I am working with Conditional Normal Distributions. In particular, I am attempting to find the conditional mean:
$$\boldsymbol{\bar \mu} = \boldsymbol\mu_1 + \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \left(\mathbf a - \boldsymbol\mu_2\right)$$
I understand that there is a general formula for applying a weighted mean. I am attempting to do something similar with the conditional mean.
Assuming:
$$\boldsymbol{x} = \begin{bmatrix}\boldsymbol{x_1}\\\boldsymbol{x_2}\end{bmatrix}$$
where $$(\boldsymbol{x_1}|\boldsymbol{x_2}=a)\sim N(\boldsymbol{\bar \mu},\boldsymbol{\bar \Sigma})$$
How would I find the weighted conditional mean? For example, if $\boldsymbol{x_2} = \begin{bmatrix}1&2\\3&4\end{bmatrix}$, is there a way to weight the $\begin{bmatrix}1&2\end{bmatrix}$ vector to have more weight than the $\begin{bmatrix}3&4\end{bmatrix}$ vector?