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This paragraph replaces a previous subsection in the article "Multivariate t-distribution".

Conditional Distribution

This was demonstrated by Muirhead though previously derived using the simpler ratio representation above, by Cornish. Let vector $X$ follow the multivariate t distribution and partition into two subvectors of $p_{1},p_{2}$ elements:

$X_{p}={\begin{bmatrix}X_{1}\\X_{2}\end{bmatrix}}\sim t_{p}\left(\mu _{p},\Sigma _{p\times p},\nu \right)$

where $p_{1}+p_{2}=p$, the known mean vector is $\mu _{p}={\begin{bmatrix}\mu _{1}\\\mu _{2}\end{bmatrix}}$ and the scale matrix is $\Sigma _{p\times p}={\begin{bmatrix}\Sigma _{11}&\Sigma _{12}\\\Sigma _{21}&\Sigma _{22}\end{bmatrix}}$.

Then

$X_{1}|X_{2}\sim t_{p_{2}}\left(\mu _{1|2},{\frac {\nu +d_{2}}{\nu +p_{2}}}\Sigma _{11|2},\nu +p_{2}\right)$

explicitly

$f(X_{1}|X_{2})={\frac {\Gamma \left[(\nu +p)/2\right]}{\Gamma (\nu /2)\nu ^{p/2}\pi ^{p/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}}\left[1+{\frac {1}{\nu }}({\mathbf {x} }-{\boldsymbol {\mu }})^{T}{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right]^{-(\nu +p)/2}$

where

$\mu _{1|2}=\mu _{1}+\Sigma _{12}\Sigma _{22}^{-1}\left(X_{2}-\mu _{2}\right)$ is the conditional mean

$\Sigma _{11|2}=\Sigma _{11}-\Sigma _{12}\Sigma _{11}^{-1}\Sigma _{21}$ is the Schur complement of $\Sigma _{22}{\text{ in }}\Sigma$

$d_{2}=(X_{2}-\mu _{2})^{T}\Sigma _{22}^{-1}(X_{2}-\mu _{2})$ is the squared Mahalanobis distance of $X_{1}$ from $\mu _{1}$ with scale matrix $\Sigma _{11}$