×
Barron, Andrew; Schervish, Mark J.; Wasserman, Larry

The consistency of posterior distributions in nonparametric problems. (English) Zbl 0980.62039

Ann. Stat. 27, No. 2, 536-561 (1999).
Summary: We give conditions that guarantee that the posterior probability of every Hellinger neighborhood of the true distribution tends to 1 almost surely. The conditions are (1) a requirement that the prior not put high mass near distributions with very rough densities and (2) a requirement that the prior put positive mass in Kullback-Leibler neighborhoods of the true distribution. The results are based on the idea of approximating the set of distributions with a finite-dimensional set of distributions with sufficiently small Hellinger bracketing metric entropy. We apply the results to some examples.

Cited in 2 Reviews
Cited in 119 Documents

MSC:

62G20 Asymptotic properties of nonparametric inference

Keywords:

exponential families; Hellinger distance; nonparametric Bayesian inference; Polya trees
Cite Review PDF
Full Text: DOI

References:

[1] ALEXANDER, K. 1984. Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Probab. 12 1041 1067. Z. · Zbl 0549.60024 · doi:10.1214/aop/1176993141
[2] BARRON, A. 1998. Information-theoretic characterization of Bayes performance and the choice Z of priors in parametric and nonparametric problems. In Bayesian Statistics 6 J.. Bernardo, J. Berger, A. Dawid and A. Smith, eds. 27 52. Oxford, New York. Z. · Zbl 0974.62020
[3] BARRON, A. R. 1988. The exponential convergence of posterior probabilities with implications for Bayes estimators of density functions. Technical Report 7, Dept. Statistics, Univ. Illinois, Champaign. Z. Z DIACONIS, P. and FREEDMAN, D. A. 1986. On the consistency of Bayes estimates with discus. sion. Ann. Statist. 14 1 26. Z. · Zbl 0595.62022 · doi:10.1214/aos/1176349830
[4] DIACONIS, P. and FREEDMAN, D. A. 1997. Consistency of Bayes estimates for nonparametric Z regression: a review. In Festschrift for Lucien Le Cam D. Pollard, E. Torgersen and G.. Yang, eds. 157 165. Springer, New York. · Zbl 0891.62026
[5] DIACONIS, P. and FREEDMAN, D. A. 1998. Consistency of Bayes estimates for nonparametric regression: normal theory. Bernoulli 4 411 444. Z. · Zbl 1037.62031 · doi:10.2307/3318659
[6] DOOB, J. L. 1949. Application of the theory of martingales. In Le Calcul des Probabilites et ses Ápplications 23 27. Colloques Internationaux du Centre National de la Recherche Scientifique, Paris. Z. · Zbl 0041.45101
[7] FABIUS, J. 1964. Asymptotic behavior of Bayes estimates. Ann. Math. Statist. 35 846 856. Z. · Zbl 0137.12604 · doi:10.1214/aoms/1177703584
[8] FREEDMAN, D. A. 1963. On the asymptotic behavior of Bayes’ estimates in the discrete case. Ann. Math. Statist. 34 1194 1216. Z. · Zbl 0253.60064 · doi:10.1214/aoms/1177703856
[9] GEMAN, S. and HWANG, C. 1982. Nonparametric maximum likelihood estimation by the method of sieves. Ann. Statist. 10 401 414. Z. · Zbl 0494.62041 · doi:10.1214/aos/1176345782
[10] GHOSHAL, S., GHOSH, J. K. and RAMAMOORTHI, R. V. 1999A. Consistency issues in Bayesian Z. nonparametrics. In Asymptotics, Nonparametrics and Time series S. Ghosh, ed. 639 667. Dekker, New York. Z. · Zbl 0984.81122 · doi:10.1016/S0550-3213(00)00377-1
[11] GHOSAL, S., GHOSH, J. K. and RAMAMOORTHI, R. V. 1999B. Consistent semiparametric Bayesian inference about a location parameter. J. Statist. Plann. Inference. To appear. Z. · Zbl 1043.62024 · doi:10.1214/aos/1013203453
[12] GHOSHAL, S., GHOSH, J. and VAN DER VAART, A. 1998. Convergence rates of posterior distribution. Technical Report 504, Dept. Mathematics, Free University, Amsterdam. Z.
[13] GRENANDER, U. 1981. Abstract Inference. Wiley, New York. Z. · Zbl 0505.62069
[14] KRAFT, C. H. 1964. A class of distribution function processes which have derivatives. J. Appl. Probab. 1 385 388. Z. JSTOR: · Zbl 0203.19702 · doi:10.2307/3211867
[15] LAVINE, M. 1992. Some aspects of Polya tree distributions for statistical modelling. Ann. Śtatist. 20 1222 1235. Z. · Zbl 0765.62005 · doi:10.1214/aos/1176348767
[16] LAVINE, M. 1994. More aspects of Polya tree distributions for statistical modelling. Ann. Śtatist. 22 1161 1176. Z. · Zbl 0820.62016 · doi:10.1214/aos/1176325623
[17] LENK, P. J. 1988. The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc. 83 509 516. Z. JSTOR: · Zbl 0648.62034 · doi:10.2307/2288870
[18] LENK, P. J. 1991. Towards a practicable Bayesian nonparametric density estimator. Biometrika 78 531 543. Z. JSTOR: · Zbl 0737.62035 · doi:10.1093/biomet/78.3.531
[19] LEONARD, T. 1978. Density estimation, stochastic processes, and prior information. J. Roy. Statist. Soc. Ser. B 40 113 146. Z. JSTOR: · Zbl 0398.62033
[20] MAULDIN, R. D., SUDDERTH, W. D. and WILLIAMS, S. C. 1992. Polya trees and random distribuťions. Ann. Statist. 20 1203 1221. Z. · Zbl 0765.62006
[21] POLLARD, D. 1991. Bracketing methods in statistics and econometrics. In Nonparametric and Semiparametric Methods in Econometrics and Statistics: Proceedings of the Fifth Z International Symposium in Economic Theory and Econometrics W. A. Barnett, J.. Powell and G. E. Tauchen, eds. 337 355. Cambridge Univ. Press. Z. · Zbl 0754.62012
[22] SCHERVISH, M. J. 1995. Theory of Statistics. Springer, New York. Z. · Zbl 0834.62002
[23] SCHWARTZ, L. 1965. On Bayes procedures. Z. Wahrsch. Verw. Gabiete 4 10 26. Z. · Zbl 0158.17606 · doi:10.1007/BF00535479
[24] SHEN, X. and WASSERMAN, L. 1998. Rates of convergence of posterior distribution. Technical Report 678, Dept. Statistics, Carnegie Mellon Univ. Z.
[25] VAN DE GEER, S. 1993. Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 21 14 44. Z. · Zbl 0779.62033 · doi:10.1214/aos/1176349013
[26] WONG, W. H. and SHEN, X. 1995. Probability inequalities for likelihood ratios and convergence rates of sieve MLE’s. Ann. Statist. 23 339 362. · Zbl 0829.62002 · doi:10.1214/aos/1176324524
[27] NEW HAVEN, CONNECTICUT 06520 CARNEGIE MELLON UNIVERSITY E-MAIL: barron@stat.yale.edu PITTSBURGH, PENNSYLVANIA 15213 E-MAIL: mark@stat.cmu.edu, larry@stat.cmu.edu
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.