The Annals of Statistics

On rates of convergence for posterior distributions in infinite-dimensional models

Stephen G. Walker, Antonio Lijoi, and Igor Prünster

Full-text: Open access

Enhanced PDF (127 KB)

Abstract

This paper introduces a new approach to the study of rates of convergence for posterior distributions. It is a natural extension of a recent approach to the study of Bayesian consistency. In particular, we improve on current rates of convergence for models including the mixture of Dirichlet process model and the random Bernstein polynomial model.

Article information

Source
Ann. Statist., Volume 35, Number 2 (2007), 738-746.

Dates
First available in Project Euclid: 5 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1183667291

Digital Object Identifier
doi:10.1214/009053606000001361

Mathematical Reviews number (MathSciNet)
MR2336866

Zentralblatt MATH identifier
1117.62047

Subjects
Primary: 62G07: Density estimation 62G20: Asymptotic properties 62F15: Bayesian inference

Keywords
Hellinger consistency mixture of Dirichlet process posterior distribution rates of convergence

Citation

Walker, Stephen G.; Lijoi, Antonio; Prünster, Igor. On rates of convergence for posterior distributions in infinite-dimensional models. Ann. Statist. 35 (2007), no. 2, 738--746. doi:10.1214/009053606000001361. https://projecteuclid.org/euclid.aos/1183667291


Export citation

References

  • Barron, A. (1988). The exponential convergence of posterior probabilities with implications for Bayes estimators of density functions. Unpublished manuscript. Available at www.stat.yale.edu/~arb4/publications.htm.
  • Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536--561.
    Mathematical Reviews (MathSciNet): MR1714718
    Digital Object Identifier: doi:10.1214/aos/1017939142
    Project Euclid: euclid.aos/1018031206
    Zentralblatt MATH: 0980.62039
  • Escobar, M. D. and West, M. (1995). Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc. 90 577--588.
    Mathematical Reviews (MathSciNet): MR1340510
    Digital Object Identifier: doi:10.2307/2291069
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0826.62021
  • Ghosal, S. (2001). Convergence rates for density estimation with Bernstein polynomials. Ann. Statist. 29 1264--1280.
    Mathematical Reviews (MathSciNet): MR1873330
    Digital Object Identifier: doi:10.1214/aos/1013203453
    Project Euclid: euclid.aos/1013203453
    Zentralblatt MATH: 1043.62024
  • Ghosal, S., Ghosh, J. K. and Ramamoorthi, R. V. (1999). Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist. 27 143--158.
    Mathematical Reviews (MathSciNet): MR1701105
    Digital Object Identifier: doi:10.1214/aos/1018031105
    Project Euclid: euclid.aos/1018031105
    Zentralblatt MATH: 0932.62043
  • Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500--531.
    Mathematical Reviews (MathSciNet): MR1790007
    Digital Object Identifier: doi:10.1214/aos/1016218228
    Project Euclid: euclid.aos/1016218228
  • 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.
    Mathematical Reviews (MathSciNet): MR1873329
    Digital Object Identifier: doi:10.1214/aos/1013203453
    Project Euclid: euclid.aos/1013203452
    Zentralblatt MATH: 1043.62025
  • Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1992245
    Zentralblatt MATH: 1029.62004
  • Lijoi, A., Prünster, I. and Walker, S. G. (2005). On consistency of nonparametric normal mixtures for Bayesian density estimation. J. Amer. Statist. Assoc. 100 1292--1296.
    Mathematical Reviews (MathSciNet): MR2236442
    Digital Object Identifier: doi:10.1198/016214505000000358
    Zentralblatt MATH: 1117.62387
  • Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351--357.
    Mathematical Reviews (MathSciNet): MR0733519
    Digital Object Identifier: doi:10.1214/aos/1176346412
    Project Euclid: euclid.aos/1176346412
    Zentralblatt MATH: 0557.62036
  • Petrone, S. (1999). Random Bernstein polynomials. Scand. J. Statist. 26 373--393.
    Mathematical Reviews (MathSciNet): MR1712051
    Digital Object Identifier: doi:10.1111/1467-9469.00155
    Zentralblatt MATH: 0939.62046
  • Petrone, S. and Wasserman, L. (2002). Consistency of Bernstein polynomial posteriors. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 79--100.
    Mathematical Reviews (MathSciNet): MR1881846
    Digital Object Identifier: doi:10.1111/1467-9868.00326
    JSTOR: links.jstor.org
    Zentralblatt MATH: 1015.62033
  • Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687--714.
    Mathematical Reviews (MathSciNet): MR1865337
    Digital Object Identifier: doi:10.1214/aos/1009210686
    Project Euclid: euclid.aos/1009210686
    Zentralblatt MATH: 1041.62022
  • Walker, S. G. (2004). New approaches to Bayesian consistency. Ann. Statist. 32 2028--2043.
    Mathematical Reviews (MathSciNet): MR2102501
    Digital Object Identifier: doi:10.1214/009053604000000409
    Project Euclid: euclid.aos/1098883780
    Zentralblatt MATH: 1056.62040
  • Zhang, T. (2004). Learning bounds for a generalized family of Bayesian posterior distributions. In Advances in Neural Information Processing Systems 16 (S. Thrun et al., eds.) 1149--1156. MIT Press, Cambridge, MA.

The Institute of Mathematical Statistics

The Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more