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On the Bernstein-von Mises phenomenon in the Gaussian white noise model. (English) Zbl 1274.62290

Summary: We study the Bernstein-von Mises (BvM) phenomenon, i.e., Bayesian credible sets and frequentist confidence regions for the estimation error coincide asymptotically, for the infinite-dimensional Gaussian white noise model governed by Gaussian prior with diagonal-covariance structure. While in parametric statistics this fact is a consequence of (a particular form of) the BvM theorem, in the nonparametric setup, however, the BvM theorem is known to fail even in some, apparently, elementary cases. In the present paper we show that BvM-like statements hold for this model, provided that the parameter space is suitably embedded into the support of the prior. The overall conclusion is that, unlike in the parametric setup, positive results regarding frequentist probability coverage of credible sets can only be obtained if the prior assigns null mass to the parameter space.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
62J05 Linear regression; mixed models
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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References:

[1] Castillo, I., A semiparametric Bernstein-von Mises Theorem for Gaussian process priors , Probability Theory and Related Fields (to appear) · Zbl 1232.62054 · doi:10.1007/s00440-010-0316-5
[2] Cox, D., An analysis of Bayesian inference for nonparametric regression , Annals of Statistics 21(2), pp. 903-923, 1993. · Zbl 0778.62003 · doi:10.1214/aos/1176349157
[3] Freedman, D., On the Bernstein-von Mises Theorem with infinite dimensional parameters , Annals of Statistics 27(4), pp. 1119-1140, 1999. · Zbl 0957.62002
[4] Ĩto, K., The topological support of Gauss measure on Hilbert space , Nagoya Mathematical Journal, 38, pp. 181-183, 1970. · Zbl 0206.43001
[5] Kim, Y. and Lee, J., A Bernstein-von Mises Theorem in the nonparametric right-censoring model , Annals of Statistics 32(4), pp. 1492-1512, 2004. · Zbl 1047.62043 · doi:10.1214/009053604000000526
[6] Kim, Y., The Bernstein-von Mises Theorem for the proportional hazard model , Annals of Statistics 34(4), pp. 1678-1700, 2006. · Zbl 1246.62050 · doi:10.1214/009053606000000533
[7] Kuelbs, J., Gaussian measures on a Banach space , Journal of Functional Analysis 5, pp. 354-367, 1970. · Zbl 0194.44703 · doi:10.1016/0022-1236(70)90014-5
[8] Kuo, H-H., Gaussian Measures in Banach Spaces , Lecture Notes in Math., no. 463, Springer-Verlag Heidelberg, 1975. · Zbl 0306.28010 · doi:10.1007/BFb0082007
[9] Lehmann, E.L. and Casella, G., Theory of Point Estimation - 2nd Edition , Springer Texts in Statistics, Springer Verlag NY, 1998. · Zbl 0916.62017
[10] Parthasarathy, K.R., Probability Measures on Metric Spaces , Academic Press New York - London, 1967. · Zbl 0153.19101
[11] Rivoirard, V. and Rousseau, J., Bernstein-von Mises Theorem for linear functionals of the density , Annals of Statistics (to appear) · Zbl 1257.62036
[12] Shen, X., Asymptotic normality of semiparametric and nonparametric posterior distributions , J. of American Stat. Association 97, pp. 222-235, 2002. · Zbl 1073.62517 · doi:10.1198/016214502753479365
[13] van der Vaart, A. W., Asymptotic Statistics , Cambridge University Press, 1998. · Zbl 0910.62001 · doi:10.1017/CBO9780511802256
[14] Zhao, L. H., Bayesian aspects of some nonparametric problems , Annals of Statistics 28(2), pp. 532-552, 2000. · Zbl 1010.62025 · doi:10.1214/aos/1016218229
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