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I am trying to find the conditional distribution of a trivariate gaussian. So here is a hypothetical trivariate gaussian: $$\mathcal{N}(\mu_{ABC},\Sigma_{ABC}),\;\mu_{ABC}=\begin{bmatrix}\mu_A \\ \mu_B \\ \mu_C \end{bmatrix},\;\Sigma_{ABC}=\begin{bmatrix} \sigma^2_{AA} & \sigma^2_{AB} & \sigma^2_{AC}\\ \sigma^2_{BA} & \sigma^2_{BB} & \sigma^2_{BC} \\ \sigma^2_{CA} & \sigma^2_{CB} & \sigma^2_{CC} \end{bmatrix}$$

I would like to find the conditional distribution parameters as follows:

$$\mu_{A}|B,C$$ $$\Sigma_{A}|B,C$$

To get the resulting distribution $$\mathcal{N}(\mu_{A},\Sigma_{A}|B,C)$$

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Write the tri-variate distribution in blocks first - the first block would have the first random variable, the next would have the remaining two variables.

Write the mean and covariance matrix of the tri-variate normal as this Conditional Normal multivariate distribution

Find the conditional mean and covariance (which is Schur complement of the right bottom 2x2 block of the whole tri-variate covariance matrix) as shown in the link.

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