Rate exact Bayesian adaptation with modified block priors. (English) Zbl 1331.62215

Summary: A novel block prior is proposed for adaptive Bayesian estimation. The prior does not depend on the smoothness of the function or the sample size. It puts sufficient prior mass near the true signal and automatically concentrates on its effective dimension. A rate-optimal posterior contraction is obtained in a general framework, which includes density estimation, white noise model, Gaussian sequence model, Gaussian regression and spectral density estimation.


62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI arXiv Euclid


[1] Barron, A. R. (1988). The Exponential Convergence of Posterior Probabilities with Implications for Bayes Estimators of Density Functions . Univ. of Illinois, Champaign.
[2] Barron, A. R. (1989). Uniformly powerful goodness of fit tests. Ann. Statist. 17 107-124. · Zbl 0674.62032
[3] 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
[4] Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398. · Zbl 0867.62022
[5] Brown, L. D., Cai, T. T., Low, M. G. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688-707. · Zbl 1029.62044
[6] Castillo, I., Kerkyacharian, G. and Picard, D. (2014). Thomas Bayes’ walk on manifolds. Probab. Theory Related Fields 158 665-710. · Zbl 1285.62028
[7] de Jonge, R. and van Zanten, J. H. (2010). Adaptive nonparametric Bayesian inference using location-scale mixture priors. Ann. Statist. 38 3300-3320. · Zbl 1204.62062
[8] Gao, C. and Zhou, H. H. (2015). Supplement to “Rate exact Bayesian adaptation with modified block priors.” .
[9] Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500-531. · Zbl 1105.62315
[10] Ghosal, S., Lember, J. and van der Vaart, A. (2008). Nonparametric Bayesian model selection and averaging. Electron. J. Stat. 2 63-89. · Zbl 1135.62028
[11] Ghosal, S. and van der Vaart, A. (2007). Convergence rates of posterior distributions for non-I.I.d. observations. Ann. Statist. 35 192-223. · Zbl 1114.62060
[12] Golubev, G. K., Nussbaum, M. and Zhou, H. H. (2010). Asymptotic equivalence of spectral density estimation and Gaussian white noise. Ann. Statist. 38 181-214. · Zbl 1181.62152
[13] Hoffmann, M., Rousseau, J. and Schmidt-Hieber, J. (2015). On adaptive posterior concentration rates. Ann. Statist. 43 2259-2295. · Zbl 1327.62306
[14] Kruijer, W., Rousseau, J. and van der Vaart, A. (2010). Adaptive Bayesian density estimation with location-scale mixtures. Electron. J. Stat. 4 1225-1257. · Zbl 1329.62188
[15] Kruijer, W. and van der Vaart, A. (2008). Posterior convergence rates for Dirichlet mixtures of beta densities. J. Statist. Plann. Inference 138 1981-1992. · Zbl 0674.62032
[16] LeCam, L. (1973). Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 38-53. · Zbl 0255.62006
[17] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399-2430. · Zbl 0867.62035
[18] Rivoirard, V. and Rousseau, J. (2012). Posterior concentration rates for infinite dimensional exponential families. Bayesian Anal. 7 311-333. · Zbl 1330.62179
[19] Rousseau, J. (2010). Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density. Ann. Statist. 38 146-180. · Zbl 1181.62047
[20] Schwartz, L. (1965). On Bayes procedures. Probab. Theory Related Fields 4 10-26. · Zbl 0158.17606
[21] Scricciolo, C. (2006). Convergence rates for Bayesian density estimation of infinite-dimensional exponential families. Ann. Statist. 34 2897-2920. · Zbl 1114.62043
[22] Shen, W., Tokdar, S. T. and Ghosal, S. (2013). Adaptive Bayesian multivariate density estimation with Dirichlet mixtures. Biometrika 100 623-640. · Zbl 1284.62183
[23] Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687-714. · Zbl 1041.62022
[24] van der Vaart, A. and van Zanten, H. (2007). Bayesian inference with rescaled Gaussian process priors. Electron. J. Stat. 1 433-448. · Zbl 1140.62066
[25] van der Vaart, A. W. and van Zanten, J. H. (2009). Adaptive Bayesian estimation using a Gaussian random field with inverse gamma bandwidth. Ann. Statist. 37 2655-2675. · Zbl 1173.62021
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