On the invariance of noninformative priors. (English) Zbl 0906.62024

Summary: H. Jeffreys’ prior [see “Theory of probability.” 3rd ed. (1961; Zbl 0116.34904)], one of the widely used noninformative priors, remains invariant under reparametrization, 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 J. O. Berger and J. M. Bernardo [J. Am. Stat. Assoc. 84, No. 405, 200-207 (1989; Zbl 0682.62018)], the reverse reference prior of J. K. Ghosh [see J. K. Ghosh and R. Mukerjee, in J. M. Bernardo et al. (eds.), Bayesian Statistics 4, 195-200 (1992)] and the probability-matching prior of H. W. Peers [J. R. Stat. Soc., Ser. B 27, 9-16 (1965; Zbl 0144.41403)] and C. M. Stein [Sequential Methods in Statistics, Banach Cent. Publ. 16, 485-514 (1985; Zbl 0662.62030)] under reparametrization. Berger and Bernardo’s \(m\)-group ordered reference prior is shown to remain invariant under a special type of reparametrization. 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 reparametrization. Also for spherically symmetric distributions, certain noninformative priors are derived using the principle of group invariance.


62F15 Bayesian inference
62A01 Foundations and philosophical topics in statistics
Full Text: DOI


[1] ANDERSON, T. W. 1984. An Introduction to Multivariate Statistical Analy sis, 2nd ed. Wiley, New York. Z.
[2] 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. · Zbl 1045.62502
[3] BERGER, J. and BERNARDO, J. M. 1989. Estimating a product of means: Bayesian analysis with reference priors. J. Amer. Statist. Assoc. 84 200 207. JSTOR: · Zbl 0682.62018 · doi:10.2307/2289864
[4] 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.
[5] BERGER, J. and BERNARDO, J. M. 1992b. Ordered group reference priors with application to the multinomial problem. Biometrika 79 25 37. Z. JSTOR: · Zbl 0763.62014 · doi:10.1093/biomet/79.1.25
[6] 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 · Zbl 0770.62054
[7] BERNARDO, J. M. 1979. Reference posterior distributions for Bayesian inference with discus. sion. J. Roy. Statist. Soc. Ser. B 41 113 147. Z. JSTOR: · Zbl 0428.62004
[8] CLARKE, B. and SUN, D. 1993. Reference priors under the chi-square distance. Unpublished manuscript. Z.
[9] CLARKE, B. and WASSERMAN, L. 1992. Information tradeoff. Technical Report 558, Carnegie Mellon Univ. Z. · Zbl 0845.62025
[10] 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. JSTOR: · Zbl 0616.62006
[11] DATTA, G. S. and GHOSH, M. 1995a. Some remarks on noninformative priors. J. Amer. Statist. Assoc. 90 1357 1363. Z. JSTOR: · Zbl 0878.62003 · doi:10.2307/2291526
[12] DATTA, G. S. and GHOSH, J. K. 1995b. On priors providing frequentist validity for Bayesian inference. Biometrika 82 37 45. Z. Z. JSTOR: · Zbl 0823.62004 · doi:10.2307/2337625
[13] 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. · Zbl 0904.62007
[14] JEFFREy S, H. 1961. Theory of Probability. Oxford Univ. Press. Z.
[15] LAPLACE, P. 1812. Theorie Analy tique des Probabilities. Courcier, Paris. Z.
[16] 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. JSTOR: · Zbl 0788.62025 · doi:10.1093/biomet/80.3.499
[17] 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. JSTOR: · Zbl 0144.41403
[18] RAO, C. R. 1973. Linear Statistical Inference and its Applications, 2nd ed. Wiley, New York. Z. · Zbl 0256.62002
[19] STEIN, C. 1959. An example of wide discrepancy between fiducial and confidence interval. Ann. Math. Statist. 30 877 880. Z. · Zbl 0093.15703 · doi:10.1214/aoms/1177706072
[20] 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. · Zbl 0662.62030
[21] TIBSHIRANI, R. 1989. Noninformative priors for one parameter of many. Biometrika 76 604 608. Z. JSTOR: · Zbl 0678.62010 · doi:10.1093/biomet/76.3.604
[22] 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. JSTOR: · Zbl 0117.14205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.