Summary
In recent years, extensive work has been done concerning the derivation of noninformative prior distributions assuring approximate frequentist validity of Bayesian inferences. This paper provides a review of matching priors obtained via quantiles andvia the distribution function. Various matching criteria are described and discussed. Emphasis is laid on a proposal of designing priors matching the true coverage probability as well as the false coverage probabilities of contiguous alternatives with the respective Bayesian counterparts. The review is not primarily meant to be a comprehensive account on the area, but, rather, to convey the main underlying ideas and point out the relationships between matching priors and other noninformative priors, such as the Jeifreys’ and the reference priors.
Similar content being viewed by others
References
Bar-Lev, S. K., Reiser, B. (1982), An exponential subfamily which admits UMPU test based on a single test statistic.The Annals of Statistics, 10, 979–989.
Berger, J. O., Bernardo, J. M. (1989), Estimating a product of means: Bayesian analysis with reference priors.Journal of the American Statistical Association, 84, 200–207.
Burger, J. O., Bernardo, J. M. (1992a), Ordered group reference priors with application to a multinomial problem.Biometrika, 79, 25–37.
Berger, J. O., Bernardo, I. M. (1992b), Reference priors in a variance components problem. InBayesian Analysis in Statistics and Econometrics (eds. P. K. Goel and N. S. Iyengar), 177–194. New York: Springer-Verlag.
Berger, J. O., Bernardo, I. M. (1992c), On the development of reference priors (with discussion). InBayesian Statistics 4 (eds. J. M. Bemardo, J. O. Berger, A. P. Dawid and A. F M. Smith), 35–60. Oxford: Oxford University Press.
Berger, J. O., Boukai, B., Wo, Y. (1999), Simultaneous Bayesian-frequentist sequential testing of nested hypotheses.Biometrika, 86, 79–92.
Berger, J. O., Brown, L. D., Wolpert, R. L. (1994), A unified conditional frequentist and Bayesian test for fixed and sequential simple hypothesis testing.The Annals of Statistics, 22, 1787–1807.
Berger, J. O., Wolpert, R. L. (1988),The Likelihood Principle, 2nd ed., (ed. S. S. Gupta), IMS, Lecture Notes-Monograph Series. Hayward California.
Bernardo, J. M. (1979), Reference posterior distributions for Bayesian inference (with discussion).Journal of the Royal Statistical Society, B, 41, 113–147.
Bickel, P. J., Ghosh, I. K. (1990), A decomposition for the likelihood ratio statistic and the Bartlett correction — a Bayesian argument.The Annals of Statitics, 18, 1070–1090.
Cox, D. R., Reid, N. (1987), Parameter orthogonality and approximate conditional inference (with discussion).Journal of the Royal Statistical Society, B, 49, 1–39.
Datta, G. S. (1996), On priors providing frequentist validity of Bayesian inference for multiple parametric functions.Biometrika, 83, 287–298.
Datta, G. S., Ghosh, J. K. (1995a), On priors providing frequentist validity for Bayesian inference.Biometrika, 82, 37–45.
Datta, G. S., Ghosh, J. K. (1995b), Noninformative priors for maximal invariant parameter in group models.Test, 4, 95–114.
Datta, G. S., Ghosh, M. (1995), Some remarks on noninformative priors.Journal of the American Statistical Association, 90, 1357–1363.
Datta, G. S., Ghosh, M. (1996), On the invariance of noninformative priors.The Annals of Statistics, 24, 141–159.
Dawid, A. P., Stone, M., Zidek, J. V. (1973), Marginalization paradoxes in Bayesian and structural inference (with discussion).Journal of the Royal Statistical Society, B, 35, 189–233.
Garvan, C. W., Ghosh, M. (1997), Noninformative priors for dispersion models.Biometrika, 84, 976–982.
Garvan, C. W., Ghosh, M. (1999), On the property of posteriors for dispersion models.Journal of Statistical Planning and Inference, 78, 229–241.
Ghosh, M., Carlin, B. P., Srivastava, M. S. (1995), Probability matching priors for linear calibration.Test, 4, 333–357.
Ghosh, J. K., Mukerjee, R. (1991), Characterization of priors under which Bayesian and frequentist Bartlett corrections are equivalent in the multiparameter case.Journal of Multivariate Analysis, 38, 385–393.
Ghosh, J. K., Mukerjee, R. (1992a), Bayesian and frequentist Bartlett corrections for likelihood ratio and conditional likelihood ratio tests.Journal of the Royal Statistical Society, B, 54, 867–875.
Ghosh, J. K., Mukerjee, R. (1992b), Non-informative priors (with discussion). InBayesian Statistics 4 (eds. J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith), 195–210. Oxford: Oxford University Press.
Ghosh, J. K., Mukerjee, R. (1993a), Frequentist validity of highest posterior density regions in the multiparameter case.Annals of the Institute of Statistical Mathematics, 45, 293–302.
Ghosh, J. K., Mukerjee, R. (1993b), On priors that match posterior and frequentist distribution functions.The Canadian Journal of Statistics; 21, 89–96.
Ghosh, J. K., Mukerjee, R. (1995), Frequentist validity of highest posterior density regions in the presence of nuisance parameters.Statistics and Decisions, 13, 131–139.
Hartigan, J. A. (1983),Bayes Theory. New York: Springer-Verlag.
Jeffreys, H. (1946), An invariant form for the prior probability in estimation problems. InProceedings of the Royal Society of London, A, 186, 453–461.
Jeffreys, H. (1961),Theory of Probability, 3rd ed. Oxford: Oxford University Press.
Johnson, R. A. (1970), Asymptotic expansions associated with posterior distributions.The Annals of Mathematical Statistics, 41, 851–864.
Kass, R. E., Wasserman, L. A. (1996), The selection of prior distributions by formal rules.Journal of the American Statistical Association, 91, 1343–1370.
Mukerjee, R., Dey, D. K. (1993), Frequentist validity of posterior quantiles in the presence of a nuisance parameter: higher order asymptotics.Biometrika, 80, 499–505.
Mukerjee, R., Ghosh, M. (1997), Second-order probability matching priors.Biometrika 84, 970–975.
Mukerjee, R., Reid, N. (1999), On a property of probability matching priors: matching the alternative coverage probabilities.Biometrika, 86, 333–340.
Neyman, J., Scorr, E. L. (1948), Consistent estimates based on partially consistent observations.Econometrica, 16, 1–32.
Nicolaou, A. (1993), Bayesian intervals with good frequentist behaviour in the presence of nuisance parameters.Journal of the Royal Statistical Society, B, 55, 377–390.
Peers, H. W. (1965), On confidence points and Bayesian probability points in the case of several parameters.Journal of the Royal Statistical Society, B, 27, 9–16.
Peers, H. W. (1968), Confidence properties of Bayesian interval estimates.Journal of the Royal Statistical Society, B, 30, 535–544.
Regazzini, E. (1983),Sulle probabilità coerenti net senso di de Finetti. Bologna: CLUEB.
Reid, N. (1995), Likelihood and Bayesian approximation methods (with discussion). InBayesian Statistics 5 (eds. J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith), 351–368. Oxford: Oxford University Press.
Rousseau, J. (1997),Étude des propértiés asymptotiques des estimateurs de Bayes. Université Paris 6, Ph.D. thesis.
Stein, C. (1965), Approximation of improper prior measures by prior probability measures. InBernoulli-Bayes-Laplace Anniversary Volume: Proceedings of an International Research Seminar Statistical Laboratory (eds. J. Neyman and L. M. Le Cam), 217- 240. New York: Springer-Verlag.
Stein, C. (1985), On the coverage probability of confidence sets based on a prior distribution. InSequential Methods in Statistics, Banach Center Publications, 16, 485–514. Warsaw: PWN-Polish Scientific Publishers.
Sun, D. (1994), Integrable expansions for posterior distributions for a two-parameter exponential family.The Annals of Statistics, 22, 1808–1830.
Sun, D. (1997), A note on noninformative priors for Weibull distributions.Journal of Statistical Planning and Inference, 61, 319–338.
Sun, D., Ghosh, M., Basu, A. P. (1998), Bayesian analysis for a stress-strength system under noninformative priors.The Canadian Journal of Statistics, 26, 323–332.
Sun, D., Ye, K. (1996), Frequentist validity of posterior quantiles for a two-parameter exponential family.Biometrika, 83, 55–65.
Sweeting, T. J. (1995), A framework for Bayesian and likelihood approximations in statistics.Biometrika, 82, 1–23.
Tibshirani, R. (1989), Noninformative priors for one parameter of many.Biometrika, 76, 604–608.
Welch, B. L. (1965), On comparisons between confidence point procedures in the case of a single parameter.Journal of the Royal Statistical Society B, 27, 1–8.
Welch, B. L., Peers, H. W. (1963), On formulae for confidence points based on integrals of weighted likelihoods.Journal of the Royal Statistical Society B, 25, 318–329.
Woodroofe, M. (1986), Very weak expansions for sequential confidence intervals.The Annals of Statistics, 14, 1049–1067.
Woodroofe, M. (1989), Very weak expansions for sequentially designed experiments: linear models.The Annals of Statistics, 17, 1087–1102.
Ye, K. (1993), Reference priors when the stopping rule depends on the parameter of interest.Journal of the American Statistical Association, 88, 360–363.
Yin, M. (1998), Asymptotic expansions for posterior probability in regression model.Statistics and Decisions, 16, 349–368.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Scricciolo, C. Probability matching priors: a review. J. Ital. Statist. Soc. 8, 83 (1999). https://doi.org/10.1007/BF03178943
DOI: https://doi.org/10.1007/BF03178943