Trevor J. Sweeting
We revisit the question of priors that achieve approximate matching of Bayesian and frequentist predictive probabilities. Such priors may be thought of as providing frequentist calibration of Bayesian prediction or simply as devices for producing frequentist prediction regions. Here we analyse the O(n−1) term in the expansion of the coverage probability of a Bayesian prediction region, as derived in [Ann. Statist. 28 (2000) 1414–1426]. Unlike the situation for parametric matching, asymptotic predictive matching priors may depend on the level α. We investigate uniformly predictive matching priors (UPMPs); that is, priors for which this O(n−1) term is zero for all α. It was shown in [Ann. Statist. 28 (2000) 1414–1426] that, in the case of quantile matching and a scalar parameter, if such a prior exists then it must be Jeffreys’ prior. In the present article we investigate UPMPs in the multiparameter case and present some general results about the form, and uniqueness or otherwise, of UPMPs for both quantile and highest predictive density matching.
References
[1] Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York.
Mathematical Reviews (MathSciNet):
MR804611
[2] Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference. J. Roy. Statist. Soc. Ser. B 41 113–147.
Mathematical Reviews (MathSciNet):
MR547240
[3] Bickel, P. J. and Ghosh, J. K. (1990). A decomposition for the likelihood ratio statistic and the Bartlett correction – a Bayesian argument. Ann. Statist. 18 1070–1090.
[4] Datta, G. S., Mukerjee, R., Ghosh, M. and Sweeting, T. J. (2000). Bayesian prediction with approximate frequentist validity. Ann. Statist. 28 1414–1426.
[5] Datta, G. S. and Sweeting, T. J (2005). Probability matching priors. In Handbook of Statistics 25. Bayesian Thinking: Modeling and Computation (D. K. Dey and C. R. Rao, eds.) 91–114. North-Holland, Amsterdam.
[6] Eaton, M. L. and Sudderth, W. D. (1998). A new predictive distribution for normal multivariate linear models. Sankhyā Ser. A 60 363–382.
[7] Ghosh, J. K. and Mukerjee, R. (1991). Characterization of priors under which Bayesian and frequentist Bartlett corrections are equivalent in the multiparameter case. J. Multivariate Anal. 38 385–393.
[8] Ghosh, J. K. and Mukerjee, R. (1993). Frequentist validity of highest posterior density regions in multiparameter case. Ann. Inst. Stat. Math. 45 293–302.
[9] Ghosh, J. K. and Mukerjee, R. (1995). Frequentist validity of highest posterior density regions in the presence of nuisance parameters. Statist. Decis. 13 131–139.
[10] Hora, R. B. and Buehler, R. J. (1966). Fiducial theory and invariant estimation. Ann. Math. Statist. 37 643–656.
Mathematical Reviews (MathSciNet):
MR199938
[11] Hora, R. B. and Buehler, R. J. (1967). Fiducial theory and invariant prediction. Ann. Math. Statist. 38 795–801.
Mathematical Reviews (MathSciNet):
MR211520
[12] Liang, F. and Barron, A. R. (2004). Exact minimax strategies for predictive density estimation, data compression and model selection. IEEE Trans. Inform. Theory 50 2708–2726.
[13] Severini, T. A., Mukerjee, R. and Ghosh, M. (2002). On an exact probability matching property of right-invariant priors. Biometrika 89 952–957.
[14] Sweeting, T. J. (2005). On the implementation of local probability matching priors for interest parameters. Biometrika 92 47–57.
[15] Sweeting, T. J., Datta, G. S. and Ghosh, M. (2006). Nonsubjective priors via predictive relative entropy loss. Ann. Statist. 34 441–468.
[16] Woodroofe, M. (1986). Very weak expansions for sequential confidence levels. Ann. Statist. 14 1049–1067.
Mathematical Reviews (MathSciNet):
MR856805
