Abstract
For the discovery of regression relationships between Y and a large set of p potential predictors , 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.
Acknowledgments
We thank the Editor, Associate Editor and referees for their many helpful suggestions.
Citation
Hugh A. Chipman. Edward I. George. Robert E. McCulloch. Thomas S. Shively. "mBART: Multidimensional Monotone BART." Bayesian Anal. 17 (2) 515 - 544, June 2022. https://doi.org/10.1214/21-BA1259
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