I often see analysts "fixing a parameter" to some value when using Stan by setting a prior on the parameter. For example, the Stan user's guide section on Item Response Models notes that in including alpha ~ std_normal() in the model{} block in a model where alpha is a latent variable, "the scale and location for alpha are fixed to ensure identifiability."
This is puzzling to me as it seems inconsistent with how we think about the parameters in Stan models where the posterior is a function of both the likelihood and the prior: does putting a non-hierarchical prior on a parameter actually "fix" it in Stan in the sense that the posterior distribution of that parameter will have (up to numerical approximation) exactly the values passed? Usually we'd expect data to change the mean and variance, although perhaps this is unique behavior for otherwise unidentified models?
Note: brms and other Stan front-ends also obviously do this as well.
