In my estimation theory textbook, the following is stated as a reminder without any further explanation:
Consider the gaussian random vector
$\bf{z} = \begin{pmatrix}\bf{x} \\ \bf{y}\end{pmatrix}$ with mean $\hat{z} = \begin{pmatrix}\hat{x} \\ \hat{y}\end{pmatrix}$ and covariance matrix $\bf{C}_z = \begin{pmatrix} \bf{C}_{xx} & \bf{C}_{xy} \\ \bf{C}_{yx} & \bf{C}_{yy} \end{pmatrix}$
If a measurement $y^*$ is given, the conditional density of $x$ conditioned on that measurement $f\left(x | y^*\right)$ is gaussian with mean
$\hat{x} + \bf{C}_{xy}\bf{C}_{yy}^{-1} \left( y^* – \hat{y} \right)$
and covariance matrix
$\bf{C}_{xx} – \bf{C}_{xy} \bf{C}_{yy}^{-1} \bf{C}_{yx}$
(Pseudo-inverses replace inverses when necessary)
Having never worked with conditional densities before, I don't see how to derive these formulas, or what the intuition behind them is.
Best Answer
A detailed proof with carefully presented computations is here, see Part b of Theorem 4.