Conditional Posterior Distribution of nu for location-scale t distribution In their book Bayesian Data Analysis, Gelman et al. provides a setup for a Gibbs Sampler for a $t_{\nu}(\mu,\sigma^2)$ distribution with $\mu$ and $\sigma^2$ unknown.  The sampler is given as (p.294):
\begin{align}
V_i|\mu,\sigma^2,\nu,y &\sim \text{Inv-}\chi^2\left(\nu+1,\frac{\nu\sigma^2+(y_i-\mu)^2}{\nu+1}\right)\\
\mu|\sigma^2,V,\nu,y &\sim \text{N}\left({\sum_{i=1}^{n}\frac{1}{V_i}y_i}\Big/{\sum_{i=1}^{n}\frac{1}{V_i}},{1}\Big/{\sum_{i=1}^{n}\frac{1}{V_i}}\right)\\
p(\sigma^2|\mu,V,\nu,y) &\propto \text{Gamma}\left(\sigma^2\Big|\frac{n\nu}{2},\frac{\nu}{2}\sum_{i=1}^{n}\frac{1}{V_i}\right)\\
\end{align}
where $V_i$ are auxiliary variables, $\mu$ is the mean and $\sigma^2$ is the square of the scaling parameter.
He then says that if $\nu$ is itself unknown, the Gibbs sampler must be expanded to include a step for sampling from the conditional posterior distribution of $\nu$.  My question is how and where can I find the conditional posterior distribution of $\nu$ for a location-scale $t$ distribution?  I'm completely lost on how to start and am not sure how to proceed.
 A: Assuming $\pi(\nu)$ is the prior distribution on $\nu$, independent from $(\mu,\sigma)$ a priori, the conditional posterior on $\nu$ satisfies
\begin{align}\pi(\nu|\mu,\sigma,\mathbf y)&\propto \pi(\nu)\prod_{i=1}^n \frac{\Gamma((\nu+1)/2)}{\sqrt{\nu}\,\Gamma(\nu/2)}\left(1+\frac{(y_i-\mu)^2}{\nu\sigma^2}\right)^{-(\nu+1)/2}\\
&\propto \pi(\nu)\left(\frac{\Gamma((\nu+1)/2)}{\sqrt{\nu}\,\Gamma(\nu/2)}\right)^n
\prod_{i=1}^n\left(1+\frac{(y_i-\mu)^2}{\nu\sigma^2}\right)^{-(\nu+1)/2}\tag{1}\end{align}
which does not simplify in any standard distribution on $\nu$.
If one uses the completion of the $y_i$'s by the $v_i$'s, as found in Gelman et al. (p.294), then
\begin{align}\pi(\nu|\mu,\sigma,\mathbf v)&\propto \pi(\nu)\prod_{i=1}^n \frac{(\nu\sigma^2)^{\nu/2}}{\Gamma(\nu/2)}{\,v_i^{-\nu/2-1}\exp\{-\nu\sigma^2}\big/2v_i\}\\
&\propto \pi(\nu)\frac{(\nu)^{n\nu/2}}{\Gamma(\nu/2)^n}{\,(v_1\cdots v_n)^{-\nu/2-1}\exp\Big\{-\nu\sigma^2}\sum_{i=1}^n1\big/2v_i\Big\}
\end{align}
which is close to a Gamma distribution, except for the prior and the  term $\Gamma(\nu/2)^{-n}$. Note that Stirling's approximation returns
$$\Gamma(\nu/2) \approx (\nu/2)^{\nu/2}e^{-\nu/2}/\sqrt{\nu/4\pi}$$
hence
$$\frac{(\nu)^{n\nu/2}}{\Gamma(\nu/2)^n}
\approx \frac{(\nu)^{n\nu/2}}{(\nu/2)^{n\nu/2}e^{-n\nu/2}}\sqrt{\nu/4\pi}^n \propto 2^{n\nu/2}e^{n\nu/2}\nu^{n/2}
$$
which has a fairly explosive behaviour.
Furthermore, in the special case $\nu$ is restricted to be an integer, the (conditional posterior) pmf (1) can be computed up to a normalising constant and hence be simulated exactly.
