Im solving Gibbs sampler related questions and Im trying to find the unconditional distribuition of $\beta | \lambda \sim N(0, \lambda^{-1}\Sigma)$ knowing that $\lambda \sim Gamma(\frac{\alpha}{2}\frac{\alpha}{2})$. I know that the resulting distribuition of beta will be a multivariate $t$ with $(\mu, \Sigma, \alpha)$, but I don't know how to get there. How do I find the distribuition of $\beta$?
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$\begingroup$ How does the Gibbs sampler come into the question? Your body text seems to be asking about how to derive that it's multivariate-t. $\endgroup$– Glen_bCommented Jul 4, 2022 at 1:53
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1$\begingroup$ Im solving questions related to the Gibbs sampler, you have a good point, I was working on it and it seems that I don't need the Gibbs sampler to solve it. $\endgroup$– OcchimaCommented Jul 4, 2022 at 2:10
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$\begingroup$ You should definitely tag this self-study and see the tag wiki info. Note that $f(\beta) = \int_0^\infty f(\beta|\lambda) f(\lambda) d\lambda$ (though you may find it easier to work with the distribution of $\lambda^{-1}$ instead). From there it's reasonably simple manipulation. Where do you get stuck? $\endgroup$– Glen_bCommented Jul 4, 2022 at 4:38
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$\begingroup$ You may find this: en.wikipedia.org/wiki/Normal-inverse-gamma_distribution useful, as well as the illustration of completing the square ... e.g. similar to what's done here: stats.stackexchange.com/questions/140540/… ... There are some derivations on site (I am sure I wrote one at some stage but there are others), so try a search. Failing that there are multiple sets of notes that discuss it such as here: web.archive.org/web/20170828154305/http://www.biostat.umn.edu/… $\endgroup$– Glen_bCommented Jul 4, 2022 at 4:56
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$\begingroup$ also see here: bookdown.org/aramir21/… $\endgroup$– Glen_bCommented Jul 4, 2022 at 5:03
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