Statistical modeling can involve a tension between assumptions and statistical identification. The law of the observable data may not uniquely determine the value of a target parameter without invoking a key assumption, and, while plausible, this assumption may not be obviously true in the scientific context at hand. Moreover, there are many instances of key assumptions which are untestable, hence we cannot rely on the data to resolve the question of whether the target is legitimately identified. Working in the Bayesian paradigm, we consider the grey zone of situations where a key assumption, in the form of a parameter space restriction, is scientifically reasonable but not incontrovertible for the problem being tackled. Specifically, we investigate statistical properties that ensue if we structure a prior distribution to assert that maybe or perhaps the assumption holds. Technically this simply devolves to using a mixture prior distribution putting just some prior weight on the assumption, or one of several assumptions, holding. However, while the construct is straightforward, there is very little literature discussing situations where Bayesian model averaging is employed across a mix of fully identified and partially identified models.
Paul Gustafson is a Professor in the Department of Statistics at the University of British Columbia. He is a Fellow of the American Statistical Association, the 2008 recipient of the CRM-SSC Prize in Statistics, and the 2020 Gold Medallist of the Statistical Society of Canada. His research interests include Bayesian methods, causal inference, evidence synthesis, measurement error, and partial identification. He has authored two books: Measurement Error and Misclassification in Statistics and Epidemiology: Impact and Bayesian Adjustments (2004, Chapman and Hall / CRC Press), and Bayesian Inference for Partially Identified Models: Exploring the Limits of Limited Data (2015, Chapman and Hall / CRC Press). He was the Editor-in-Chief of the Canadian Journal of Statistics (2007-2009), and is currently the Special Editor for Statistical Methods for the journal Epidemiology. Paul served as a founding Co-director of the Master of Data Science program at UBC, and currently serves as Head of the Department of Statistics.