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Let's say we have a distribution with PDF described by the product of Gamma and Student-t distributions. This is equivalent to a generative model, in which precision is first drawn from Gamma, and the mean is then drawn from Student-t with some known number of degrees of freedom, mean, and sigma = sqrt(1/precision). Now, I want to find the marginal distribution of mean, integrating out the precision parameter. I'm hoping to obtain another Student-t, but with a different number of degrees of freedom. Any advice?

P.S. The model in which the original PDF is the product of Gamma and Normal does produce Student-t. I'm hoping that I can repeat this process analytically for a chain of interlieved Normal and Gamma distributions in a graphical model, and still obtain a closed form solution.

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  • $\begingroup$ It looks like you are using "mean" in two ways. $\endgroup$
    – Douglas Zare
    Commented Dec 14, 2015 at 22:47
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    $\begingroup$ I am rather skeptical that it works out. Have you tried falsifying the hypothesis by numerical computation? (You might have to manually compute the derivatives to parameters of the student-t during numerical optimization, though this could potentially be skipped if you use something like autograd.) $\endgroup$
    – Yi Liu
    Commented Dec 15, 2015 at 2:56
  • $\begingroup$ @YiLiu Thanks for your reply. Haven't tried that. Will investigate more today. $\endgroup$
    – akuz
    Commented Dec 15, 2015 at 7:16

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