What is the correct order of these hierarchical priors?

I'm quite new to Bayesian data analysis, so this is most likely am easy question.

I have the following model: a function f has two exponential rate parameters $\lambda_1$ and $\lambda_2$ and for some input $x$ it returns a set of $p_i$which are success rates for binomially distributed data at different level $x$. So f works similar to an inverse link function in glms.

Parameter estimation works fine. Now I would like to set up some alternative models to test, e.g. one in which both $\lambda$'s are equal. They should however reflect independent processes, so I cannot set a common prior from which both are drawn. Instead I would like to add another hierarchical layer. I want the $\lambda's$ to have gamma pirors, so I thought I need an hyper prior that allows for each gamma having independent shape parameters which however have the same mean (drawn from a mormal) (a). But I wonder whether it shouldn't be the other way around, with gamma at the top and drawing from the normals on the second layer gives some independence to the $\lambda$'s (b).

Which way would be right to express that the rates are the same but govern independent processes?

Many thanks in advance

Jan

• P.S. I know I cold probably answer whether the rates are the same by just estimating their difference. However, I would like to add further aspects to the model layer and and try out model selection techniques with this simple example. Dec 5, 2014 at 23:00
• Maybe if you could describe your data and your research questions we could help you better.
– Tim
Dec 5, 2014 at 23:08
• The data follows a psychometric function... It goes in a sigmoidal manner from approx. 0 to approx. max and contains at each level the number of successes y out of max. Typically you would model it with a logistic function a cumulative Gaussian or Weibull. However, we derived the function I called f above from a theory to take over the role of the link function. I also try to put together the whole model in a Bayesian way (replacing f by stochastic processes, this however leads to different problems). So I would like to try this first. Dec 5, 2014 at 23:19
• Check out "Bayesian Cognitive Modelling" book (bayesmodels.com) for more info on this kind of models and an intro to Bayesian estimation.
– Tim
Dec 5, 2014 at 23:33
• Thanks Tim, I already have this on my to-read list. I think the question is rather general, as this situation may appear in all kinds of domains... so maybe someone here has a suggestion... Dec 5, 2014 at 23:35

Normal is not a good prior for Gamma because shape parameters both have to be greater than zero. Maybe you could think of Log-Normal or truncated Normal?

I would imagine the model (in Stan) rather like this:

data {...}
parameters {...}
model {
alpha ~ lognormal(...);
beta ~ lognormal(...);
lambda ~ gamma(alpha, beta);
p <- binomial(N, lambda * x);
}


so hyperpriors for $\lambda$'s are drawn from common distributions. If you want $\lambda$'s to have Gamma distributions, Normal (case b) is not a choice for you. Gamma is also rather chosen as a prior for scale parameter in Normal distribution, then for location (case b). So case a looks better, but Normal distribution doesn't sound like a good prior for parameters for $\lambda$'s.

If you provided more information or pasted your code maybe I could elaborate a little bit more.