out. Thus, given $Y$, $E[X \mid Y]$ is a constant.

$$ E\left[ E \left[ X \mid Y \right] \mid Y, Z\right] = E[X \mid Y] $$

For $E[X \mid Y]$ to be equal to $E[X \mid Y, Z]$, $X|Y$ has to be independent of $Z$, which may or may not be true.

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