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2021 mBART: Multidimensional Monotone BART
Hugh A. Chipman, Edward I. George, Robert E. McCulloch, Thomas S. Shively
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Bayesian Anal. Advance Publication 1-30 (2021). DOI: 10.1214/21-BA1259


For the discovery of regression relationships between Y and a large set of p potential predictors x1,,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.

Funding Statement

The authors gratefully acknowledge support from the National Science Foundation (grants DMS-1944740 and DMS-1916233), from the Natural Sciences and Engineering Research Council of Canada (NSERC) and from a Simons Fellowship from the Isaac Newton Institute at the University of Cambridge.


We thank the Editor, Associate Editor and referees for their many helpful suggestions.


Download Citation

Hugh A. Chipman. Edward I. George. Robert E. McCulloch. Thomas S. Shively. "mBART: Multidimensional Monotone BART." Bayesian Anal. Advance Publication 1 - 30, 2021.


Published: 2021
First available in Project Euclid: 16 April 2021

Digital Object Identifier: 10.1214/21-BA1259

Primary: 62F15
Secondary: 62G08


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