Yale University
Department of Statistics
Seminar

Wednesday, September 6, 2000
Xiao-Li Meng
Department of Statistics
The University of Chicago

Seminar to be held in Room 107, 24 Hillhouse at 4:15 pm


SINGLE OBSERVATION UNBIASED PRIORS

We call an improper prior for a parameter a Single Observation Unbiased Prior (SOUP) if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter. Such priors are desirable when we seek ``vague" priors for Bayesian predictions that will be utilized under aggregation, as in our motivating example of performing imputation for census undercount. Furthermore, we show that, under very mild regularity conditions, any amalgamation invariant ``default" prior for a convolution parameter must be a SOUP. 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 the notion of $\pi$-completeness, which is a necessary and sufficient condition for a prior distribution, proper or improper, to be uniquely characterized by the corresponding posterior mean. We show that the uniqueness of SOUP is determined by the $\pi$-completeness of the dual family. We also show that for a natural exponential family, the $\pi$-completeness is implied by its completeness. This renders the insight that the well-known Diaconis-Ylvisaker (1979) theorem on characterizing conjugate prior for exponential families via linear posterior expectation is a direct consequence of the completeness of the natural exponential family. For most of the examples we have examined, the inverse of the variance function is the SOUP for the mean parameter of the corresponding variable, suggesting that Hartigan's (1965) results on asymptotic unbiasedness can be generalized to some families with discrete parameters. Other desirable properties of SOUP are helpful in imputing block totals in our census application: for example, the use of SOUP mutually calibrates Bayesian and frequentist inferences for aggregates of convolution parameters across many small areas. Open problems include a possible extension of Berger's (1990) result on the inadmissibility of unbiased estimators.

A preprint, Meng and Zaslavsky (1999) is available by emailing meng@galton.uchicago.edu.