Skip to main content
Log in

Non parametric mixture priors based on an exponential random scheme

  • Statisticial Methods
  • Published:
Statistical Methods and Applications Aims and scope Submit manuscript

Abstract

We propose a general procedure for constructing nonparametric priors for Bayesian inference. Under very general assumptions, the proposed prior selects absolutely continuous distribution functions, hence it can be useful with continuous data. We use the notion ofFeller-type approximation, with a random scheme based on the natural exponential family, in order to construct a large class of distribution functions. We show how one can assign a probability to such a class and discuss the main properties of the proposed prior, namedFeller prior. Feller priors are related to mixture models with unknown number of components or, more generally, to mixtures with unknown weight distribution. Two illustrations relative to the estimation of a density and of a mixing distribution are carried out with respect to well known data-set in order to evaluate the performance of our procedure. Computations are performed using a modified version of an MCMC algorithm which is briefly described.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Altomare F, Campiti M (1994)Korovkin-type approximation theory and its application. W. de Gruyter, Berlin

    Google Scholar 

  • Antoniak CE (1974) Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems.Ann. Statist. 2, 1152–1174

    MATH  MathSciNet  Google Scholar 

  • Billingsley P (1968) Convergence of probability measure. Wiley, New York

    Google Scholar 

  • Cifarelli DM, Muliere P, Secchi P (1999) Prior processes for Bayesian nonparametrics.Technical Report no. 377/P, Dip. Mat. F. Brioschi, Politecnico di Milano

  • Consonni G, Veronese P (1995) A Bayesian method for combining results from several binomial experiments.J. Amer. Statist. Assoc. 90, 935–944

    Article  MATH  MathSciNet  Google Scholar 

  • Dalal SR, Hall WJ (1983) Approximating priors by mixtures of natural conjugate priors.J. Roy. Statist. Soc. Ser. B 45, 278–286

    MATH  MathSciNet  Google Scholar 

  • Diaconis P, Ylvisaker D (1985) Quantifying prior opinion. In: Bernardo JM, deGroot MH, Lindley DV, Smith AFM (eds)Bayesian statistics 2, pp 133–156. Elsevier, North Holland

    Google Scholar 

  • Escobar MD, West M (1995) Bayesian density estimation and inference using mixtures.J. Amer. Statist. Assoc. 90, 577–587

    Article  MATH  MathSciNet  Google Scholar 

  • Feller W (1971) An introduction to probability theory and its applications, Vol. II, Wiley, New York

    MATH  Google Scholar 

  • Ferguson TS (1983) Bayesian density estimation by mixtures of normal distributions. In: Rizvi H, Rustagi J (eds) Recent advances in statistics, pp 287–302. Academic Press, New York

    Google Scholar 

  • George EI (1986) Combining minimax shrinkage estimators.J. Amer. Statist. Assoc. 81, 437–445

    Article  MATH  MathSciNet  Google Scholar 

  • Ghosal S, Ghosh JK, Ramamoorthi RV (1999) Posterior consistency of Dirichlet mixtures in density estimation.Ann. Statist. 27, 143–158

    Article  MATH  MathSciNet  Google Scholar 

  • Green PJ, Richardson S (1999) Modelling heterogeneity with and without the Dirichlet process.Technical report, University of Bristol. http://www.stats.bris.ac.uk/~peter/

  • Ishwaran H, Zarepour M (2000) Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models.Biometrika 87, 371–390

    Article  MATH  MathSciNet  Google Scholar 

  • Lo AY (1984) On a class of Bayesian nonparametric estimates, density estimates.Ann. Statist. 12, 351–357

    MATH  MathSciNet  Google Scholar 

  • Muliere P, Tardella L (1998) Approximating distributions of functionals of Ferguson-Dirichlet priors.Can. J. Statist. 26, 283–297

    MATH  MathSciNet  Google Scholar 

  • Petrone S (1999) Random Bernstein polynomials.Scand. J. Statist. 26, 373–393

    Article  MATH  MathSciNet  Google Scholar 

  • Petrone S, Veronese P (2001) Feller operators based on natural exponential families.Studi Statistici 60, Istituto Metodi Quantitativi, Università L. Bocconi, Milano

    Google Scholar 

  • Petrone S, Wasserman L (2002) Consistency of Bernstein posterior.J. Roy. Stat. Soc., Ser. B, 64, 79–100

    Article  MATH  MathSciNet  Google Scholar 

  • Prakasa Rao BLS (1983) Nonparametric Functional Estimation. Academic Press, Orlando

    MATH  Google Scholar 

  • Roeder K (1990) Density estimation with confidence sets exemplified by superclusters and voids in the galaxies.J. Amer. Statist. Assoc. 85, 617–624

    Article  MATH  Google Scholar 

  • Sethuraman J (1994) A constructive definition of Dirichlet priors.Statist. Sinica 4, 639–650

    MATH  MathSciNet  Google Scholar 

  • Smith B (1999)Bayesian output analysis program (BOA). Department of Biostatistics. School of Public Health, University of Iowa

  • Tenbusch A (1995)Nonparametric curve estimation with Bernstein estimates. Universitätsverlag Rasch, Osnabrück, Germany

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Petrone, S., Veronese, P. Non parametric mixture priors based on an exponential random scheme. Statistical Methods & Applications 11, 1–20 (2002). https://doi.org/10.1007/BF02511443

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02511443

Key words

Navigation