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How to prove that $E[Y]=E[E[E[Y|X_1, X_2]]]$ ?

PS. I don't see how $E[E(Y|X_{1},X_{2})|X_{1}]=Y[Y|X_{1}]$ and $E[Y]=E[E(Y|X_{1})]$ can be used here. But it feels close. Please help, I'm stuck PPS. Thanks @Deep North. Let's consider the case when X1 is independent of X2

den2042
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  • http://stats.stackexchange.com/questions/95947/a-generalization-of-the-law-of-iterated-expectations read here – Deep North Sep 23 '15 at 12:20
  • I am also wondering if your question is missing some conditions. – Deep North Sep 23 '15 at 12:23
  • http://stats.stackexchange.com/questions/95947/a-generalization-of-the-law-of-iterated-expectations does not answer E[Y]=E[E[E[Y|X1,X2]]] – den2042 Sep 23 '15 at 12:27
  • I am trying to understand Appendix A in http://www.ncbi.nlm.nih.gov/pubmed/?term=17999182 In principle, they may consider the case when X1 is independent of X2. But I reckon, it is not the case there, I am not sure – den2042 Sep 23 '15 at 12:36
  • Ok, I read the appendix very quickly, I think the question is not about law of iterated expectations. I think the key to understand Appendix A is that $E(Y|x_1,x_2)$ is a function of $x_1,x_2$ – Deep North Sep 23 '15 at 12:52
  • ...E(Y|x) is a function of x... That's clear. But still I don't get the very first line in the very first formula in the Appendix A – den2042 Sep 23 '15 at 12:56
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    The notation is ambiguous. By definition, $E[Y|X_1,X_2]$ is a function of $(X_1,X_2)$. But exactly what distributions correspond to the outer two expectations? – whuber Sep 23 '15 at 13:01

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$E[Y \mid X_1, X_2]$ is a random variable that is a function $g(X_1, X_2)$ of the random variables $X_1$ and $X_2$. How do we find the expected value of a function of random variable(s)? Well, simply speaking (that is, without dragging in measure theory and abstract formulations), the law of the unconscious statistician says that we multiply $g(X_1,X_2)$ by the (joint) density (or mass function) of $(X_1, X_2)$ and integrate (or sum) the product. The law of iterated expectation tells us that

$$E[g(X_1,X_2)] = E\left[ E[Y \mid X_1, X_2]\right] = E[Y],\tag{1}$$ that is, this function of $X_1$ and $X_2$ that seemingly has nothing to do with $Y$ if we look only at the expectation on the left side of $(1)$ happens to have the same expected value as $Y$.

Remember that $E[Y]$ is just a constant, say $\mu_Y$, and thus $E[\mu_Y] = \mu_Y$ (and var$(\mu_Y) = 0$); that is how we statisticians incorporate into our math the unreasonable beliefs of our clients who insist that they expect constants to have the same value at all times and not vary in any way!

Now, you want to show that $$E[Y] = E\big[ E[E[Y \mid X_1, X_2]] \big]$$ which is straightforward: that expression inside the bigger square brackets on the right is a constant whose value is $\mu_Y = E[Y]$, and we have just agreed (I hope) that $E[\mu_Y] = \mu_Y = E[Y]$.

It is possible that what the OP is asking about is a proof of $$E[Y] = E\bigr[ E\big[E[Y \mid X_1, X_2]\mid X_1 \big] \bigr]\tag{2}$$ which lets us exercise the iterated part of the law of iterated expectation some more.
We have already noted that $E[Y \mid X_1, X_2]$ is a random variable $g(X_1, X_2)$ whose expected value just happens to equal $E[Y]$. But what about the conditional expected value of $g(X_1,X_2)$ given $X_1$? Well, $E[g(X_1,X_2)\mid X_1]$ is a random variable that happens to be a function of $X_1$, say $h(X_1)$, with the useful property $E[h(X_1)]$ equals the unconditional expected value $E[g(X_1,X_2)]$ of $g(X_1,X_2)$ and so we have that $$ E\bigr[h(X_1)\bigr] = E\bigr[E\big[g(X_1,X_2)\mid X_1\big]\bigr] = E\bigr[E\big[E[Y\mid X_1,X_2] \mid X_1\big]\bigr]$$ upon substituting $E\big[g(X_1,X_2)\mid X_1\big]$ for $h(X_1)$ and then substituting $E[Y\mid X_1,X_2]$ for $g(X_1,X_2)$.

So we have shown that the right side of $(2)$ equals $E[h(X_1)]$ which equals $E[g(X_1,X_2)]$ which equals $E[Y]$, and we are done.

Dilip Sarwate
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