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I want to show (if possible) that $$\mathrm{E}[\mathrm{Var(Y|X_1, X_2)}] - \mathrm{E}[\mathrm{Var(Y|X_1)}] \geq \mathrm{E}[\mathrm{Var(Y|X_1, X_2, X_3)}] - \mathrm{E}[\mathrm{Var(Y|X_1, X_3)}] \tag 1$$

So how I can show that if the given inequality above holds or not? Any leads to the solution will be greatly appreciated. Thanks in advance!


Assumptions: For simplification purposes, we can also assume the following case:

  • $X_1$, $X_2$, and $X_3$, and $Y$ are jointly Gaussian and each has standard distribution with $\mathcal{N}(0, 1)$.

Alternative Version

Using the definition of conditional variance, (1) can also be expressed as below: $$\mathrm{Var}(\mathrm{E}[Y | X_1]) - \mathrm{Var}(\mathrm{E}[Y | X_1, X_2]) \geq \mathrm{Var}(\mathrm{E}[Y | X_1, X_3]) - \mathrm{Var}(\mathrm{E}[Y | X_1, X_2, X_3]) \tag 2$$

Instead of expectation of conditional variances, now we have variance of conditinional expectations.

What I Have: Using the definition of conditional variance, it is easy to show that $$\mathrm{E}[\mathrm{Var}(Y | X_1)] \geq \mathrm{E}[\mathrm{Var}(Y | X_1, X_3)]$$ and $$\mathrm{E}[\mathrm{Var}(Y | X_1, X_2)] \geq \mathrm{E}[\mathrm{Var}(Y | X_1, X_2, X_3)]$$

However, knowing these two inequalities does not really help.

Also, I think the both sides of the inequality in (1) are constants. So intuitively there should be a way to compare them.

Edit1: I think it is not easy to show the inequalities in (1) or (2) hold directly. So, instead, it might be easier to express $\mathrm{Var}(\mathrm{E}[Y | X_1]) - \mathrm{Var}(\mathrm{E}[Y | X_1, X_2])$ in a closed form using the assumption above. Does this expression have a closed form? How to proceed from here? Thanks!

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  • $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. $\endgroup$ – gung - Reinstate Monica Sep 26 '16 at 20:25
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    $\begingroup$ Thanks for the suggestion. But this question is neither from a course book nor some homework. I am self-studying conditional variance and expectation and trying to understand the effect of observing one more random variable $X_2$ when we already observed $X_1$, compared to the case when we already observed $X_1$ and $X_3$. $\endgroup$ – amipima Sep 26 '16 at 20:40

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