I am trying to use a Hierarchical model where there I have a normal distribution with random mean and precision:

$$ y \sim N(\mu, \tau)\\ \mu \sim N(M, T)\\ \tau \sim Gamma(\alpha, \beta) $$

I'm trying to marginalize out $\mu$ and $\tau$ to find $p(y | M, T, \alpha, \beta)$. I think I can marginalize out $\mu$: $$ \int d\mu d\tau N(y | \mu,\tau) N(\mu | M,T) Gamma(\tau | \alpha, \beta)\\ =\int d\tau N\left(y | M, \frac{1}{1/\tau + 1/T}\right)Gamma(\tau |\alpha, \beta)\\ $$ I'm not sure how to complete the integral over $\tau$. I know that if T=0 the distribution would be a student-t but other than that I'm stuck.

  • $\begingroup$ The model is not hierarchical. The integral cannot be solved in closed form. $\endgroup$ – Xi'an Apr 6 '19 at 12:09
  • $\begingroup$ okay good to know thanks. Just out of curiosity why do you say it's not hierarchical? My definition seems pretty similar to what is found in en.wikipedia.org/wiki/Bayesian_hierarchical_modeling. I just haven't included the subscripts and in this case and I haven't included a prior. Maybe you would prefer the term compound probability distribution. $\endgroup$ – ablanch5 Apr 7 '19 at 14:29
  • $\begingroup$ A hierarchical model has a second prior specified over the parameters of a first prior. $\endgroup$ – Xi'an Apr 8 '19 at 5:45

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