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Suppose:

$x \sim \mathcal{N}(x; \mu, \Sigma) \;\;\;$ st. $\;\;\; \mu \sim T_{v}(\mu; k, M)$

Where $T$ is the $t$-distribution with v degrees of freedom, location $k$, and shape $M$.
Then, is there a clean posterior predictive for $x$ and/or a clean distribution for $p([x, \mu])$?

I have: \begin{equation} \begin{aligned} p(x, \mu) &= p(\mu)p(x|\mu)\\ &= T_{v}(\mu; k, M)\mathcal{N}(x; \mu, \Sigma) \end{aligned} \end{equation} and \begin{equation} \begin{aligned} p(x) &= \int p(\mu)p(x|\mu)\;d\mu\\ &= \int T_{v}(\mu; k, M)\mathcal{N}(x; \mu, \Sigma) \;d\mu \end{aligned} \end{equation}

However, I am not sure these can be simplified or have standard forms.

As, the normal is the conjugate prior to the mean of the normal, these have clean solutions when $\mu \sim \mathcal{N}$. However, I am not sure there will be closed form in the above case.

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  • $\begingroup$ I was hoping to find the analytic solution. Not approximate. It may be the case though that I must use numeric methods comparing the t(v)-distribution and Normal as you seem to be eluding to. $\endgroup$ Commented Jan 5 at 5:20
  • $\begingroup$ I don't believe you understand the question. Again, I'm just asking if there's an analytic solution out there. I'm not about to pretend a T-distribution is a gaussian with n>30 $\endgroup$ Commented Jan 5 at 6:01
  • $\begingroup$ Have you tried expressing the T distribution as a mixture of Normal distributions over their variance? $\endgroup$
    – Xi'an
    Commented Jan 5 at 18:41
  • $\begingroup$ @xI have tried expressing the T distribution as a normal with inverse-wishart distributed variance. This leads for my purposes to:\ $\int W^{-1}(M; T, v) \mathcal{N}(x; k, M + \Sigma) dM$\ However, I can't simplify this either... It is discouraging, but simply deriving the T-distribution is hard enough, so its not impossible I just haven't cracked it... $\endgroup$ Commented Jan 5 at 19:19

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