Semiparametric Bernstein-von Mises theorem and bias, illustrated with Gaussian process priors

Sankhya: The Indian Journal of Statistics

Volumn 74 A, Issue PART 2, 2012, Pages 194-221

  Castillo, Ismaël ^a  

^a CNRS   (France)

Author keywords

Bayesian nonparametrics; Bernstein von Mises theorem; Gaussian process priors; Semiparametric models

Indexed keywords

EID: 84877104942     PISSN: 09727671     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (26)

1. BICKEL, P.J. and KLEIJN, B.J.K. (2012). The semiparametric Bernstein-von Mises theorem. Ann. Statist., 40, 206-237.
2. Boucheron, S. and GASSIAT, E. (2009). A Bernstein-von Mises theorem for discrete probability distributions. Electron. J. Stat., 3, 114-148.
3. Castillo, I. (2008). Lower bounds for posterior rates with Gaussian process priors. Electron. J. Stat., 2, 1281-1299.
4. CASTILLO, I. (2012). A semiparametric Bernstein-von Mises theorem for Gaussian process priors. Probab. Theory Related Fields, 152, 53-99.
5. CASTILLO, I. and CATOR, E. (2011). Semiparametric shift estimation based on the cumulated periodogram for non-regular functions. Electron. J. Stat., 5, 102-126.
6. COX, D.D. (1993). An analysis of Bayesian inference for nonparametric regression. Ann. Statist, 21, 903-923.
7. DIACONIS, P. and Freedman, D. (1986). On the consistency of Bayes estimates. Ann. Statist., 14, 1-67. With a discussion and a rejoinder by the authors.
8. FREEDMAN, D. (1999). On the Bernstein-von Mises theorem with infinite-dimensional parameters. Ann. Statist., 27, 1119-1140.
9. Gamboa, F., LOUBES, J.-M. and Maza, E. (2007). Semi-parametric estimation of shifts. Electron. J. Stat., 1, 616-640.
10. GHOSAL, S. (2000). Asymptotic normality of posterior distributions for exponential families when the number of parameters tends to infinity. J. Multivariate Anal., 74, 49-68.
11. GHOSAL, S., GHOSH, J.K. and Van Der Vaart, A.W. (2000). Convergence rates of posterior distributions. Ann. Statist., 28, 500-531.
12. GHOSAL, S., Van Der Vaart, A.W. (2007). Convergence rates of posterior distributions for non-I.I.D. observations. Ann. Statist., 35, 192-223.
13. r-metrics, 1 < r < ∞. Ann. Statist., 39, 2883-2911.
14. Ibragimov, I.A. and Has'minskiǐ, R.Z. (1981). Statistical estimation. In Applications of Mathematics, (volume 16). Springer, New York.
15. Johnstone, I. (2010). High dimensional Bernstein-von Mises: simple examples. In Festschrift for Lawrence D. Brown. Inst. Math. Stat. Collect., (volume 6, pages 87-98).
16. KIM, Y. (2006). The Bernstein-von Mises theorem for the proportional hazard model. Ann. Statist., 34, 1678-1700.
17. McNeney, B. and Wellner, J.A. (2000). Application of convolution theorems in semi- parametric models with non-i.i.d. data. J. Statist. Plann. Inference, 91, 441-480.
18. Rousseau, J. and Rivoirard, V. (2012). Bernstein-von Mises theorem for linear functionals of the density. Ann. Statist., 40, 1489-1523.
19. Shen, X. (2002). Asymptotic normality of semiparametric and nonparametric posterior distributions. J. Amer. Statist. Assoc., 97, 222-235.
20. Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist., 29, 687-714.
21. Van Der Vaart, A. (2002). The statistical work of Lucien Le Cam. Ann. Statist., 30, 631-682.
22. Van Der Vaart, A.W. (1998). Asymptotic statistics. In Cambridge Series in Statistical and Probabilistic Mathematics (volume 3). Cambridge University Press, Cambridge.
23. Van Der Vaart, A.W. and Van Zanten, H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist., 36, 1435-1463.
24. Van Der Vaart, A.W. and Van Zanten, H. (2008). Reproducing kernel Hilbert spaces of Gaussian priors. Inst. Math. Stat. Collect., 3, 200-222.
25. Van Der Vaart, A.W. and Wellner, J.A. (1996). Weak convergence and empirical processes. Springer Series in Statistics. Springer, New York.
26. Wu, Y. and Ghosal, S. (2008). Posterior consistency for some semi-parametric problems. Sankhya, 70, 267-313.