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I have the following marginal posterior of a vector $\phi$ ($p$ by $1$):

$$p(\phi | Y) \propto \left[1+\frac{1}{h}\left(\phi - \tilde{\phi} \right)' \Gamma \left(\phi - \tilde{\phi} \right) \right]^{-\frac{n+p}{2}}$$

where $Y$ denotes the data, $h$ is a constant, $\tilde{\phi}$ ($p$ by $1$) is a vector of constants, $\Gamma$ is a $p$ by $p$ matrix of constants, and $n$ is a constant which is different from $h$.

I'm told that the marginal posterior of $\phi$ is a multivariate $t$-distribution, however, checking wikipedia (http://en.wikipedia.org/wiki/Multivariate_t-distribution), although it appears to be almost the same as the multivariate $t$-distribution, there is a small difference. Namely, the $\frac{1}{h}$ should be a $\frac{1}{n}$, but here $h$ and $n$ are different.

Could someone clarify this?

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So ... if you take a factor of $n/h$ into $\Gamma$ ($\Gamma^*=\frac{n}{h}\Gamma$, say), what does it look like then?

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  • $\begingroup$ Ah, can't believe I didn't see that haha, thanks @Glen_b! $\endgroup$
    – TrueTears
    Jan 12, 2015 at 12:48

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