Electronic Journal of Statistics

Transformations and Bayesian density estimation

Andrew Bean, Xinyi Xu, and Steven MacEachern

Full-text: Open access


Dirichlet-process mixture models, favored for their large support and for the relative ease of their implementation, are popular choices for Bayesian density estimation. However, despite the models’ flexibility, the performance of density estimates suffers in certain situations, in particular when the true distribution is skewed or heavy tailed. We detail a method that improves performance in a variety of settings by initially transforming the sample, choosing the transformation to facilitate estimation of the density on the new scale. The effectiveness of the method is demonstrated under a variety of simulated scenarios, and in an application to body mass index (BMI) observations from a large survey of Ohio adults.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 3355-3373.

Received: January 2016
First available in Project Euclid: 16 November 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Bean, Andrew; Xu, Xinyi; MacEachern, Steven. Transformations and Bayesian density estimation. Electron. J. Statist. 10 (2016), no. 2, 3355--3373. doi:10.1214/16-EJS1158. https://projecteuclid.org/euclid.ejs/1479287225

Export citation


  • [1] M. D. Escobar and M. West. Bayesian Density Estimation and Inference Using Mixtures., Journal of the American Statistical Association, 90(430):577–588, 1995.
  • [2] T. S. Ferguson. A Bayesian Analysis of Some Nonparametric Problems., The Annals of Statistics, 1(2):209–230, 1973.
  • [3] T. S. Ferguson. Bayesian Density Estimation by Mixtures of Normal Distributions, 1983.
  • [4] C. Fernandez and Mark F. J. Steel. On Bayesian Modeling of Fat Tails and Skewness., Journal of the American Statistical Association, 93(441):359–371, 1998.
  • [5] S. Ghosal, J. K. Ghosh, and R. V. Ramamoorthi. Posterior consistency of Dirichlet mixtures in density estimation., Annals of Statistics, 27(1):143–158, 1999.
  • [6] S. Ghosal and A. W. Van Der Vaart. Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities., Annals of Statistics, 29(5) :1233–1263, 2001.
  • [7] J. E. Griffin. Default priors for density estimation with mixture models., Bayesian Analysis, 5(1):45–64, 2010.
  • [8] T. Iwata, D. Duvenaud, and Z. Ghahramani. Warped mixtures for nonparametric cluster shapes., arXiv preprint arXiv:1206.1846, 2012.
  • [9] M. C. Jones and S. J. Sheather. Using non-stochastic terms to advantage in kernel-based estimation of integrated squared density derivatives., Statistics & Probability Letters, 11(6):511–514, Jun 1991.
  • [10] A. Y. Lo. On a Class of Bayesian Nonparametric Estimates: I. Density Estimates., The Annals of Statistics, 12(1):351–357, 1984.
  • [11] S. N. MacEachern. Estimating normal means with a conjugate style dirichlet process prior., Communications in Statistics – Simulation and Computation, 23(3):727–741, jan 1994.
  • [12] F. J. Rubio and M. F. J. Steel. Inference in two-piece location-scale models with Jeffreys priors., Bayesian Analysis, 9(1):1–22, 2014.
  • [13] D. Ruppert and M. P. Wand. Correcting for Kurtosis in Density Estimation., Australian Journal of Statistics, 34(March 1991):19–29, 1992.
  • [14] S. J. Sheather and M. C. Jones. A Reliable Data-Based Bandwidth Selection Method for Kernel Density Estimation., Journal of the Royal Statistical Society. Series B (Methodological), 53(3):683–690, 1991.
  • [15] M. P. Wand., KernSmooth: Functions for Kernel Smoothing Supporting Wand and Jones (1995), 2015. R package version 2.23-14.
  • [16] M. P. Wand and M. C. Jones., Kernel Smoothing. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. Taylor & Francis, 1994.
  • [17] M. P. Wand, J. S. Marron, and D. Ruppert. Transformations in density estimation., Journal of the American Statistical Association, 86(414):343–353, 1991.
  • [18] L. Yang. Root-n convergent transformation-kernel density estimation., Journal of Nonparametric Statistics, 12(4):447–474, 2000.
  • [19] L. Yang and J. S. Marron. Iterated transformation-kernel density estimation., Journal of the American Statistical Association, 94(446):580–589, 1999.
  • [20] I. K. Yeo and R. A. Johnson. A New Family of Power Transformations to Improve Normality or Symmetry., Biometrika, 87(4):954–959, 2000.