Context of the question: You can take everything below as given.
$E_2$ is a $k$ by $1$ matrix and $V_{22}$ is a $k$ by $k$ matrix. Let $X$ denote the data. I have derived so far the joint posterior of $E_2$ and $V_{22}$ is:
$$p(E_2, V_{22}|X) \propto |V_{22}|^{-(T+k+1)/2} \exp\left(-\frac{T}{2} tr\hat{V}_{22} V_{22}^{-1} - \frac{T}{2}tr\left(E_2 - \hat{E}_2 \right)\left(E_2 - \hat{E}_2 \right)'V_{22}^{-1} \right)$$
where $T$ is just a constant, $\hat{V}_{22}$ and $\hat{E}_2$ are just functions of the data, $X$.
It clearly follows that:
$$E_2|V_{22}, X \sim N\left(\hat{E}_2, \frac{V_{22}}{T} \right) \text{and } V_{22} | X \sim W^{-1}\left(T\hat{V}_{22}, T-1 \right)$$
where $W^{-1}(\Psi, \nu)$ denotes the inverse-Wishart distribution following wikipedia's notation: http://en.wikipedia.org/wiki/Inverse-Wishart_distribution.
Question:
I am trying to derive the marginal posterior of $E_2$ but I seem to be running into problems towards the very end. This is what I've done:
$$p(E_2|X) = \int_{V_{22}} p(E_2, V_{22}|X) dV_{22} \propto \int_{V_{22}} |V_{22}|^{-(T+k+1)/2} \exp\left(-\frac{1}{2} tr\left[T\hat{V}_{22} + T(E_2 - \hat{E}_2)(E_2 - \hat{E}_2)' \right]V_{22}^{-1} \right)dV_{22}$$
Note that the integrand is just simply a kernel of the inverse-Wishart distribution (again following wikipedia's notation) with:
$\nu = T$, $p=k$, and $\Psi = T\hat{V}_{22} + T(E_2 - \hat{E}_2)(E_2 - \hat{E}_2)'$
So the integral is:
$$\left(\frac{|\Psi|^{\frac{\nu}{2}}}{2^{\frac{\nu p }{2}} \Gamma_p\left(\frac{\nu}{2}\right)} \right)^{-1} = \frac{2^{\frac{T k }{2}} \Gamma_k\left(\frac{T}{2}\right)}{|T\hat{V}_{22} + T(E_2 - \hat{E}_2)(E_2 - \hat{E}_2)'|^{\frac{T}{2}}}$$
I have a feeling that the resulting distribution is the multivariate t-distribution (http://en.wikipedia.org/wiki/Multivariate_t-distribution) but I am not sure how to manipulate the above to that of a multivariate t-distribution. Help would be appreciated.
