I am trying to understand how $Gamma(\nu/2;\nu/2)$ can in anyway be related to a $\chi_{\nu}^2$ distribution, maybe even $\chi_{\nu}^2/\nu$.
This comes up in Expectation-Maximization algorithms when trying to use MLE to find the parameters of a multivariate t-distribution (in my case, bivariate) [1,2,3,4,5]
I am aware of this thread Relationship between gamma and chi-squared distribution.
But this does not seem to be the same thing, since in that thread, they are discussing a random variable $X_i$ ~ $\mathcal{N}(0,\sigma^2)$ but where $X_1, X_2,...X_N$ all have the same $\sigma^2$. This is not the same as constructing the t-distribution since the tdist can viewed as a mixture of normal distributions with DIFFERENT $\sigma^2$. Also, I still don't see how this gets to $Gamma(\nu/2;\nu/2)$.
To be even more specific, in [1] they describe that, if $Y$ follows the multivariate t-distribution $t_p(\mu,\Sigma,\nu)$ then, given the weight $\tau$, $Y$ has the multivariate Normal distribution and $\nu \tau$ is $\chi_{\nu}^2$ distributed, therefore $\tau$ is Gamma distributed. So:
$Y|\mu,\Sigma,\nu,\tau$ ~ $\mathcal{N}(\mu,\Sigma/\tau)$ and $\tau|\mu,\Sigma,\nu$ ~ $Gamma(\nu/2;\nu/2)$
What I can't figure out is how $\tau$ has that distribution if $\nu \tau$ is $\chi_{\nu}^2$. Is $\chi_{\nu}^2/\nu$ equivalent to $Gamma(\nu/2;\nu/2)$ ? How?
Thanks for the help!
References:
[1] ML ESTIMATION OF THE t DISTRIBUTION USING EM AND ITS EXTENSIONS, ECM AND ECME - Chuanhai Liu and Donald B. Rubin, Statistica Sinica 5(1995), 19-39 (This is the link given in the first post)
[2] Robust mixture modelling using the t distribution - D. PEEL and G. J. MCLACHLAN, Statistics and Computing (2000) 10, 339–348
[3] The EM Algorithm--An Old Folk-Song Sung to a Fast New Tune - Xiao-Li Meng and David van Dyk, Blackwell Publishing for the Royal Statistical Society
[4] A Novel Parameter Estimation Algorithm for the Multivariate t-Distribution and Its Application to Computer Vision - Chad Aeschliman1, Johnny Park1, and Avinash C. Kak, European Conference on Computer Vision 2010
[5] The mixtures of Student’s t-distributions as a robust framework for rigid registration - Demetrios Gerogiannis 1, Christophoros Nikou *, Aristidis Likas, Image and Vision Computing 27 (2009) 1285–1294
