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.
- : dimensional vector
- : dimensional mean vector
- : dimensional covariance matrix. It is positive semi-definite: for all .
- : dimensional matrix.
if all linear combinations of are univariate normal, i.e.
for any .
PDF and MGF
Linear Transformations of MVN
Let , where
- Uncorrelation implies independence:
- Conditional: , where , .