Brazilian Journal of Probability and Statistics

On default priors and approximate location models

D. A. S. Fraser and N. Reid

Full-text: Open access

Abstract

A prior for statistical inference can be one of three basic types: a mathematical prior originally proposed in Bayes [Philos. Trans. R. Soc. Lond. 53 (1763) 370–418; 54 (1764) 269–325], a subjective prior presenting an opinion, or a truly objective prior based on an identified frequency reference. In this note we consider a method for deriving a mathematical prior based on approximate location models. This produces a mathematical posterior, and any practical interpretation of such a posterior is in terms of exact or approximate confidence under the postulated model. We describe how a proposed prior can be simply checked for consistency with confidence methods, using expansions about the maximum likelihood estimator.

Article information

Source
Braz. J. Probab. Stat., Volume 25, Number 3 (2011), 353-361.

Dates
First available in Project Euclid: 22 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1313973398

Digital Object Identifier
doi:10.1214/11-BJPS147

Mathematical Reviews number (MathSciNet)
MR2832890

Zentralblatt MATH identifier
1236.62002

Keywords
Conditioning confidence default prior Jeffreys prior noninformative prior objective prior reference prior subjective prior

Citation

Fraser, D. A. S.; Reid, N. On default priors and approximate location models. Braz. J. Probab. Stat. 25 (2011), no. 3, 353--361. doi:10.1214/11-BJPS147. https://projecteuclid.org/euclid.bjps/1313973398


Export citation

References

  • Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. Philos. Trans. R. Soc. Lond. 53, 370–418; 54 (1764), 296–325. Reprinted in Biometrika 45 (1958), 293–315.
  • Bédard, M., Fraser, D. A. S. and Wong, A. (2007). Higher accuracy for Bayesian and frequentist inference: Large sample theory for small sample likelihood. Statist. Sci. 22, 301–321.
  • Dawid, A. P., Stone, M. and Zidek, J. V. (1973). Marginalization paradoxes in Bayesian and structural inference. J. R. Stat. Soc. Ser. B 35, 189–233.
  • Fraser, A. M., Fraser, D. A. S. and Fraser, M. J. (2010a). Parameter curvature revisited and the Bayesian frequentist divergence. J. Statist. Res. 44, 335–346.
  • Fraser, A. M., Fraser, D. A. S. and Staicu, A.-M. (2010b). Second order ancillary: A differential view with continuity. Bernoulli 16, 1208–1223.
  • Fraser, D. A. S. (2003). Likelihood for component parameters. Biometrika 90, 327–339.
  • Fraser, D. A. S.(2010a). Is Bayes posterior just quick and dirty confidence? Statist. Sci. To appear.
  • Fraser, D. A. S. (2010b). Bayesian analysis or evidence based statistics? In International Encyclopedia of Statistical Science (M. Lovric, ed.) 90–97. Berlin: Springer-Verlag.
  • Fraser, D. A. S. (2010c). Bayesian inference: An approach to statistical Iinference. In Wiley Interdisciplinary Reviews: Computational Statistics 2, 487–496. DOI:10.1002/wics.102.
  • Fraser, D. A. S. (2010d). Default priors: Measuring Bayes bias for interest parameters. Manuscript.
  • Fraser, D. A. S. and Reid, N. (1995). Ancillaries and third order significance. Util. Math. 47, 33–53.
  • Fraser, D. A. S. and Reid, N. (2002). Strong matching of frequentist and Bayesian parametric inference. J. Statist. Plann. Inference 103, 263–285.
  • Fraser, D. A. S., Reid, N., Marras, E. and Yi, G. Y. (2010c). Default priors for Bayesian and frequentist inference. J. R. Statist. Soc. Ser. B 75, 631–654.
  • Fraser, D. A. S., Reid, N., Wong, A. and Yun Yi, G. (2003). Direct Bayes for interest parameters. Valencia 7, 529–533.
  • Fraser, D. A. S. and Sun, Y. (2009). Some corrections for Bayes curvature. Pakistan J. Statist. 25, 351–370.
  • Fraser, D. A. S., Wong, A. and Sun, Y. (2009). Three enigmatic examples and inference from likelihood. Canad. J. Statist. 37, 161–181.
  • Fraser, D. A. S. and Yi, G. Y. (2003). Location reparameterization and default priors for statistical analysis. J. Iran. Stat. Soc. 1, 55–78.
  • Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. Ser. A 222, 309–368.
  • Fisher, R. A. (1930). Inverse probability. Proc. Camb. Phil. Soc. 26, 528–535.
  • Lindley, D. V. (1958). Fiducial distribution and Bayes theorem. J. R. Stat. Soc. Ser. B 20, 102–107.
  • Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 236, 333–380.
  • Welch, B. L. and Peers, H. W. (1963). On formulae for confidence points based on integrals of weighted likelihoods. J. R. Stat. Soc. Ser. B 25, 318–329.