Needles and straw in a haystack: Posterior concentration for possibly sparse sequences

Annals of Statistics

Volumn 40, Issue 4, 2012, Pages 2069-2101

  Castillo, Ismaël ^a   Van Der Vaart, Aad ^b  

^a University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

^b VU UNIVERSITY AMSTERDAM   (Netherlands)

Author keywords

Asymptotics; Bayesian estimators; Contraction; Gaussian sequence model; Mixture priors; Sparsity

Indexed keywords

EID: 84871997969     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/12-AOS1029     Document Type: Article

References (22)

1. ABRAMOVICH, F.,BENJAMINI, Y.,DONOHO, D. L. and JOHNSTONE, I. M. (2006). Adapting to unknown sparsity by controlling the false discovery rate. Ann. Statist. 34 584-653. MR2281879
2. ABRAMOVICH, F., GRINSHTEIN, V. and PENSKY, M. (2007). On optimality of Bayesian testimation in the normal means problem. Ann. Statist. 35 2261-2286. MR2363971
3. BICKEL, P. J., RITOV, Y. and TSYBAKOV, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705-1732. MR2533469
4. BIRGÉ, L. and MASSART, P. (2001). Gaussian model selection. J. Eur. Math. Soc. (JEMS) 3 203-268. MR1848946
5. BROWN, L. D. andGREENSHTEIN, E. (2009). Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means. Ann. Statist. 37 1685-1704. MR2533468
6. CAI, T. T., JIN, J. and LOW, M. G. (2007). Estimation and confidence sets for sparse normal mixtures. Ann. Statist. 35 2421-2449. MR2382653
7. CANDES, E. and TAO, T. (2007). The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Statist. 35 2313-2351. MR2382644
8. CASTILLO, I. (2008). Lower bounds for posterior rates with Gaussian process priors. Electron. J. Stat. 2 1281-1299. MR2471287
9. CASTILLO, I. and VAN DER VAART, A. W. (2012). Supplement to "Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences." DOI:10.1214/12- AOS1029SUPP.
10. DONOHO, D. L. and JOHNSTONE, I. M. (1994). Minimax risk over lp-balls for lq -error. Probab. Theory Related Fields 99 277-303.
11. DONOHO, D. L., JOHNSTONE, I. M., HOCH, J. C. and STERN, A. S. (1992). Maximum entropy and the nearly black object. J. R. Stat. Soc. Ser. B Stat. Methodol. 54 41-81.With discussion and a reply by the authors. MR1157714
12. GEORGE, E. I. and FOSTER, D. P. (2000). Calibration and empirical Bayes variable selection. Biometrika 87 731-747. MR1813972
13. GOLUBEV, G. K. (2002). Reconstruction of sparse vectors in white Gaussian noise. Problemy Peredachi Informatsii 38 75-91. MR2101314
14. HUANG, J., MA, S. and ZHANG, C.-H. (2008). Adaptive Lasso for sparse high-dimensional regression models. Statist. Sinica 18 1603-1618. MR2469326
15. JIANG, W. and ZHANG, C.-H. (2009). General maximum likelihood empirical Bayes estimation of normal means. Ann. Statist. 37 1647-1684. MR2533467
16. JOHNSTONE, I.M. and SILVERMAN, B. W. (2004). Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences. Ann. Statist. 32 1594-1649. MR2089135
17. JOHNSTONE, I. M. and SILVERMAN, B. W. (2005). Empirical Bayes selection of wavelet thresholds. Ann. Statist. 33 1700-1752. MR2166560
18. SCOTT, J. G. and BERGER, J. O. (2010). Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem. Ann. Statist. 38 2587-2619. MR2722450
19. YUAN, M. and LIN, Y. (2005). Efficient empirical Bayes variable selection and estimation in linear models. J. Amer. Statist. Assoc. 100 1215-1225. MR2236436
20. ZHANG, C.-H. (2005). General empirical Bayes wavelet methods and exactly adaptive minimax estimation. Ann. Statist. 33 54-100. MR2157796
21. ZHANG, C.-H. and HUANG, J. (2008). The sparsity and bias of the LASSO selection in highdimensional linear regression. Ann. Statist. 36 1567-1594. MR2435448
22. ZOU, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418-1429. MR2279469