It is well known that bivariate normal pdf can be written in terms of univariate pdfs: $$ \frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-\rho^2}}\exp\left(-\frac{(\xi_1^2+\xi_2^2-2\rho \xi_1\xi_2)}{2(1-\rho^2)}\right)=\frac{1}{\sigma_x\sigma_y\sqrt{1-\rho^2}}\phi\left(\frac{\xi_1-\rho\xi_2}{\sqrt{1-\rho^2}}\right)\phi\left(\xi_2\right) $$ Is there a similar result for bivariate Student t distribution, that is, can a bivariate Student t distribution be written in terms of univariate Student t densities?
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Reading through this paper, if $(\boldsymbol X_1,\boldsymbol X_2)$ is distributed from a $p$ dimensional multivariate Student's $\mathfrak{t}$ distribution $\mathfrak{t}_{p}(\nu,\boldsymbol\mu,\boldsymbol\Sigma)$ [LaTeX copied from Wikipedia] $$ \frac{\Gamma\left[(\nu+p)/2\right]}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}\left|{\boldsymbol\Sigma}\right|^{1/2}}\left[1+\frac{1}{\nu}({\mathbf x}-{\boldsymbol\mu})^{\rm T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right]^{-(\nu+p)/2} $$ then both the marginal and the conditional distributions of $\boldsymbol X_1$ given $\boldsymbol X_2$ are also $p_1$ dimensional multivariate Student's $\mathfrak{t}$ distributions: $$\boldsymbol X_1 \sim \mathfrak{t}_{p_1}(\nu,\boldsymbol\mu_1,\boldsymbol\Sigma_{11})$$ and \begin{align}\boldsymbol X_1|\boldsymbol X_2 \sim \mathfrak{t}_{p_1}\big(&\nu+p_2,\boldsymbol\mu_1+\boldsymbol\Sigma_{12}\boldsymbol\Sigma^{-1}_{22}(\boldsymbol X_2−\boldsymbol \mu_2),\\&\dfrac{\nu+(\boldsymbol X_2-\boldsymbol\mu_2)^\text{T}\boldsymbol \Sigma^{−1}_{22}(\boldsymbol X_2-\boldsymbol\mu_2)}{\nu+p_2}\boldsymbol \Sigma_{11|2}\Big) \end{align} where$$\boldsymbol \Sigma_{11|2}=\boldsymbol \Sigma_{11}−\boldsymbol \Sigma_{21}\boldsymbol \Sigma^{−1}_{22}\boldsymbol \Sigma_{12}$$ This is easily proved by using the demarginalisation of the Student's $\mathfrak{t}$ as a mixture of a Normal variate with a chi-squared variate: $$\boldsymbol X|q\sim\mathcal N_p(\boldsymbol\mu,\boldsymbol\Sigma/q),\qquad q\sim\chi^2_\nu/\nu$$
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1$\begingroup$ Thank you! I derived it for the bivariate case: $ t_{\nu,2}(x,y;0,C) = \frac{1}{\sqrt{(1-\rho^2)\frac{x^2+\nu}{\nu+1}}} \times t_{\nu,1}(x)\times t_{\nu+1,1}\left(\frac{y-\rho x}{\sqrt{(1-\rho^2)\frac{x^2+\nu}{\nu+1}}}\right). $ $\endgroup$ – Alex May 22 at 4:39
