What Are the Limits of Posterior Distributions Arising From Nonidentified Models, and Why Should We Care?

In health research and other fields, the observational data available to researchers often fall short of the data that ideally would be available, due to the inherent limitations of study design and data acquisition. Were they available, these ideal data might be readily analyzed via straightforward statistical models with such desirable properties as parameter identifiability. Conversely, realistic models for the available data that incorporate uncertainty about the link between ideal and available data may be nonidentified. While there is no conceptual difficulty in implementing Bayesian analysis with nonidentified models and proper prior distributions, it is important to know to what extent data can be informative about parameters of interest. Determining the large-sample limit of the posterior distribution is one way to characterize the informativeness of data. In some nonidentified models, it is relatively straightforward to determine the limit via a particular reparameterization of the model; however, in other nonidentified models there is no such obvious approach. Thus we have developed an algorithm for determining the limiting posterior distribution for at least some such more difficult models. The work is motivated by two specific nonidentified models that arise quite naturally, and the algorithm is applied to reveal how informative the data are for these models. This article has supplementary material online.