Single observation unbiased priors

• 成果类型: Article
• 署名作者: Meng, XL; Zaslavsky, AM
• 署名单位: Harvard University; Harvard University; Harvard Medical School
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2002
• 页码: 1345-1375
• 关键词: quadratic variance functions; natural exponential-families; multiple-imputation; conjugate priors; estimators

摘要

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 Diaconis-Ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation 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 Hartigan's results on asymptotic unbiasedness can be generalized to some families with discrete parameters. We also discuss a possible extension of Berger's result on the inadmissibility of unbiased estimators, as the nonexistence of SOUP can be a first step in establishing such inadmissibility.