The Annals of Statistics

Quantile pyramids for Bayesian nonparametrics

Nils Lid Hjort and Stephen G. Walker

Full-text: Open access

Abstract

Pólya trees fix partitions and use random probabilities in order to construct random probability measures. With quantile pyramids we instead fix probabilities and use random partitions. For nonparametric Bayesian inference we use a prior which supports piecewise linear quantile functions, based on the need to work with a finite set of partitions, yet we show that the limiting version of the prior exists. We also discuss and investigate an alternative model based on the so-called substitute likelihood. Both approaches factorize in a convenient way leading to relatively straightforward analysis via MCMC, since analytic summaries of posterior distributions are too complicated. We give conditions securing the existence of an absolute continuous quantile process, and discuss consistency and approximate normality for the sequence of posterior distributions. Illustrations are included.

Article information

Source
Ann. Statist. Volume 37, Number 1 (2009), 105-131.

Dates
First available in Project Euclid: 16 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1232115929

Digital Object Identifier
doi:10.1214/07-AOS553

Mathematical Reviews number (MathSciNet)
MR2488346

Zentralblatt MATH identifier
1360.62124

Subjects
Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 62F15: Bayesian inference

Keywords
Consistency Dirichlet process nonparametric Bayes Bernshteĭn–von Mises theorem quantile pyramids random quantiles

Citation

Hjort, Nils Lid; Walker, Stephen G. Quantile pyramids for Bayesian nonparametrics. Ann. Statist. 37 (2009), no. 1, 105--131. doi:10.1214/07-AOS553. https://projecteuclid.org/euclid.aos/1232115929


Export citation

References

  • Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536–561.
  • Bernshteĭn, S. (1917). Теория Вероятностей [Theory of Probability]. Gostekihzdat, Moskva.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • De Blasi, P. and Hjort, N. L. (2007). Bayesian survival analysis in proportional hazard models with logistic relative risk. Scand. J. Statist. 34 229–257.
  • Doksum, K. A. (1974). Empirical probability plots and statistical inference for nonlinear models in the two-sample case. Ann. Statist. 2 267–277.
  • Doss, H. and Gill, R. D. (1992). An elementary approach to weak convergence for quantile processes, with applications to censored survival data. J. Amer. Statist. Assoc. 87 869–877.
  • Dubins, L. E. and Freedman, D. A. (1967). Random distribution functions. Proc. 5th Berkeley Symp. Math. Statist. Probab. 2 183–214. Univ. California Press, Berkeley.
  • Dunson, D. B. and Taylor, J. A. (2005). Approximate Bayesian inference for quantiles. J. Nonparametr. Statist. 17 385–400.
  • Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615–629.
  • Freedman, D. A. (1999). On the Bernstein–von Mises theorem with infinite-dimensional parameter. Ann. Statist. 27 1119–1140.
  • Ghosal, S., Ghosh, J. K. and Ramamoorthi, R. V. (1999a). Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist. 27 143–158.
  • 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.
  • Ghosal, S. (2000). Asymptotic normality of posterior distributions for exponential families when the number of parameters tends to infinity. J. Multivariate Anal. 74 49–68.
  • Hjort, N. L. (2003). Topics in nonparametric Bayesian statistics (with discussion). In Highly Structured Stochastic Systems (P. J. Green, N. L. Hjort and S. Richardson, eds.) 455–487. Oxford Univ. Press.
  • Hjort, N. L. (2007). Model selection for cube root asymptotics. In Report No. 50/2007 from the Mathematisches Forschungsinstitut Oberwolfach, Reassessing the Paradigms of Statistical Model-Building 33–36.
  • Hjort, N. L. and Petrone, S. (2006). Nonparametric quantile inference with the Dirichlet process prior. In Advances in Statistical Modeling and Inference: Festschrift for Kjell Doksum (V. Nair, ed.) 463–492.
  • Jeffreys, H. (1967). Theory of Probability, 3rd ed. Clarendon, Oxford.
  • Kalbfleisch, J. (1978). Likelihood methods and nonparametric testing. J. Amer. Statist. Assoc. 83 167–170.
  • Kim, Y. and Lee, J. (2004). A Bernstein–von Mises theorem in the nonparametric right-censoring model. Ann. Statist. 32 1492–1512.
  • Kottas, A. and Gelfand, A. (2001). Bayesian semiparametric median regression modeling. J. Amer. Statist. Assoc. 96 1458–1468.
  • Kraft, C. H. (1964). A class of distribution function processes which have derivatives. J. Appl. Probab. 1 385–388.
  • Lavine, M. (1992). Some aspects of Polya tree distributions for statistical modelling. Ann. Statist. 20 1222–1235.
  • Lavine, M. (1994). More aspects of Polya tree distributions for statistical modelling. Ann. Statist. 22 1161–1176.
  • Lavine, M. (1995). On an approximate likelihood for quantiles. Biometrika 82 220–222.
  • von Mises, R. (1931). Wahrscheinlichkeitsrechnung. Springer, Berlin.
  • Monahan, J. F. and Boos, D. D. (1992). Proper likelihoods for Bayesian analysis. Biometrika 79 271–278.
  • Parzen, E. (1979). Nonparametric statistical data modeling (with discussion). J. Amer. Statist. Assoc. 74 105–131.
  • Parzen, E. (2004). Quantile probability and statistical data modeling. Statist. Sci. 19 652–662.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes. Wiley, New York.
  • Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701–1762.
  • Walker, S. G., Damien, P., Laud, P. W. and Smith, A. F. M. (1999). Bayesian nonparametric inference for random distributions and related functions (with discussion). J. Roy. Statist. Soc. Ser. B 61 485–528.
  • Walker, S. G. (2003). On sufficient conditions for Bayesian consistency. Biometrika 90 482–488.