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.
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.
"mBART: Multidimensional Monotone BART." Bayesian Anal. Advance Publication 1 - 30, 2021. https://doi.org/10.1214/21-BA1259