On Model Expansion, Model Contraction, Identifiability, and Prior Information: Two Illustrative Scenarios involving Mismeasured Variables

When a candidate model for data is nonidentifiable, conventional wisdom dictates that the model must be simplified somehow, in order to gain identifiability. We explore two scenarios involving mismeasured variables where in fact model expansion, as opposed to model contraction, might be used to obtain an identifiable model. We compare the merits of model contraction and model expansion. We also investigate whether it is necessarily a good idea to alter the model for the sake of identifiability. In particular, we compare the properties of estimators obtained from identifiable models to those of estimators obtained from nonidentifiable models in tandem with crude prior information. Both asymptotic theory and simulations with MCMC-based estimators are used to draw comparisons. A technical point which arises is that the asymptotic behaviour of a posterior mean from a nonidentifiable model can be investigated using standard asymptotic theory, once the posterior mean is described in terms of the identifiable part of the model only.