The Annals of Statistics

Single observation unbiased priors

Xiao-Li Meng and Alan M. Zaslavsky

Full-text: Open access


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.

Article information

Ann. Statist. Volume 30, Number 5 (2002), 1345-1375.

First available in Project Euclid: 28 October 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference
Secondary: 62C15: Admissibility

Admissibility affine duality amalgamation invariance Bayes linear prediction census undercount completeness $O$-completeness conjugate priors duality generalized Bayes estimators multiple imputation beta binomial $F$ gamma negative binomial Poisson location families scale families


Meng, Xiao-Li; Zaslavsky, Alan M. Single observation unbiased priors. Ann. Statist. 30 (2002), no. 5, 1345--1375. doi:10.1214/aos/1035844979.

Export citation


  • BERGER, J. O. (1985). Statistical Decision Theory and Bayesian Analy sis, 2nd ed. Springer, New York.
  • BERGER, J. O. (1990). On the inadmissibility of unbiased estimators. Statist. Probab. Lett. 9 381-384.
  • BERGER, J. O. and SRINIVASAN, C. (1978). Generalized Bay es estimators in multivariate problems. Ann. Statist. 6 783-801.
  • BICKEL, P. J. and MALLOWS, C. L. (1988). A note on unbiased Bay es estimates. Amer. Statist. 42 132-134.
  • BROWN, L. D. (1971). Admissible estimates, recurrent diffusions, and insoluble boundary value problems. Ann. Math. Statist. 42 855-903.
  • BROWN, L. D. (1979). A heuristic method for determining admissibility of estimators, with applications. Ann. Statist. 7 960-994.
  • BROWN, L. D. (1980). A necessary condition for admissibility. Ann. Statist. 8 540-544.
  • CONSONNI, G. and VERONESE, P. (1992). Conjugate priors for exponential families having quadratic variance functions. J. Amer. Statist. Assoc. 87 1123-1127.
  • CONSONNI, G. and VERONESE, P. (1993). Unbiased Bay es estimates and improper priors. Ann. Inst. Statist. Math. 45 303-315.
  • DIACONIS, P. and YLVISAKER, D. (1979). Conjugate priors for exponential families. Ann. Statist. 7 269-281.
  • DIACONIS, P. and YLVISAKER, D. (1985). Quantifying prior opinion. In Bayesian Statistics 2 (J. M. Bernardo, D. V. Lindley and A. F. M. Smith, eds.) 133-156. North-Holland, Amsterdam.
  • FIRTH, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika 80 27-38. [Correction (1995) 82 667.]
  • GUTIÉRREZ-PEÑA, E. and SMITH, A. F. M. (1995). Conjugate parameterizations for natural exponential families. J. Amer. Statist. Assoc. 90 1347-1356. [Correction (1996) 91 1757.]
  • GUTIÉRREZ-PEÑA, E. and SMITH, A. F. M. (1997). Exponential and Bayesian conjugate families: Review and extensions (with discussion). Test 6 1-90.
  • HARTIGAN, J. (1964). Invariant prior distributions. Ann. Math. Statist. 35 836-845.
  • HARTIGAN, J. (1965). The asy mptotically unbiased prior distribution. Ann. Math. Statist. 36 1137-1152.
  • HARTIGAN, J. (1998). The maximum likelihood prior. Ann. Statist. 26 2083-2103.
  • HOGAN, H. (1993). The 1990 post-enumeration survey: Operations and results. J. Amer. Statist. Assoc. 88 1047-1060.
  • HSIEH, H. K., KORWAR, R. M. and RUKHIN, A. L. (1990). Inadmissibility of the maximum likelihood estimator of the inverse Gaussian mean. Statist. Probab. Lett. 9 83-90.
  • HUDSON, H. M. (1978). A natural identity for exponential families with applications in multiparameter estimation. Ann. Statist. 6 473-484.
  • HWANG, J. T. (1982). Improving upon standard estimators in discrete exponential families with applications to Poisson and negative binomial cases. Ann. Statist. 10 857-867.
  • JOHNSON, N. L. (1957). Uniqueness of a result in the theory of accident proneness. Biometrika 44 530-531.
  • KASS, R. E. and WASSERMAN, L. (1996). The selection of prior distributions by formal rules. J. Amer. Statist. Assoc. 91 1343-1370.
  • KENDALL, M., STUART, A. and ORD, J. K. (1987). Kendall's Advanced Theory of Statistics 1. Distribution Theory, 5th ed. Oxford, New York.
  • LEHMANN, E. L. (1983). Theory of Point Estimation. Wiley, New York.
  • MENG, X. L. (1994). Multiple-imputation inferences with uncongenial sources of input (with discussion). Statist. Sci. 9 538-573.
  • MORRIS, C. N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10 65-80.
  • MORRIS, C. N. (1983). Natural exponential families with quadratic variance functions: Statistical theory. Ann. Statist. 11 515-529.
  • NICOLAOU, A. (1993). Bayesian intervals with good frequentist behavior in the presence of nuisance parameters. J. Roy. Statist. Soc. Ser. B 55 377-390.
  • RUBIN, D. B. (1987). Multiple Imputation for Nonresponse in Survey s. Wiley, New York.
  • RUBIN, D. B. (1996). Multiple imputation after 18+ years (with discussion). J. Amer. Statist. Assoc. 91 473-520.
  • RUKHIN, A. L. (1995). Admissibility: A survey of a concept in progress. Inter. Statist. Rev. 63 95-115.
  • SCHIRM, A. L. and PRESTON, S. H. (1992). Comment on "Should we have adjusted the U.S. Census of 1980?" by D. A. Freedman and W. C. Navidi. Survey Methodology 18 35-43.
  • STEIN, C. (1986). On the coverage probability of confidence sets based on prior distributions. In Sequential Methods in Statistics, 3rd ed. (G. B. Wetherbil and K.D. Glazebrook, eds.) 485-514. Chapman and Hall, London.
  • STIGLER, S. M. (1972). Completeness and unbiased estimation. Amer. Statist. 26 28-29.
  • VOINOV, V. G. and NIKULIN, M. S. (1993). Unbiased Estimators and Their Applications 1. Univariate Case. Kluwer, Dordrecht.
  • WELCH, B. L. and PEERS, H. W. (1963). On formulae for confidence points based on integrals of weighted likelihoods. J. Roy. Statist. Soc. Ser. B 25 318-329.
  • ZASLAVSKY, A. M. (1989). Representing the census undercount: Reweighting and imputation methods. Ph.D. dissertation, Dept. Mathematics, MIT.