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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} $$