Conditional distribution of a multivariate normal
Theorem taught in Covariance and the multivariate normal, part of Statistical learning
Conditional distribution of a multivariate normal
| Type | Theorem |
| Taught in | Covariance and the multivariate normal |
| Part | Statistical learning |
If \mathbf{X}=(\mathbf{X}_A,\mathbf{X}_B)\sim\operatorname{N}_p(\boldsymbol{\mu},\Sigma) is partitioned as above, then
\mathbf{X}_A\mid(\mathbf{X}_B=\mathbf{x}_B) \sim\operatorname{N}_q\bigl(\boldsymbol{\mu}_{A\mid B}(\mathbf{x}_B),\,\Sigma_{A\mid B}\bigr),
with
\boldsymbol{\mu}_{A\mid B}(\mathbf{x}_B)=\boldsymbol{\mu}_A+\Sigma_{AB}\Sigma_{BB}^{-1}(\mathbf{x}_B-\boldsymbol{\mu}_B)
and
\Sigma_{A\mid B}=\Sigma_{AA}-\Sigma_{AB}\Sigma_{BB}^{-1}\Sigma_{AB}^\top.
Used in
- Taught in Covariance and the multivariate normal, where the derivation and examples are.
- All 8 questions on Covariance and the multivariate normal
See also
Joint distribution and density · Mean vector and covariance matrix · Correlation · Multivariate normal distribution · Squared Mahalanobis distance · Linear images of a normal vector · Partial correlation · Random vectors