I am doing a lot of modeling with models that require some parameters to be positive by design. However, I am struggling to figure out which approach works best when I try to use multilevel modeling with these types of models.

My current approach ist to draw the individual parameters from a lognormal distribution, to ensure the resulting parameters are always positive. I.e.

$\theta_i \sim lognormal(\mu_\theta,\sigma_\theta^2)$

However, then I need a method to also determine the group-level parameters $\mu_\theta$ and $\sigma_\theta^2$. In my modeling experiments, I have tried drawing either the parameter $\mu_\theta$ from a normal, or drawing the group-level mode, mean or median from a lognormal or gamma distribution as a hyperprior and then translating this to the distribution parameter. This often gives different results and I am not sure which results should be trusted more.

Are there any guidelines on how to model such purely positive parameters in a multilevel model and which distributions to use a hyperprior?


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