Multivariate Gaussians
A vector-valued random variable $x\in \mathbb{R}^n$ is said to have a multivariate gaussian ( or normal) distribution with mean $\mu\in \mathbb{R}^n$ and covariance matrix $\Sigma \in \mathbb{S}_{++}^n$ if
$$p(x; \mu, \Sigma)=\frac{1}{(2\pi)^{n/2}|\Sigma |^{1/2}}\text{exp}(-\frac{1}{2}(x-\mu)^T \Sigma ^{-1}(x-\mu)). $$
We write this as $x \sim \mathcal N(\mu, \Sigma)$
Consider a random vector $x \sim \mathcal N(\mu, \Sigma)$, Suppose that the variable in $x$ have been partitioned into two sets $x_A=[x_1 \cdots x_r]^T \in \mathbb{R}^r$ and $x_B =[x_{r+1} \cdots x_n]^T \in \mathbb{R}^{n-r} $, such that
\begin{equation*} x= \left [ \begin{array}{c} x_A \\ x_B \end{array} \right ] \qquad \mu= \left [ \begin{array}{c} \mu_A \\ \mu_B \end{array} \right ] \qquad \Sigma= \left [ \begin{array}{cc} \Sigma_{AA} & \Sigma_{AB} \\ \Sigma_{BA} & \Sigma_{BB} \end{array} \right ] \end{equation*}
Here, $ \Sigma_{AB} = \Sigma_{BA}^T $ since $ \Sigma = E[(x-\mu)(x-\mu)^T] = \Sigma^T $. The following properties hold:
1. Normalization
$$ \int_{x\in \mathbb{R}^n} p(x; \mu, \Sigma) dx = 1$$
2.Marginalization
The marginal densities are gaussian:
\begin{equation*} p(x_A) = \int_{x_B \in \mathbb{R}^{n-r}} p(x_A, x_B; \mu \Sigma) dx_B \end{equation*}
$$p(x_B) = \int_{x_A \in \mathbb{R}^r} p(x_A, x_B; \mu \Sigma) dx_A$$
$$x_A \sim \mathcal N(\mu_A, \Sigma_{AA})$$
$$x_B \sim \mathcal N(\mu_B, \Sigma_{BB})$$
3. Conditioning
The conditional densities are also gaussian
$$p(x_A | x_B) = \frac{p(x_A, x_B; \mu, \Sigma)}{p(x_B)}$$
$$p(x_B | x_A) = \frac{p(x_A, x_B; \mu, \Sigma)}{p(x_A)}$$
$$x_A | x_B \sim \mathcal N(\mu_A + \Sigma_{AB} \Sigma_{BB}^{-1}(x_B-\mu_B), \Sigma_{AA} – \Sigma_{AB} \Sigma_{BB}^{-1} \Sigma_{BA} )$$
$$x_B | x_A \sim \mathcal N(\mu_B + \Sigma_{BA} \Sigma_{AA}^{-1}(x_A-\mu_A), \Sigma_{BB} – \Sigma_{BA} \Sigma_{AA}^{-1} \Sigma_{AB} )$$
4. Summation
The sum of independent Gaussian random variables (with same dimensionality), $y \sim \mathcal N(\mu_x. \Sigma_{xx})$ and $z \sim \mathcal N(\mu_{z}, \Sigma_{zz}) $, is also Gaussian:
$$ y+z \sim \mathcal N(\mu_x + \mu_z, \Sigma_{xx}+\Sigma_{zz}) $$
\begin{equation} x=y+z \end{equation}
