# Stefan Siegert

## The multivariate Normal distribution in canonical form

$$\def\fat#1{\boldsymbol{#1}}$$

Let $$\fat{x}$$ be a random vector with multivariate Normal distribution with expectation $$\fat{\mu}$$ and variance matrix $$\Sigma$$ \begin{aligned} & \fat{x} \sim \mathcal{N}(\fat{\mu}, \Sigma) \end{aligned} The pdf of $$\fat{x}$$ is given by \begin{aligned} p(\fat{x}) = |2\pi\Sigma|^{-1/2} \exp-\frac{1}{2}\left\{ (\fat{x} - \fat{\mu})'\Sigma^{-1}(\fat{x} - \fat{\mu}) \right\}& \end{aligned}

Defining the precision matrix $$Q=\Sigma^{-1}$$, the log pdf of $$\fat{x}$$ can be written as \begin{aligned} \log p(\fat{x}) & = const - \frac{1}{2} (\fat{x} - \fat\mu)'Q(\fat{x}-\fat\mu)\\ & = const' - \frac{1}{2} \fat{x}'Q\fat{x} + (Q\fat{\mu})'\fat{x} \end{aligned} where the additive constants do not depend on $$\fat{x}$$.

That is whenever we have a random vector $$\fat{x}$$ whose log-pdf can be written in the form \begin{aligned} & \log p(\fat{x}) = const - \frac{1}{2} \fat{x}'Q\fat{x} + \fat{b}' \fat{x}, \end{aligned} for some constant (positive-definite) matrix $$Q$$, and some constant vector $$\fat{b}$$, then we know that $$\fat{x}$$ has a multivariate Normal distribution with expectation $$Q^{-1}\fat{b}$$ and variance matrix $$Q^{-1}$$ \begin{aligned} & \fat{x} \sim \mathcal{N}(Q^{-1}\fat{b}, Q^{-1}). \end{aligned}

The Normal distribution written in terms of $$Q$$ and $$\fat{b}$$ is said to be in canonical form. For a Normal random vector with density given in canonical form with parametrs $$Q$$ and $$\fat{b}$$ we write \begin{aligned} & \fat{x} \sim \mathcal{N}_c(\fat{b}, Q). \end{aligned}
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