By Garrick Lee Wallstrom
University of Minnesota
June 25, 1997
The choice of a suitable reference prior is a problem often faced by practitioners of Bayesian methods. Stone's notion of strong inconsistency between a model and an inference (e.g., a formal posterior) can be used to exclude some improper priors from consideration. An inference is said to be consistent when it is not strongly inconsistent with the model. We discuss the phenomenon of strong inconsistency and provide examples of consistent and strongly inconsistent inferences.
One technique to establish consistency of an inference requires approximating it with proper Bayes inferences. When approximating a formal Bayes inference, truncations of the improper prior serve as natural choices for the proper priors. An example shows that truncations may not suffice to approximate a consistent inference. We show, however, that under mild regularity conditions, truncations can be used in the approximation. This result is then applied in a discussion of the relevance of the version of the formal posterior chosen.
We also study which features of an improper prior determine whether strong inconsistency arises. For translation families, Heath and Sudderth (1989) and Wetzel (1993) identified that the tails of the improper prior and of the model play a key role. We extend their work and apply the results in several examples.