Bernoulli

  • Bernoulli
  • Volume 10, Number 4 (2004), 651-663.

Extending Doob's consistency theorem to nonparametric densities

Antonio Lijoi, Igor Prünster, and Stephen G. Walker

Full-text: Open access

Abstract

We extend Doob's well-known result on Bayesian consistency. The extension covers the case where the nonparametric prior is fully supported by densities. However, our use of martingales differs from that of Doob. We also consider rates.

Article information

Source
Bernoulli, Volume 10, Number 4 (2004), 651-663.

Dates
First available in Project Euclid: 23 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1093265634

Digital Object Identifier
doi:10.3150/bj/1093265634

Mathematical Reviews number (MathSciNet)
MR2076067

Zentralblatt MATH identifier
1055.62053

Keywords
consistency Hellinger distance martingale rate of convergence

Citation

Lijoi, Antonio; Prünster, Igor; Walker, Stephen G. Extending Doob's consistency theorem to nonparametric densities. Bernoulli 10 (2004), no. 4, 651--663. doi:10.3150/bj/1093265634. https://projecteuclid.org/euclid.bj/1093265634


Export citation

References

  • [1] Barron, A., Schervish, M.J. and Wasserman, L. (1999) The consistency of posterior distributions in nonparametric problems. Ann. Statist., 27, 536-561. Abstract can also be found in the ISI/STMA publication
  • [2] Diaconis, P. and Freedman, D. (1986) On the consistency of Bayes estimates. Ann. Statist., 14, 1-26. Abstract can also be found in the ISI/STMA publication
  • [3] Doob, J.L. (1949) Application of the theory of martingales. In Le Calcul des Probabilités et ses Applications, Colloques Internationaux du Centre National de la Recherche Scientifique 13, pp. 23-27. Paris: CNRS.
  • [4] Ghosal, S., Ghosh, J.K. and Ramamoorthi, R.V. (1999a) Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist., 27, 143-158. Abstract can also be found in the ISI/STMA publication
  • [5] Ghosal, S., Ghosh, J.K. and Ramamoorthi, R.V. (1999b) Consistent semiparametric Bayesian inference about a location parameter. J. Statist. Plann. Inference, 77, 181-193. Abstract can also be found in the ISI/STMA publication
  • [6] Ghosal, S., Ghosh, J.K. and van der Vaart, A. (2000) Convergence rates of posterior distributions. Ann. Statist., 28, 500-531. Abstract can also be found in the ISI/STMA publication
  • [7] Grey, D.R. (2001) A note on convergence of probability measures. J. Appl. Probab., 38, 1055-1058. Abstract can also be found in the ISI/STMA publication
  • [8] Kraft, C.H. (1964) A class of distribution function processes which have derivatives. J. Appl. Probab., 1, 385-388.
  • [9] Lavine, M. (1992) Some aspects of Pólya tree distributions for statistical modelling. Ann. Statist., 20, 1222-1235.
  • [10] Lavine, M. (1994) More aspects of Pó lya tree distributions for statistical modelling. Ann. Statist., 22, 1161-1176.
  • [11] Lenk, P.J. (1988) The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc., 83, 509-516. Abstract can also be found in the ISI/STMA publication
  • [12] Lenk, P.J. (1991) Towards a practicable Bayesian nonparametric density estimator. Biometrika, 78, 531-543. Abstract can also be found in the ISI/STMA publication
  • [13] Leonard, T. (1978) Density estimation, stochastic processes and prior information. J. Roy. Statist. Soc. Ser. B, 40, 113-146.
  • [14] Lo, A.Y. (1984) On a class of Bayesian nonparametric estimates: I. Density estimates. Ann. Statist., 12, 351-357.
  • [15] Mauldin, R.D., Sudderth, W.D. and Williams, S.C. (1992) Po´ lya trees and random distributions. Ann. Statist., 20, 1203-1221.
  • [16] Petrone, S. and Wasserman, L. (2002) Consistency of Bernstein polynomial posteriors. J. Roy. Statist. Soc. Ser. B, 64, 79-100. Abstract can also be found in the ISI/STMA publication
  • [17] Schwartz, L. (1965) On Bayes procedures. Z. Wahrscheinlichkeitstheorie Verw. Geb., 4, 10-26.
  • [18] Shen, X. and Wasserman, L. (2001) Rates of convergence of posterior distributions. Ann. Statist., 29, 687-714. Abstract can also be found in the ISI/STMA publication