This is a general question, not strictly about a specific Stan model. (But it’s a spin-off question from this).
Can it be that if I’m dealing with an unidentified complex model, I get biased posterior distributions? That is, I generate data with known true values for the parameters and I get (too) precise 95% CrI that exclude the true values? (Or does this mean that I have an error somewhere else in the model?)
I understand that the most common scenario would be that the model doesn’t converge because the posteriors have many modes and the chains get stuck in different modes. Or if I have good priors then the posteriors of the unidentified model would be more or less the priors. But I faced this problem of biased posteriors a couple of times, (always with super complex models, last time here) and I couldn’t figure out what’s wrong.
You might be able to troubleshoot this by checking whether the incorrect estimation persists even if you initialize your chains at the known true values. If multiple chains so initialized converge to the wrong values, then it’d at least be worth undertaking an extra-careful search for bugs. If (at least some of) these chains find the correct value, then it seems very likely to be a problem with either unidentifiability or multimodality.
Edit: Or if the sampler diagnostics are bad it could just be a problem with the parameterization.
First and foremost, even when fitting the true model there is no general guarantee that any single posterior will have any relationship to the true value. Having a model where nearly every posterior distribution is close to the true value is a nice property to have, but it’s one that has to be verified in every application (for example with Bayesian calibration).
If the observational model is non-identified, such that every possible likelihood function doesn’t concentrate around a single point no matter the size of the observation, then computational problems are always an additional issue. It’s hard to accurately quantify a posterior that stretches to infinity with only finite computation!