Multivariate Non-Central t distribution

$x$ is a $P\times 1$ random vector and has a multivariate non-central t distribution, denoted by $t(\mu,\sigma^2 \Sigma,\nu)$, if its density is given by:

$$\mathcal{T}(x;\mu,\sigma^2 \Sigma,\nu)= \frac{\Gamma[(\nu+P)/2]}{... ...} \left(1+\frac{(x-\mu)^T\Sigma^{-1}(x-\mu)}{\sigma^2 \nu} \right)^{-(\nu+P)/2}$$ (24)

where $\Gamma(a)$ is the single-parameter Gamma function. The non-central t distribution has mean$=\mu$ and covariance$=\sigma^2\Sigma\nu/(\nu-2)$ for $\nu>2$.

We can represent a multivariate non-central t distribution using a two-parameter gamma distribution and a multivariate Normal distribution in a Bayesian framework. If we introduce a variable $\tau$, and specify a joint posterior over $x$ and $\tau$ as:

$$p(\tau,x\vert\mu,\sigma^2\Sigma,\nu)\propto p(x\vert\tau,\mu,\sigma^2\Sigma)p(\tau\vert\nu)$$ (25)

$$x\vert\tau,\mu,\sigma^2\Sigma \sim N(\mu,(\sigma^2\Sigma / \tau))$$

$$\tau\vert\nu \sim Ga(\nu/2,\nu/2)$$

then the marginal posterior for $x$ is a multivariate non-central t distribution, i.e.:

$$p(x\vert\mu,\sigma^2\Sigma,\nu) = \int p(\tau,x\vert\mu,\sigma^2\Sigma,\nu) d\tau$$ (26)

$$x\vert\mu,\sigma^2\Sigma,\nu \sim t(\mu,\sigma^2\Sigma,\nu)$$