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Convergence rates of posterior distributions. (English) Zbl 1105.62315
Summary: We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinite-dimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, log-spline models, Dirichlet processes and interval censoring.

MSC:
62F15 Bayesian inference
62E20 Asymptotic distribution theory in statistics
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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[1] Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536-561. · Zbl 0980.62039
[2] Birgé, L. (1983). Approximation dans les espaces métriques et théorie de l’estimation. Z. Wahrsch. Verw. Gebiete 65 181-238. · Zbl 0506.62026
[3] Birgé, L. (1984). Sur un théor eme de minimax et son application aux tests. Probab. Math. Statist. 3 259-282. · Zbl 0571.62036
[4] Birgé, L. and Massart, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113-150. · Zbl 0805.62037
[5] Birgé, L. and Massart, P. (1997). From model selection to adaptive estimation. In Festschrift for Lucien Le Cam (G. Yang and D. Pollard, eds.) 55-87. Springer, New York. · Zbl 0920.62042
[6] Birgé, L. and Massart, P. (1998). Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 329-375. · Zbl 0954.62033
[7] de Boor, C. (1978). A Practical Guide to Splines. Springer, New York. · Zbl 0406.41003
[8] Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates (with discussion). Ann. Statist. 14 1-67. · Zbl 0595.62022
[9] Doob, J. L. (1949). Le Calcul des Probabilités et ses Applications. Coll. Int. du CNRS 13 23-27.
[10] Dudley, R. M. (1984). A course on empirical processes. Lectures Notes in Math. 1097 2-141. Springer, Berlin. · Zbl 0554.60029
[11] Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230. · Zbl 0255.62037
[12] Ferguson, T. S. (1974). Prior distribution on the spaces of probability measures. Ann. Statist. 2 615-629. · Zbl 0286.62008
[13] Freedman, D. A. (1963). On the asymptotic behavior of Bayes’ estimates in the discrete case. Ann. Math. Statist. 34 1194-1216.
[14] Freedman, D. A. (1965). On the asymptotic behavior of Bayes’ estimates in the discrete case II. Ann. Math. Statist. 36 454-456. · Zbl 0137.12603
[15] Ghosal, S., Ghosh, J. K. and Ramamoorthi, R. V. (1997). Non-informative priors via sieves and packing numbers. In Advances in Statistical Decision Theory and Applications (S. Panchapakeshan and N. Balakrishnan eds.) 129-140. Birkhäuser, Boston. Ghosal, S., Ghosh, J. K. and Ramamoorthi R. V. (1999a). Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist. 27 143-158. Ghosal, S., Ghosh, J. K. and Ramamoorthi, R. V. (1999b). Consistency issues in Bayesian nonparametrics. In Asymptotics, Nonparametrics and Time Series: A Tribute to Madan Lal Puri (Subir Ghosh, ed.) 639-667. Dekker, New York. · Zbl 0932.62043
[16] Ibragimov, I. A. and Has’minskii, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, New York.
[17] Kolmogorov, A. N. and Tikhomirov, V. M. (1961). Epsilon-entropy and epsilon-capacity of sets in function spaces. Amer. Math. Soc. Trans. Ser. 2 17 277-364. · Zbl 0133.06703
[18] Le Cam, L. M. (1973). Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 38-53. · Zbl 0255.62006
[19] Le Cam, L. M. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York. · Zbl 0605.62002
[20] Le Cam, L. M. and Yang, G. (1990). Asymptotics in Statistics: Some Basic Concepts. Springer, New York. · Zbl 0719.62003
[21] Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA and Amer. Statist. Assoc., Alexandria, VA. · Zbl 0741.60001
[22] Schwartz, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete 4 10-26. · Zbl 0158.17606
[23] Shen, X. and Wasserman, L. (1999). Rates of convergence of posterior distributions. · Zbl 1041.62022
[24] Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590-606. · Zbl 0603.62050
[25] Stone, C. J. (1990). Large-sample inference for log-spline models. Ann. Statist. 18 717-741. · Zbl 0712.62036
[26] Stone, C. J. (1994). The use of polynomial splines and their tensor products in multivariate function estimation (with discussion). Ann. Statist. 22 118-184. · Zbl 0827.62038
[27] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[28] Wasserman, L. (1998). Asymptotic properties of nonparametric Bayesian procedures. Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statist. 133 293-304. Springer, New York. · Zbl 0918.62045
[29] Wong, W. H. and Shen, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 339-362. · Zbl 0829.62002
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