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

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Principal causal effect identification and surrogate end point evaluation by multiple trials (2015, background): “Without identifiability, Bayesian inference is sensitive to prior distributions (Gustafson, 2009).”
Sign Restrictions, Structural Vector Autoregressions, and Useful Prior Information ∗ (2015, background): “(5) We propose that researchers need to explicitly acknowledge and defend the role of prior beliefs in influencing structural conclusions and illustrate how this could be done using a simple model of the U.S. labor market.”
Delta-method inference for a class of set-identified SVARs (2018, background): “A practical concern with Bayesian analysis in set-identified SVARs is that posterior inference continues to be influenced by prior beliefs even if the sample size is infinite (Poirier (1998), Gustafson (2009), Moon and Schorfheide (2012)).”