Multivariate Normal Distribution#

Multivariate, Continuous, Symmetric, Unbounded

The multivariate normal distribution, also known as the multivariate Gaussian distribution, is a generalization of the univariate normal distribution to multiple dimensions. A random vector is said to follow a \(k\)-dimensional multivariate normal distribution if every linear combination of its \(k\) components follows a univariate normal distribution.

The multivariate normal distribution is often used to describe the joint distribution of a set of correlated random variables.

In Bayesian modeling, a Gaussian process is a generalization of the multivariate normal distribution to infinite dimensions, and it is used as a prior distribution over functions.

Key properties and parameters#

Support	\(x \in \mathbb{R}^k\)
Mean	\(\mu\)
Variance	\(T^{-1}\)

Parameters:

\(\mu\) : (array of floats) Mean vector of length \(k\).
\(\Sigma\) : (array of floats) Covariance matrix of shape \(k \times k\).
\(T\) : (array of floats) Precision matrix, the inverse of the covariance matrix.

Alternative parameterization:

The MvNormal has 2 alternative parameterizations. In terms of the mean and the covariance matrix, or in terms of the mean and the precision matrix.

The link between the 2 alternatives is given by:

\[ T = \Sigma^{-1} \]

Probability Density Function (PDF)#

\[ f(x \mid \mu, T) = \frac{|T|^{1/2}}{(2\pi)^{k/2}} \exp\left\{ -\frac{1}{2} (x-\mu)^{\prime} T (x-\mu) \right\} \]

Parameters \(\mu\) and \(\Sigma\)

Parameters \(\mu\) and \(T\)

Cumulative Distribution Function (CDF)#

The multivariate normal joint CDF does not have a closed-form analytic solution in the general case, due to the complexity of integrating its density over \(\mathbb{R}^k\). However, each marginal distribution is a univariate normal distribution with a well-defined CDF that can be computed directly.