Thanks @bbbales2, @stijn and @betanalpha for the insightful answers and comments. This is great and helped clarifying some very fundamental concepts!
Just one last thing; Michael, you wrote that in
the log normal location parameter has to be positive so the original model as written isn’t consistent.
Perhaps I misunderstood, but I had always understood the log-normal to be valid for shape parameters \mu \in \mathbb{R}; the Stan documentation also seems to suggest as much. So positivity of the location parameter shouldn’t be a requirement for lognormal likelihoods. Since the location parameters of the lognormal distribution are on the log-scale, this shouldn’t be an issue. Or am I mistaken?
The reason why I’m asking is that the bigger context is a hierarchical dose response model that I’m trying to fit in RStan; some of the model’s parameters denote count-like (strictly positive) quantities which I assume to be lognormally distributed with location parameters which can be <0 (on the log-scale).