## The Annals of Statistics

### On the invariance of noninformative priors

#### Abstract

Jeffreys' prior, one of the widely used noninformative priors, remains invariant under reparameterization, but does not perform satisfactorily in the presence of nuisance parameters. To overcome this deficiency, recently various noninformative priors have been proposed in the literature.

This article explores the invariance (or lack thereof) of some of these noninformative priors including the reference prior of Berger and Bernardo, the reverse reference prior of J. K. Ghosh and the probability-matching prior of Peers and Stein under reparameterization. Berger and Bernardo's m-group ordered reference prior is shown to remain invariant under a special type of reparameterization. The reverse reference prior of J. K. Ghosh is shown not to remain invariant under reparameterization. However, the probability-matching prior is shown to remain invariant under any reparameterization. Also for spherically symmetric distributions, certain noninformative priors are derived using the principle of group invariance.

#### Article information

Source
Ann. Statist. Volume 24, Number 1 (1996), 141-159.

Dates
First available: 26 September 2002

http://projecteuclid.org/euclid.aos/1033066203

Mathematical Reviews number (MathSciNet)
MR1389884

Digital Object Identifier
doi:10.1214/aos/1033066203

Zentralblatt MATH identifier
0906.62024

Subjects
Primary: 62F15: Bayesian inference 62A05

#### Citation

Datta, Gauri Sankar; Ghosh, Malay. On the invariance of noninformative priors. The Annals of Statistics 24 (1996), no. 1, 141--159. doi:10.1214/aos/1033066203. http://projecteuclid.org/euclid.aos/1033066203.

#### References

• ANDERSON, T. W. 1984. An Introduction to Multivariate Statistical Analy sis, 2nd ed. Wiley, New York. Z.
• BERGER, J. 1992. Discussion of Non-informative priors,'' by J. K. Ghosh and R. Mukerjee. In Z Bayesian Statistics 4 J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith,. eds. 205 206. Oxford Univ. Press. Z.
• BERGER, J. and BERNARDO, J. M. 1989. Estimating a product of means: Bayesian analysis with reference priors. J. Amer. Statist. Assoc. 84 200 207.
• BERGER, J. and BERNARDO, J. M. 1992a. On the development of reference priors with discus. Z sion. In Bayesian Statistics 4 J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M.. Smith, eds. 35 60. Oxford Univ. Press. Z.
• BERGER, J. and BERNARDO, J. M. 1992b. Ordered group reference priors with application to the multinomial problem. Biometrika 79 25 37. Z.
• BERGER, J. and BERNARDO, J. M. 1992c. Reference priors in a variance components problem. In Z. Bayesian Analy sis in Statistics and Econometrics P. K. Goel and N. S. Iy engar, eds. 177 194. Springer, New York. Z. Z
• BERNARDO, J. M. 1979. Reference posterior distributions for Bayesian inference with discus. sion. J. Roy. Statist. Soc. Ser. B 41 113 147. Z.
• CLARKE, B. and SUN, D. 1993. Reference priors under the chi-square distance. Unpublished manuscript. Z.
• CLARKE, B. and WASSERMAN, L. 1992. Information tradeoff. Technical Report 558, Carnegie Mellon Univ. Z.
• COX, D. R. and REID, N. 1987. Orthogonal parameters and approximate conditional inference Z. with discussion. J. Roy. Statist. Soc. Ser. B 49 1 39. Z.
• DATTA, G. S. and GHOSH, M. 1995a. Some remarks on noninformative priors. J. Amer. Statist. Assoc. 90 1357 1363. Z.
• DATTA, G. S. and GHOSH, J. K. 1995b. On priors providing frequentist validity for Bayesian inference. Biometrika 82 37 45. Z. Z.
• GHOSH, J. K. and MUKERJEE, R. 1992. Non-informative priors with discussion. In Bayesian Z. Statistics 4 J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds. 195 210. Oxford Univ. Press. Z.
• JEFFREy S, H. 1961. Theory of Probability. Oxford Univ. Press. Z.
• LAPLACE, P. 1812. Theorie Analy tique des Probabilities. Courcier, Paris. Z.
• MUKERJEE, R. and DEY, D. K. 1993. Frequentist validity of posterior quantiles in the presence of a nuisance parameter: higher order asy mptotics. Biometrika 80 499 505. Z.
• PEERS, H. W. 1965. On confidence points and Bayesian probability points in the case of several parameters. J. Roy. Statist. Soc. Ser. B 27 9 16. Z.
• RAO, C. R. 1973. Linear Statistical Inference and its Applications, 2nd ed. Wiley, New York. Z.
• STEIN, C. 1959. An example of wide discrepancy between fiducial and confidence interval. Ann. Math. Statist. 30 877 880. Z.
• STEIN, C. 1985. On coverage probability of confidence sets based on a prior distribution. In Sequential Methods in Statistics 485 514. PWN, Polish Scientific Publishers, Warsaw. Z.
• TIBSHIRANI, R. 1989. Noninformative priors for one parameter of many. Biometrika 76 604 608. Z.
• WELCH, B. L. and PEERS, H. W. 1963. On formulae for confidence points based on intervals of weighted likelihoods. J. Roy. Statist. Soc. Ser. B 25 318 329.
• ATHENS, GEORGIA 30602-1952 P.O. BOX 118545 UNIVERSITY OF FLORIDA
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