I am interested in mathematical techniques frequently used in finance literature.
If $\tilde{x}$ and $\tilde{y}$ are jointly normally distributed, we can use the so-called projection theorem.
$E[\tilde{x}\mid \tilde{y} ] = E[\tilde{x}] + \frac{Cov(\tilde{x},\tilde{y})}{Var(\tilde{y})}(\tilde{y}-E(\tilde{y}))$
and
$Var[\tilde{x}\mid \tilde{y}] = Var(\tilde{x})-\frac{Cov^2(\tilde{x},\tilde{y})}{Var(\tilde{y})}$
I would like to know the projection theorem for multi-variables.
Let us assume the following:
$x\sim N(0,\sigma_{x}^{-1})$
$\alpha\sim N(\bar\alpha,\sigma_{\alpha}^{-1})$
$\beta\sim N(\bar\beta,\sigma_{\beta}^{-1})$
$\tilde{y}=\tilde{x}+\alpha$
$\tilde{z}=\tilde{x}+\beta$
Can we apply any useful formula in this case? If $\bar\alpha=0$ and $\bar\beta=0$, we can use the following formula.
$E[\tilde{x}\mid \tilde{y}, \tilde{z}] = \frac{\sigma_{\alpha}y+\sigma_{\beta}z}{\sigma_{x}+\sigma_{\alpha}+\sigma_{\beta}}$
$Var[\tilde{x}\mid \tilde{y}, \tilde{z}]=\frac{1}{\sigma_{x}+\sigma_{\alpha}+\sigma_{\beta}}$
How about the case when $\bar\alpha\neq0$ and $\bar\beta\neq0$? I appreciate any help.