I am fitting a model to estimate the parameters of the distribution of my observations. My observations are proportions measured in the continuous scale, hence they are bounded in the interval [0, 1]. The values are well distributed around 0.5 with a bell-curve shape. I am also modelling the random effect for each subject. I was wondering if I could improve my model by parametrizing it in a non-centered way. The (centered) model I devised so far is:

y_{likelihood}=Beta(a, b), with a=\mu\cdot\phi, b=(1-\mu)\cdot\phi, and \phi is the precision parameter, which I assume to be common to all observations. Then \mu=logit^{-1}(\mathbf{X}\beta+\mathbf{Z}\gamma), where the design matrix \mathbf{X} has dimension (n_{observations}, 1), i.e. is intercept only, and \mathbf{Z} has dimension (n_{observations}, n_{subjects}).

\beta is the main effect, and I set the prior \beta \sim \mathcal{N}(0,10). \gamma_{s} is the random effect for each subject (the deviation of each subject around the intercept for the population \beta), and I set the prior \gamma_{s} \sim \mathcal{N}(0,\sigma) for s=1...n_{subjects}, with \sigma\sim Half \mathcal{N}(10) common to the all the subjects. The model seems to work fine, I do not get divergences, and the posterior check looks good.

I tried to implement the non-centered version in a naive way with gamma_{offest, s}\sim\mathcal{N}(0, 1) and the deterministic \gamma_s=\gamma_{offset, s}\cdot\sigma (as suggested in Wiecki’s blog post here), but I get really odd values for gamma_{offest, s} (like reeeeaaaaally wide and flat distributions). I feel like this is not the right approach in this case, as I need to take into account the support of the beta distribution, and how the values in my linear regression get squashed by the inverse logit transformation.

Do you have any suggestions?