How to calculate conditional probability on student multivariate distribution I have a multivariate student distribution fitted on some data on 4 dimensions (so I know the parameters).
I am trying to calculate the $P(X_4\le x_4| X_1=x_1, X_2=x_2, X_3=x_3)$ but falling short.
I found a paper stating that conditional distribution of multivariate students are student distributions but that's about it.
Is there a close form formula that I should use or do I have to run a Monte-Carlo kind of process?
 A: The $p$-dimensional $t$ distribution has its density given by
$$f_p(\mathbf x;\nu,\boldsymbol\mu,\boldsymbol\Sigma) =\frac{\Gamma\left[(\nu+p)/2\right]}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}\left|{\boldsymbol\Sigma}\right|^{1/2}}\left[1+\frac{1}{\nu}({\mathbf x}-{\boldsymbol\mu})^{\rm T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right]^{-(\nu+p)/2}$$
Hence
\begin{align}f(x_4|x_1,x_2,x_3)&\propto f_4(\mathbf x;\nu,\boldsymbol\mu,\boldsymbol\Sigma)\\
&\propto \left[1+\frac{1}{\nu}({\mathbf x}-{\boldsymbol\mu})^{\rm T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right]^{-(\nu+4)/2}\\
&\propto \left[1+\frac{1}{\nu}(a(x_4-\mu_4)^2+b(x_4-\mu_4)+c)\right]^{-(\nu+4)/2} \end{align}
the last term being obtained by expanding$$({\mathbf x}-{\boldsymbol\mu})^{\rm T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})$$as a second degree polynomial in terms of $x_4-\mu_4$. With $a,b,c$ depending on $(x_1-\mu_1,x_2-\mu_2,x_3-\mu_3)$ as well as $\boldsymbol\Sigma$. Since
\begin{align}\frac{1}{\nu}(a(x_4-\mu_4)^2+b(x_4-\mu_4)+c)
&=\frac{1}{\nu}\{a(x_4-\mu_4+{b}/{2a})^2+c-b^2/4a\}
\end{align}
the conclusion is that
$$f(x_4|x_1,x_2,x_3)\propto \left[1+\frac{1}{\nu+3}
\frac{(x_4-\eta_4)^2}{\sigma_4^2}\right]^{-(\nu+4)/2}$$
where
$$\eta_4=\mu_4-\frac{b}{2a}$$
and
$$\sigma_4^2=\frac{1}{\nu+3}\frac{\nu+c-\frac{b^2}{4a}}{a}$$
is indeed the density of a $t$ distribution with $\nu_4=\nu+3$ degrees of freedom.
