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Let random vectors $x,u,v$ have joint Gaussian distribution, and $u,v$ be independent. Show that $E(x|u,v)=E(x|u)+E(x|v)-E(x)$.

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Write $x=au+bv+w$ for some numbers $a$ and $b$, with $(u,v,w)$ gaussian and independent. –  Did Nov 3 '11 at 11:38
    
Are random vectors the same as random variables? or do $x,u,v$ all have multivariate jointly Gaussian distributions (that is, the components of each are not necessarily independent Gaussian random variables) but the joint distribution of $u$ and $v$ factors into the product of the jointly Gaussian distributions of vectors $u$ and of $v$? –  Dilip Sarwate Nov 3 '11 at 16:35
    
"Are random vectors the same as random variables?" I really want to assume they are the same here. –  kkk Nov 4 '11 at 7:12
    
A random vector is simply a vector-valued random variable. –  Michael Hardy Nov 4 '11 at 22:01
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2 Answers

Write $x=au+bv+w$ for some numbers $a$ and $b$, with $(u,v,w)$ gaussian and independent.

Thus $y=E(x|u,v)$ is $y=au+bv+E(w)$. Since $(u,v)$ is independent, \begin{align} & E(x|u) = E(y|u) = au+bE(v)+E(w), \\ & E(x|v) = E(y|v) = aE(u)+bv+E(w) \\ & E(x) = E(y) = aE(u)+bE(v)+E(w). \end{align} This yields \begin{align} E(x|u)+E(x|v) & = au+aE(u)+bv+bE(v)+2E(w) \\ & = y+aE(u)+bE(v)+E(w) \end{align} hence $$ E(x|u)+E(x|v)=E(x|u,v)+E(x). $$

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Who says you can write $x = au + bv + w$ for some $a$ and $b$? –  TMM Jan 13 '13 at 19:47
    
@TMM: Who says...? Well, the definition of being multivariate gaussian... –  cardinal Jan 13 '13 at 23:22
    
@cardinal It's probably me, but I still don't see it. If $X,Y$ are random variables and $X$ depends on $Y$, you cannot say $X = aY + Z$ either (with $Z$ independent from $Y$). –  TMM Jan 14 '13 at 0:00
    
@TMM: Use the linearity property in the definition and note that if $U$ and $V$ are independent then they are generated from orthogonal vectors corresponding to a subspace of dimension two. Consider an orthonormal basis extension to the full space in this coordinate system. Everything should then be clear. –  cardinal Jan 14 '13 at 0:29
    
@TMM If $(x,y)$ is gaussian, then there exists some (explicit) coefficients $(a,b)$ such that $E(x\mid y)=ay+b$. You are right that this is a specific property of gaussian families, which does not hold in general. –  Did Jan 14 '13 at 9:36
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How far $u$ is above the expected value $\mu_u$ of $u$ is $\dfrac{x-\mu_u}{\sigma_u}$ where $\sigma_u$ is the standard deviation of $u$.

How many $x$-standard deviations one should "expect" $x$ to be above the expected value of $x$, given the observed value of $u$, is that expression above multiplied by the correlation between $x$ and $u$; thus it is $\rho_{x,u}\left(\dfrac{x-\mu_u}{\sigma_u}\right)$. So $$ E(x\mid u) = \mu_x + \sigma_x \rho_{x,u}\left(\dfrac{x-\mu_u}{\sigma_u}\right) $$ where $\mu_x$ and $\sigma_x$ are the expected value and standard deviation of $x$ respectively. And of course a similar result applies to $v$.

So $$ \begin{align} & {} \quad E(X\mid u) + E(x\mid c) - E(x) \\ \\ & = \mu_x + \sigma_x \rho_{x,u}\left(\frac{x-\mu_u}{\sigma_u}\right) + \mu_x + \sigma_x \rho_{x,v}\left(\frac{x-\mu_v}{\sigma_v}\right) - \mu_x \\ \\ & = \mu_x + \sigma_x \rho_{x,u}\left(\frac{x-\mu_u}{\sigma_u}\right) + \sigma_x \rho_{x,v}\left(\frac{x-\mu_v}{\sigma_v}\right) \\ \\ \\ & = \mu_x + \frac{\rho_{x,u}\sigma_x\sigma_u}{\sigma_u^2}\left(x-\mu_u\right) + \frac{\rho_{x,v}\sigma_x\sigma_v}{\sigma_u^2}\left(x-\mu_v\right) \\ \\ \\ & = \mu_x + \frac{\operatorname{cov}(x,u)}{\sigma_u^2}\left(x-\mu_u\right) + \frac{\operatorname{cov}(x,v)}{\sigma_u^2}\left(x-\mu_v\right) \end{align} $$

The whole matrix of covarinaces, which is the variance of the random vector $(x,u,v)^\top$ is $$ \begin{bmatrix} \sigma_x^2, & \rho_{x,u}\sigma_x\sigma_u, & \rho_{x,v}\sigma_x\sigma_v \\ \rho_{x,u}\sigma_x\sigma_u, & \sigma_u^2, & 0 \\ \rho_{x,v}\sigma_x\sigma_v, & 0,& \sigma_v^2 \end{bmatrix}. $$ Via the usual formulas for conditional expectations in a multivariate normal distribution, we have $$ E\left(x \mid \begin{bmatrix} u \\ v \end{bmatrix}\right) = \mu_x + \begin{bmatrix} \operatorname{cov}(x,u), & \operatorname{cov}(x,v) \end{bmatrix} \begin{bmatrix} \sigma_u^2 & 0 \\ 0 & \sigma_v^2 \end{bmatrix}^{-1} \begin{bmatrix} u-\mu_u \\ v-\mu_v \end{bmatrix}. $$

Since the matrix is diagonal, the inversion is simple: just invert the diagonal elements. Multiply the matrices, and you have the desired result.

If $u$ and $v$ had not been independent, this would have been more complicated.

Later edit: I see I treated these three things as scalar-valued. If they're vectors, and not necessarily all of the same number of scalar components, then where I wrote $\dfrac{\operatorname{cov}(x,u)}{\sigma_u^2}$, we would need $\operatorname{cov}(x,u)(\sigma_u^2)^{-1}$ where the two things being multiplied are matrices, in that order. Otherwise it's the same.

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Even if $x$,$u$,$,v$ are vectors as indicated in the questions, we could still write $E(x)$ as a scalar by thinking $x$ is some random variables taking different values? –  kkk Nov 4 '11 at 7:12
    
@kkk I'm not sure I understand your last question. Certainly every random vector has components that are scalar-valued random variables. And their joint distribution amounts to the distribution of the random vector. –  Michael Hardy Nov 4 '11 at 22:00
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