Bayesian Analysis

Invariant {HPD} credible sets and {MAP} estimators

Pierre Druilhet and Jean-Michel Marin

Full-text: Open access

Abstract

MAP estimators and HPD credible sets are often criticized in the literature because of paradoxical behaviour due to a lack of invariance under reparametrization. In this paper, we propose a new version of MAP estimators and HPD credible sets that avoid this undesirable feature. Moreover, in the special case of non-informative prior, the new MAP estimators coincide with the invariant frequentist ML estimators. We also propose several adaptations in the case of nuisance parameters.

Article information

Source
Bayesian Anal., Volume 2, Number 4 (2007), 681-691.

Dates
First available in Project Euclid: 22 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ba/1340370710

Digital Object Identifier
doi:10.1214/07-BA227

Mathematical Reviews number (MathSciNet)
MR2361970

Zentralblatt MATH identifier
1331.62136

Keywords
Bayesian statistics HPD MAP Jeffreys measure nuisance parameters reference prior

Citation

Druilhet, Pierre; Marin, Jean-Michel. Invariant {HPD} credible sets and {MAP} estimators. Bayesian Anal. 2 (2007), no. 4, 681--691. doi:10.1214/07-BA227. https://projecteuclid.org/euclid.ba/1340370710


Export citation

References

  • Berger, J. (1985). Statistical Decision Theory and Bayesian Analysis. New York: Springer, 2 edition.
  • Berger, J. and Bernardo, J. (1989). "Estimating a Product of Means: Bayesian Analysis With Reference Priors." J. American Statist. Assoc., 84(405): 200–207.
  • –- (1992). "Ordered Group reference Priors With Application to Multinomial and Variance Component Problems." Biometrika, 79: 25–37.
  • Bernardo, J. (1979). "Reference Posterior Distributions for Bayesian Inference." J. Royal Statist. Soc. Series B, 41(2): 113–147.
  • –- (2005). "Intrinsic Credible Regions: An Objective Bayesian Approach to Interval Estimation." Test, 14(2): 317–384.
  • Crowder, M. and Sweeting, T. (1989). "Bayesian inference for a bivariate binomial." Biometrika, 76: 599–604.
  • Datta, G. and Ghosh, M. (1996). "On invariance of noninformative priors." Ann. Statist., 24(1): 141–159.
  • Deemer, W. and Votaw, D. (1955). "Estimation of parameters of truncated or censored exponential distributions." Ann. Math. Statist., 26: 498–504.
  • Halperin, M. (1952). "Maximum likelihood estimation in truncated samples." Ann. Math. Statistics, 23: 226–238.
  • Harville, D. (1974). "Bayesian inference for variance components using only error contrasts." Biometrika, 61: 383–385.
  • Jeffreys, H. (1961). Theory of Probability. Oxford University Press, 3 edition.
  • Kass, R. and Wasserman, L. (1996). "The selection of prior distributions by formal rules." J. American Statist. Assoc., 431(91): 1343–1370.
  • Lehmann, E. and Romano, J. (2005). Testing Statistical Hypotheses. New York: Springer, 3 edition.
  • Polson, N. and Wasserman, L. (1990). "Prior distribution for the bivariate binomial." Biometrika, 77(4): 901–904.
  • Rousseau, J. and Robert, C. (2005). "Discussion on a paper of J. Bernardo: Intrinsic Credible Regions: An Objective Bayesian Approach to Interval Estimation." Test, 14(2): 367–369.
  • Sun, D. and Berger, J. (1998). "Reference priors with partial information." Biometrika, 77(4): 901–904.