Multivariate Gaussian Distribution

Definition

A generalization of the Gaussian Distribution to the \(d\)-dimensional case, with \(\mathcal{X} \in \mathbb{R}^d\). It is defined as:

\[ p(\mathbf{x}) = \frac{1}{\sqrt{ (2\pi)^{d} \ |\boldsymbol{\Sigma}| }}\exp\left( -\frac{1}{2}(\mathbf{x} - \boldsymbol{\mu})^{\top} \Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu}) \right) \]

Note that \(|\boldsymbol{\Sigma}|\) is the determinant of the covariance matrix \(\boldsymbol{\Sigma}\in \mathbb{R}^{d\times d}\). This determinant reflects the overall variance; if the variance of the distribution is large in most directions, \(|\boldsymbol{\Sigma}|\) will also be large.

References

Gaussian (Normal) Probability Distribution Covariance Matrix