Multivariate Gaussian

Creator
Creator
Seonglae ChoSeonglae Cho
Created
Created
2024 Oct 14 9:29
Editor
Edited
Edited
2025 Apr 27 21:37
Refs

Multivariate Normal Distribution, Joint Normal Distribution

It is a single distribution and GMM is multiple.
Each random variable normally distributed, at the same time joint multi-variable normally distributed
Covariance Matrix
affects exponential part of
Probability Density Function
Marginals
Posterior
where

Conditional Mean

How the mean of shifts when is given
generalized form with minus except notation
generalized form with range notation
multi to multi with dimension analysis if we choose k variables for left side
Generalized version with a set notation

Conditional Variance

generalized form with minus except notation
generalized form with range notation
multi to multi with dimension analysis if we choose k variables for left side
Generalized version with a set notation

Inverse matrix

If the covariance matrix is singular, it means that the random variables are fully constrained or deterministic, not truly random. For example, if , then the distribution of would collapse into a plane, not a proper 3D Gaussian distribution.

2-variables

3-variables

 
 
 
 
 
 
 
 

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