Brazilian Journal of Probability and Statistics

On default priors and approximate location models

D. A. S. Fraser and N. Reid

Full-text: Open access


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

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

First available in Project Euclid: 22 August 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

Export citation


  • 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.