# Conditional Distributions from some Elliptical Vectors

This winter, in my ACT8595 course, I asked my students (that was some homework) to prove that it was possible to derive the conditional distribution when we have a Student-t random vector (and to get the analytical expression of the later). But before, let us recall a standard result about the Gaussian vector. If $\boldsymbol{X}=(\boldsymbol{X}_1,\boldsymbol{X}_2)$ is a Gaussian random vector, i.e.

$\begin{pmatrix}\boldsymbol{X}_1\\ \boldsymbol{X}_2\end{pmatrix}\sim \mathcal{N}\left(\begin{pmatrix}\boldsymbol{\mu}_1\\ \boldsymbol{\mu}_2\end{pmatrix},\left(\begin{array}{cc}\boldsymbol{\Sigma}_{11}& \boldsymbol{\Sigma}_{12}\\\boldsymbol{\Sigma}_{21}& \boldsymbol{\Sigma}_{22}\end{array}\right)\right)$

then $\boldsymbol{X}_1\vert \boldsymbol{X}_2=\boldsymbol{x}_2$ has a Gaussian distribution. More precisely, it is a $\mathcal{N}({\boldsymbol\mu}_{1\vert 2},{\boldsymbol\Sigma}_{1\vert 2})$ distribution, with

${\boldsymbol\mu}_{1\vert 2}=\boldsymbol\mu_1 + \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1}\left(\boldsymbol{x}_2 - \boldsymbol\mu_2\right)$

and ${\boldsymbol\Sigma}_{1\vert 2}$ is the Schur complement of the block ${\boldsymbol\Sigma}_{11}$ of the matrix ${\boldsymbol\Sigma}$,

${\boldsymbol\Sigma}_{1\vert 2}=\boldsymbol\Sigma_{11} - \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \boldsymbol\Sigma_{21}$

Observe that $\boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1}$ is also related to well known quantity: in the bivariate case, where $\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$ are univariate Gaussian variables,

$\boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1}=\frac{\text{cov}(X_1,X_2)}{\text{var}(X_2)}=\text{corr}(X_1,X_2)\sqrt{\frac{\text{var}(X_1)}{\text{var}(X_2)}}$

which is the slope in the linear regression of $X_1$ on $X_2$.

In the case of the Student-t distribution, the conditional distrubution will not be a Student-t distribution anymore, but it will still be an elliptical distribution, and some interpretations of various quantities can actually be obtained.

The density of the multivariate centred Student-t distribution, with unit variance, and parameters $\nu$ and $\boldsymbol{R}$ is

$f(\boldsymbol{x})= \frac{\Gamma([d+\nu]/2)}{(\nu\pi)^{d/2} \Gamma(\nu/2)\vert\boldsymbol{R}\vert^{1/2}} \left( 1+\frac{1}{\nu}\boldsymbol{x}'\boldsymbol{R}^{-1}\boldsymbol{x} \right)^{-(d+\nu)/2}$

If we consider the following blocks,

$\boldsymbol{R}= \left( \begin{array}{cc} \boldsymbol{R}_{11}& \boldsymbol{R}_{12}\\ \boldsymbol{R}_{21}& \boldsymbol{R}_{22} \end{array} \right)$

then we can get that marginal distributions have a centred Student-t distribution, with unit variance, and parameters $\nu$ and $\boldsymbol{R}_{22}$,

$f_2(\boldsymbol{x}_2)= \frac{\Gamma([d_2+\nu]/2)}{(\nu\pi)^{d_2/2} \Gamma(\nu/2)\vert\boldsymbol{R}_{22}\vert^{1/2}} \left( 1+\frac{1}{\nu}\boldsymbol{x}_2'\boldsymbol{R}_{22}^{-1}\boldsymbol{x}_2 \right)^{-(d_2+\nu)/2}$

Then, to derive the conditional density, we can use Bayes formula,

$f_{1\vert 2}(\boldsymbol{x}_1\vert \boldsymbol{x}_2)= \frac{f(\boldsymbol{x}_1,\boldsymbol{x}_2)}{f_2(\boldsymbol{x}_2)}$

One can write (as in Section 9.1 in Tong, 1990, The Multivariate Normal Distribution, but other expressions can be found in Section 2.5 in Fang, Ng and Kotz, 1989, Symmetric multivariate and related distributions, or in Section 1.11 in Kotz and Nadarajah, 2004, Multivariate t distributions and their applications) this conditional density as

$f_{1\vert 2}(\boldsymbol{x}_1\vert \boldsymbol{x}_2)=\kappa \left(1+\frac{1}{\nu}\boldsymbol{x}_2'\boldsymbol{R}_{22}^{-1}\boldsymbol{x}_2\right)^{(d_2+\nu)/2} \left(1+\frac{1}{\nu}\left[\boldsymbol{x}_2'\boldsymbol{R}_{22}^{-1}\boldsymbol{x}_2+\alpha(\boldsymbol{x}_1,\boldsymbol{x}_2)\right]\right)^{-(d_1+\nu)/2}$

with

$\kappa=\frac{\Gamma([d+\nu]/2)}{(\nu\pi)^{d_1/2} \Gamma([d_2+\nu]/2)}\frac{1}{\vert\boldsymbol{R}_{11}-\boldsymbol{R}_{12}\boldsymbol{R}_{22}^{-1}\boldsymbol{R}_{21}\vert^{1/2}}$

and

$\alpha(\boldsymbol{x}_1,\boldsymbol{x}_2)=(\boldsymbol{x}_1-\boldsymbol{R}_{12}\boldsymbol{R}_{22}^{-1}\boldsymbol{x}_{2})' [\boldsymbol{R}_{11}-\boldsymbol{R}_{12}\boldsymbol{R}_{22}^{-1}\boldsymbol{R}_{21}]^{-1}(\boldsymbol{x}_1-\boldsymbol{R}_{12}\boldsymbol{R}_{22}^{-1}\boldsymbol{x}_{2})$

This conditional distribution is elliptical, but it is not a Student-t distribution, except in the case where $\boldsymbol{x}_2\in\{(\pm1,\cdots,\pm 1)\}$, or when the correlation matrix $\boldsymbol{R}$ is the identity.

Now, if we look at the components of this densiy, we can observe that we have

$(\boldsymbol{x}_1-\boldsymbol{R}_{12}\boldsymbol{R}_{22}^{-1}\boldsymbol{x}_{2})$

which was mentioned previously, in the Gaussian case: the term on the right is the conditional mean,

$\mathbb{E}(\boldsymbol{X}_1\vert \boldsymbol{X}_2=\boldsymbol{x}_2)=\boldsymbol{\mu}_{1\vert 2}$

and the bloc that appears at several places is the conditional variance,

$\text{Var}(\boldsymbol{X}_1\vert \boldsymbol{X}_2=\boldsymbol{x}_2)=\boldsymbol{R}_{1\vert 2}=\boldsymbol{R}_{11}-\boldsymbol{R}_{12}\boldsymbol{R}_{22}^{-1}\boldsymbol{R}_{21}$

Now, if we want to visualize that conditional density, let us plot it. The code below is based on Bayes formula

> library(mnormt)
> r=.6
> R=matrix(c(1,r,r,1),2,2)
> nu=4
> f2=function(x2) dt(x2,df=nu)
> f =function(x) dmt(x,S=R,df=nu)
> f1.2=function(x1,x2) f(c(x1,x2))/f2(x2)

In order to compare that conditional density with a Student-t one, let us define the density of a non-centred Student-t random variable,

> dstd=function(x,mu,s,nu) gamma((nu+1)/2)/
+ (gamma(nu/2)*s*sqrt(pi*nu))*
+ (1+1/nu*(x-mu)^2/(s^2))^(-(nu+1)/2)

Here is the function we can use to plot those two densities,

> graphdensity=function(x2=-1.5){
+ vectx1=seq(-3,3,length=251)
+ y=Vectorize(function(x) f1.2(x,x2))(vectx1)
+ plot(vectx1,y,type="l",col="red",ylim=c(0,.5),
+ xlab="",ylab="")
+ abline(v=r*x2,lty=2)
+ lines(vectx1,dstd(vectx1,x2*r,sqrt(1-r^2),nu),col="blue",lty=2)}
> graphdensity(-1.5)

In the case where $x_2=\pm1$, the two lines are rather close (the difference migth come from computational issues)

> graphdensity(-1)

and just to conclude, a last one

> graphdensity(0)