# Multivariate Gaussian with 3 Partitions

Given that we partition a Gaussian random vector $\textbf{x}$ into three groups, $\textbf{x}_a$, $\textbf{x}_b$, and $\textbf{x}_c$, with a corresponding partitioning of the mean vector $\mu$ and of the covariance matrix $\Sigma$ in the form

$\mu = \begin{pmatrix} \mu_a\\ \mu_b\\ \mu_c\\ \end{pmatrix}, \enspace \Sigma=\begin{pmatrix} \Sigma_{aa} & \Sigma_{ab} & \Sigma_{ac} \\ \Sigma_{ba} & \Sigma_{bb} & \Sigma_{bc} \\ \Sigma_{ca} & \Sigma_{cb} & \Sigma_{cc} \\ \end{pmatrix}$

What is the expression for the conditional distribution $P(x_a|x_b)$ in which $x_c$ has been marginalized out? (Note the 3 groups are also multivariate)

Marginalising out $x_c$ means that you can forget about the last row and column of the covariance and $\mu_c$ in the mean vector.
http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions $$p(x_a|x_b)=\mathcal{N}\left(\mu_{a|b},\Sigma_{a|b}\right)$$
Where $\mu_{a|b},\Sigma_{a|b}$ is given in the above link.