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Let's say my data of dimension $3$ has a multivariate normal distribution with known mean vector $\mu$ and covariance matrix $\Sigma$

I then observe a sample from the distribution but with $1$ value missing $[ ?, x_2, x_3 ]$

How can I find the most likely value of $x_1$ given $x_2$ and $x_3$?

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Condition on $(x_2, x_3)$, $P(x_1|x_2,x_3)$ is still a Gaussian distribution.

Its mean (which coincides with its mode) is

$$\mu_1+\begin{bmatrix}\Sigma_{12} & \Sigma_{12} \end{bmatrix}\begin{bmatrix}\Sigma_{22} & \Sigma_{23} \\ \Sigma_{32} & \Sigma_{33}\end{bmatrix}^{-1}\begin{bmatrix} x_2 - \mu_2 \\ x_3-\mu_3 \end{bmatrix}. $$

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