mBART: Multidimensional Monotone BART

Hugh A. Chipman, Edward I. George, Robert E. McCulloch, Thomas S. Shively

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


For the discovery of regression relationships between Y and a large set of p potential predictors x 1,…, xp, the flexible nonparametric nature of BART (Bayesian Additive Regression Trees) allows for a much richer set of possibilities than restrictive parametric approaches. However, subject matter considerations sometimes warrant a minimal assumption of monotonicity in at least some of the predictors. For such contexts, we introduce mBART, a constrained version of BART that can flexibly incorporate monotonicity in any predesignated subset of predictors using a multivariate basis of monotone trees, while avoiding the further confines of a full parametric form. For such monotone relationships, mBART provides (i) function estimates that are smoother and more interpretable, (ii) better out-of-sample predictive performance, and (iii) less post-data uncertainty. While many key aspects of the unconstrained BART model carry over directly to mBART, the introduction of monotonicity constraints necessitates a fundamental rethinking of how the model is implemented. In particular, the original BART Markov Chain Monte Carlo algorithm relied on a conditional conjugacy that is no longer available in a monotonically constrained space. Various simulated and real examples demonstrate the wide ranging potential of mBART.

Original languageEnglish (US)
Pages (from-to)515-544
Number of pages30
JournalBayesian Analysis
Issue number2
StatePublished - Jun 2022


  • Bayesian nonparametrics
  • MCMC algorithm
  • ensemble model
  • isotonic regression
  • multidimensional nonparametric regression
  • shape constrained inference

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics


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