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Single observation unbiased priors. (English) Zbl 1019.62024

Summary: This paper studies a class of default priors, which we call single observation unbiased priors (SOUP). A prior for a parameter is a SOUP if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter. We prove that, under mild regularity conditions, a default prior for a convolution parameter is “noninformative” in the sense of yielding a posterior inference invariant under amalgamation only if it is a SOUP. Therefore, when amalgamation invariance is desirable, as in our motivating example of performing imputation for census undercount, SOUP is the only possible coherent “noninformative” prior for Bayesian predictions that will be utilized under aggregation.
The use of SOUP also mutually calibrates Bayesian and frequentist inferences for aggregates of convolution parameters across many small areas. We describe approaches that identify SOUPs in many common models, in particular a constructive duality method that identifies SOUPs in pairs of distribution families. We introduce O-completeness, a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean. Uniqueness of the SOUP is determined by the O-completeness of the dual family. O-completeness of a natural exponential family is implied by its completeness. Hence, the P. Diaconis and D. Ylvisaker [Ann. Stat. 7, 269-281 (1979; Zbl 0405.62011)] characterization of the conjugate prior for natural exponential families via linear posterior expectations is a direct consequence of the completeness of such families.
For most of the examples we have examined, the inverse of the variance function is the SOUP for the mean parameter of the corresponding family, suggesting that J. A. Hartigan’s [Ann. Math. Stat. 36, 1137-1152 (1965; Zbl 0133.42106)] results on asymptotic unbiasedness can be generalized to some families with discrete parameters. We also discuss a possible extension of J. O. Berger’s [Stat. Probab. Lett. 9, No. 5, 381-384 (1990; Zbl 0693.62005)] result on the inadmissibility of unbiased estimators, as the nonexistence of SOUP can be a first step in establishing such inadmissibility.

MSC:

62F15 Bayesian inference
62C15 Admissibility in statistical decision theory
62C10 Bayesian problems; characterization of Bayes procedures
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[39] CAMBRIDGE, MASSACHUSETTS 02138 E-MAIL: meng@stat.harvard.edu DEPARTMENT OF HEALTH CARE POLICY HARVARD MEDICAL SCHOOL 180 LONGWOOD AVE BOSTON, MASSACHUSETTS 02115-5899 E-MAIL: zaslavsky@hcp.med.harvard.edu
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