Skip to main content
SpringerLink
Log in

Bayesian Kantorovich Deconvolution in Finite Mixture Models

  • Conference paper
  • First Online:
Book cover New Statistical Developments in Data Science (SIS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 288))

Included in the following conference series:

Abstract

This chapter addresses the problem of recovering the mixing distribution in finite kernel mixture models, when the number of components is unknown, yet bounded above by a fixed number. Taking a step back to the historical development of the analysis of this problem within the Bayesian paradigm and making use of the current methodology for the study of the posterior concentration phenomenon, we show that, for general prior laws supported over the space of mixing distributions with at most a fixed number of components, under replicated observations from the mixed density, the mixing distribution is estimable in the Kantorovich or \(L^1\)-Wasserstein metric at the optimal pointwise rate \(n^{-1/4}\) (up to a logarithmic factor), n being the sample size.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Chen, J.: Optimal rate of convergence for finite mixture models. Ann. Stat. 23(1), 221–233 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dall’Aglio, G.: Sugli estremi dei momenti delle funzioni di ripartizione doppia. (Italian) Ann. Scuola Norm. Sup. Pisa 3(10), 35–74 (1956)

    Google Scholar 

  3. Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. 27(3), 642–669 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  4. Efron, B.: Empirical Bayes deconvolution estimates. Biometrika 103(1), 1–20 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gao, F., van der Vaart, A.: Posterior contraction rates for deconvolution of Dirichlet-Laplace mixtures. Electron. J. Stat. 10(1), 608–627 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ghosal, S.: Convergence rates for density estimation with Bernstein polynomials. Ann. Stat. 29(5), 1264–1280 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ghosal, S., Ghosh, J.K., van der Vaart, A.W.: Convergence rates of posterior distributions. Ann. Stat. 28(2), 500–531 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ghosal, S., van der Vaart, A.W.: Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. Ann. Stat. 29(5), 1233–1263 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Heinrich, P., Kahn, J.: Strong identifiability and optimal minimax rates for finite mixture estimation. Ann. Stat. 46(6A), 2844–2870 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ishwaran, H.: Exponential posterior consistency via generalized Pólya urn schemes in finite semiparametric mixtures. Ann. Stat. 26(6), 2157–2178 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ishwaran, H., James, L.F., Sun, J.: Bayesian model selection in finite mixtures by marginal density decompositions. J. Am. Stat. Assoc. 96(456), 1316–1332 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. LeCam, L.: Convergence of estimates under dimensionality restrictions. Ann. Stat. 1(1), 38–53 (1973)

    Article  MathSciNet  Google Scholar 

  13. Massart, P.: The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18(3), 1269–1283 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nguyen, X.: Convergence of latent mixing measures in finite and infinite mixture models. Ann. Stat. 41(1), 370–400 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Scricciolo, C.: On rates of convergence for Bayesian density estimation. Scand. J. Stat. 34(3), 626–642 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Scricciolo, C.: Posterior rates of convergence for Dirichlet mixtures of exponential power densities. Electron. J. Stat. 5, 270–308 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Scricciolo, C.: Adaptive Bayesian density estimation in \(L^{p}\)-metrics with Pitman-Yor or Normalized Inverse-Gaussian process kernel mixtures. Bayesian Anal. 9(2), 475–520 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Scricciolo, C.: Bayes and maximum likelihood for \(L^1\)-Wasserstein deconvolution of Laplace mixtures. Stat. Methods Appl. 27(2), 333–362 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shen, X., Wasserman, L.: Rates of convergence of posterior distributions. Ann. Stat. 29(3), 687–714 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shorack, G.R., Wellner, J.A.: Empirical Processes with Applications to Statistics. Wiley, New York (1986)

    MATH  Google Scholar 

  21. Wong, W.H., Shen, X.: Probability inequalities for likelihood ratios and convergence rates of sieve MLES. Ann. Stat. 23(2), 339–362 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author gratefully acknowledges financial support from MIUR, grant n\(^\circ \) 2015SNS29B “Modern Bayesian nonparametric methods”.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Economiche, Università degli Studi di Verona, Via Cantarane 24, 37129, Verona (VR), Italy

    Catia Scricciolo

Authors
  1. Catia Scricciolo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Catia Scricciolo .

Editor information

Editors and Affiliations

  1. Dipartimento di Statistica, Informatica, Applicazioni ‘G. Parenti’ (DiSIA), Università degli Studi di Firenze, Florence, Italy

    Alessandra Petrucci

  2. Dipartimento Scienze Statistiche, Sapienza Università di Roma, Rome, Italy

    Filomena Racioppi

  3. Dipartimento di Matematica e Fisica, Università della Campania ‘Luigi Vanvitelli’, Caserta, Italy

    Rosanna Verde

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Scricciolo, C. (2019). Bayesian Kantorovich Deconvolution in Finite Mixture Models. In: Petrucci, A., Racioppi, F., Verde, R. (eds) New Statistical Developments in Data Science. SIS 2017. Springer Proceedings in Mathematics & Statistics, vol 288. Springer, Cham. https://doi.org/10.1007/978-3-030-21158-5_10

Download citation

Publish with us

Policies and ethics

Search

Navigation