How can a Bayesian linear hierarchical random intercept model with normally distributed priors for coefficients represent a non-normally DV?

Question

Suppose you have a hierarchical random intercept model with a dependent variable that is zero inflated. The link function is linear and the priors for the coefficients are normally distributed. In BRMS the model looks like this:

model <- brm(DV ~ (1 | Level_1) + (1|Level_1:Level_2) +(1|Level_1:Level_2:Level_3)

I ran such a model and obtained a normally distributed error term. I would like to better understand the statistical process behind this. Given that the DV is zero inflated, how can the normally distributed priors for the coefficients lead to a normally distributed error term? Does this mean that the posterior predictive values for the DV are also zero inflated (they are, when I look at them, but do they HAVE to be to obtain a normally distributed error term)?

Answer 1

One approach could be to choose a different model set-up. The question is whether this is always necessary or how the model set-up depends on the data generating mechanism. Suppose for example that the IVs are also zero inflated but their coefficients are normally distributed. In such a case one should get a zero inflated distribution (and a normally distributed error term), even though all coefficients are normally distributed and the link function is linear and normally distributed.

My hunch is that what model to chose depends on the data generating mechanism including the distributions of the DVs as well as the exact hierarchical structure of the model. Of course this is easier said than done, since we don’t know the exact data generating mechanism.

The question then become when to chose what type of modelling structure, such as a semiparametric model. Should one start with the simplest approach (the linear model with normally distributed priors) and look at the posterior predictive distribution of the DV and the error term (and maybe also extreme values in the posterior predictive distributions)? Should one go with a more complex model straight away? What would be evidence to decide to go down one rout rather than another (given that one does not have a theory about the data generating mechanism) ?