## 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
\[\begin{aligned}
& \fat{x} \sim \mathcal{N}_c(\fat{b}, Q).
\end{aligned}\]