Heteroscedastic BART via Multiplicative Regression Trees

M. T. Pratola, H. A. Chipman, E. I. George, R. E. McCulloch

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


Bayesian additive regression trees (BART) has become increasingly popular as a flexible and scalable nonparametric regression approach for modern applied statistics problems. For the practitioner dealing with large and complex nonlinear response surfaces, its advantages include a matrix-free formulation and the lack of a requirement to prespecify a confining regression basis. Although flexible in fitting the mean, BART has been limited by its reliance on a constant variance error model. Alleviating this limitation, we propose HBART, a nonparametric heteroscedastic elaboration of BART. In BART, the mean function is modeled with a sum of trees, each of which determines an additive contribution to the mean. In HBART, the variance function is further modeled with a product of trees, each of which determines a multiplicative contribution to the variance. Like the mean model, this flexible, multidimensional variance model is entirely nonparametric with no need for the prespecification of a confining basis. Moreover, with this enhancement, HBART can provide insights into the potential relationships of the predictors with both the mean and the variance. Practical implementations of HBART with revealing new diagnostic plots are demonstrated with simulated and real data on used car prices and song year of release. Supplementary materials for this article are available online.

Original languageEnglish (US)
Pages (from-to)405-417
Number of pages13
JournalJournal of Computational and Graphical Statistics
Issue number2
StatePublished - Apr 2 2020


  • Applied statistical inference
  • Big data
  • Nonparametric regression
  • Uncertainty quantification

ASJC Scopus subject areas

  • Statistics and Probability
  • Discrete Mathematics and Combinatorics
  • Statistics, Probability and Uncertainty


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