Multivariate Normal Cheatsheet

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Multivariate normal (MVN) is used everywhere in machine learning, from simple regressions, linear discriminant analysis, Kalman filters to gaussian processes. But very few textbooks summarize the important characteristics in a concise manner. So here they are.

Definition

if .

  • : dimensional vector
  • : dimensional mean vector
  • : dimensional covariance matrix. It is positive semi-definite: for all .
  • : dimensional matrix.
  • where

Equivalent Definition

if all linear combinations of are univariate normal, i.e.

for any .

PDF and MGF

  • PDF:
  • MGF:

Linear Transformations of MVN

If ,

Within MVN…

Let , where
and

Then,

  • Uncorrelation implies independence:
  • Marginal:
  • Conditional: , where , .

Joint Distribution

then

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