Adaptive bayesian estimation using a gaussian random field with inverse gamma bandwidth

Annals of Statistics

Volumn 37, Issue 5 B, 2009, Pages 2655-2675

  Van Der Vaart, A W ^a   Van Zanten, J H ^a  

^a VU UNIVERSITY AMSTERDAM   (Netherlands)

Author keywords

Adaptation; Bayesian inference; Classication; Gaussian process priors; Nonparametric density estimation; Nonparametric regression; Posterior distribution; Rate of convergence

Indexed keywords

EID: 69049092744     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/08-AOS678     Document Type: Article

References (50)

1. Barron, A., Birgé, L. and Massart, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301-413. MR1679028
2. Barron, A.R. and Cover, T. M. (1991). Minimum complexity density estimation. IEEE Trans. Inform. Theory 37 1034-1054. MR1111806
3. Bauer, H. (1981). Probability Theory and Elements of Measure Theory, 2nd ed. Academic Press, London. MR0636091
4. Belitser, E. and Levit, B. (2001). Asymptotically local minimax estimation of infinitely smooth density with censored data. Ann. Inst. Statist. Math. 53 289-306. MR1841137
5. Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 207216. MR0399402
6. Cai, T. T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach. Ann. Statist. 27 898-924. MR1724035
7. Chaudhuri, N., Ghosal, S. and Roy, A. (2007). Nonparametric binary regression using a Gaussian process prior. Stat. Methodol. 4 227-243. MR2368147
8. Choi, T. (2007). Alternative posterior consistency results in nonparametric binary regression using Gaussian process priors. J. Statist. Plann. Inference 137 2975-2983. MR2323804
9. Choi, T. and Schervish, M. J. (2007). Posterior consistency in nonparametric regression problem under Gaussian process prior. J. Multivariate Anal. MR2396949
10. Donoho, D.L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? J. Roy. Statist. Soc. Ser. B 57 301-369. MR1323344
11. Donoho, D.L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539. MR1394974
12. Efromovich, S. Y. and Pinsker, M. S. (1984). Learning algorithm for nonparametric filtering. Autom. Remote Control 11 1434-1440.
13. Efromovich, S. (1999). Nonparametric Curve Estimation. Springer, New York. MR1705298
14. Ghosal, S. and Roy, A. (2006). Posterior consistency in nonparametric regression problem under Gaussian process prior. Ann. Statist. 34 2413-2429. MR2291505
15. Ghosal, S. and van der Vaart, A. W. (2001). Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. Ann. Statist. 29 1233-1263. MR1873329
16. Ghosal, S. and van der Vaart, A. W. (2007). Convergence rates for posterior distributions for non-i.i.d. observations. Ann. Statist. 35 697-723. MR2336864
17. Ghosal, S., Ghosh, J.K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500-531. MR1790007
18. Ghosal, S., Lember, J. and van der Vaart, A. (2008). Nonparametric Bayesian model selection and averaging. Electron. J. Stat. 2 63-89. MR2386086
19. Golubev, G. K. (1987). Adaptive asymptotically minimax estimates for smooth signals. Problemy Peredachi Informatsii 23 57-67. MR0893970
20. Golubev, G. K. and Levit, B. Y. (1996). Asymptotically efficient estimation for analytic distributions. Math. Methods Statist. 5 357-368. MR1417678
21. HUANG, T.-M. (2004). Convergence rates for posterior distributions and adaptive estimation. Ann. Statist. 32 1556-1593. MR2089134
22. lBRAGIMOV, I. A. and HAS'MINSKIǏ, R. Z. (1980). An estimate of the density of a distribution. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 98 61-85, 161-162. MR0591862
23. lBRAGIMOV, I. A. and KHAS'MINSKIǏ, R. Z. (1982). An estimate of the density of a distribution belonging to a class of entire functions. Teor. Veroyatnost. i Primenen. 27 514-524. MR0673923
24. KlMELDORF, G. S. and Wahba, G. (1970). A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Ann. Math. Statist. 41 495-502. MR0254999
25. KOLMOGOROV, A. N. and TIHOMIROV, V. M. (1961). £-entropy and £-capacity of sets in functional space. Amer. Math. Soc. Transl. (2) 17 277-364. MR0124720
26. KUELBS, J. and Li, W. V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 133-157. MR1237989
27. KUELBS, J., Li, W.V. and LINDE, W. (1994). The Gaussian measure of shifted balls. Probab. Theory Related Fields 98 143-162. MR1258983
28. LEMBER, J. and VAN DER Vaart, A. W. (2007). On universal Bayesian adaptation. Statist. Decisions 25 127-152. MR2388859
29. LENK, P. J. (1988). The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc. 83 509-516. MR0971380
30. LENK, P. J. (1991). Towards a practicable Bayesian nonparametric density estimator. Bio- metrika 78 531-543. MR1130921
31. LEONARD, T. (1978). Density estimation, stochastic processes and prior information. J. Roy. Statist. Soc. Ser. B 40 113-146. MR0517434
32. LEPSKI, O. V. and Levit, B. Y. (1998). Adaptive minimax estimation of infinitely differentiable functions. Math. Methods Statist. 7 123-156. MR1643256
33. Lepskiǐ, O. V. (1990). A problem of adaptive estimation in Gaussian white noise. Teor. Veroyatnost. i Primenen. 35 459-470. MR1091202
34. Lepskiǐ, O. V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Teor. Veroyatnost. i Primenen. 36 645-659. MR1147167
35. Lepskiǐ, O. V. (1992). Asymptotically minimax adaptive estimation. II. Schemes without optimal adaptation. Adaptive estimates. Teor. Veroyatnost. i Primenen. 37 468-481. MR1214353
36. Li, W.V. and LINDE, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 1556-1578. MR1733160
37. NUSSBAUM, M. (1985). Spline smoothing in regression models and asymptotic efficiency in L2. Ann. Statist. 13 984-997. MR0803753
38. PARTHASARATHY, K. R. (2005). Introduction to Probability andMeasure. Texts and Readings in Mathematics 33. Hindustan Book Agency, New Delhi. MR2190360
39. PISIER, G. (1989). The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics 94. Cambridge Univ. Press, Cambridge
40. RASMUSSEN, C. E. and WILLIAMS, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA.
41. STEIN, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30. Princeton Univ. Press, Princeton, NJ. MR0290095
42. STONE, C. J. (1984). An asymptotically optimal window selection rule for kernel density estimates. Ann. Statist. 12 1285-1297. MR0760688
43. Tokdar, S.T. and Ghosh, J. K. (2005). Posterior consistency of Gaussian process priors in density estimation. J. Statist. Plann. Inference 137 34-42. MR2292838
44. Tomczak-Jaegermann, N. (1987). Dualité des nombres d'entropie pour des opérateurs à valeurs dans un espace de Hilbert. C. R. Acad. Sci. Paris Ser. I Math. 305 299-301. MR0910364
45. van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. MR1385671
46. van der Vaart, A.W.andvan Zanten, J. H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist. 36. MR2418663
47. van der Vaart, A.W.andvan Zanten, J. H. (2007). Bayesian inference with rescaled Gaussian process priors. Electron. J. Stat. 1 433-448. MR2357712
48. van der Vaart, A.W.andvan Zanten, J. H. (2008). Reproducing kernel Hilbert spaces of Gaussian priors. IMS Collections 3 200-222.
49. Wahba, G. (1978). Improper priors, spline smoothing and the problem of guarding against model errors in regression. J. Roy. Statist. Soc. Ser. B 40 364-372. MR0522220
50. Whitney, H. (1934). Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36 63-89. MR1501735