I have been working for a month to model a serie of negative binomial distribution, and I couldn't come with stable priors. Obviously I lack some more fundamental knowledge.
if I have:
y[n,g] ~ negative_binomial_2(mu[g], sigma[g]);
mu[g] ~ gamma(alpha[g], beta[g]);
to use this in a mixed model in which alphas and betas have different relations and value ranges I need to find the priors for alphas and betas
alpha[g] ~ ?(par1, par2)
beta[g] ~ ?(par3, par4)
Even with bad modelling I observed that for two classes of data I have two distinct distributions of
par1,2,3,4 (class 1)
par1,2,3,4 (class 2)
and alphas and betas are correlated possibly with a quadratic distribution
What hyper distributions can I use? I have tried to stick with real alpha and vector beta giving beta a gamma prior and many other things but a correct prior seem to exist only if alpha or beta are known, but here they are both parameters.
And stack exchange propose a solution if alpha beta both unknown, but at the end doesn't lead to anything I can practically implement in stan https://stats.stackexchange.com/questions/2635/conjugate-prior-for-a-gamma-distribution
There is a conjugate prior for the Gamma distribution developed by Miller (1980) whose details you can find on Wikipedia and also in the pdf linked in footnote 6. Checkout section 3.2 on page 25 of this paper, there is a prior with four parameters: p, q, r, & s
Thanks a lot. I'm getting lost after a lot of work.