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

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


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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 Unpublished manuscript.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.
  • Escobar, M. D. and West, M. (1995). Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc. 90 577--588.
  • Ghosal, S. (2001). Convergence rates for density estimation with Bernstein polynomials. Ann. Statist. 29 1264--1280.
  • Ghosal, S., Ghosh, J. K. and Ramamoorthi, R. V. (1999). Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist. 27 143--158.
  • Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500--531.
  • 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.
  • Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer, New York.
  • 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.
  • Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351--357.
  • Petrone, S. (1999). Random Bernstein polynomials. Scand. J. Statist. 26 373--393.
  • Petrone, S. and Wasserman, L. (2002). Consistency of Bernstein polynomial posteriors. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 79--100.
  • Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687--714.
  • Walker, S. G. (2004). New approaches to Bayesian consistency. Ann. Statist. 32 2028--2043.
  • 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.