Multivariate Normals

Why

We generalize the normal density to $d$-dimensional space.

Definition

Let $f: \R ^d \to \R $ be a density such that

\[ f(x) = \mgaussiandensity{x}{\mu }{\Sigma } \]

where $\mu \in \R ^d$, $\Sigma \in \mathbfsf{S} ^d$, and $\Sigma \succ 0$. We call $f$ a multivariate normal density. A multivariate normal density with $d = 1$ is a normal density, so we refer to multivariate normal densities as normal densities without ambiguity. We frequently use the word normal as a substantive, and refer to normals when we mean multivariate normal densities. Many people call a multivariate normal distribution a multivariate gaussian distribution and speak of gaussians instead of normals.¹

We call $\mu $ the mean and $\Sigma $ the covariance matrix. We call $\Sigma ^{-1}$ the precision matrix.

Maximum

The maximum of a normal density is its mean, $\mu \in \R ^d$.

We avoid this usage in accordance with the project’s policy on historical names. ↩︎