mBART: Multidimensional Monotone BART

TL;DR

mBART extends the flexible Bayesian Additive Regression Trees model by incorporating monotonicity constraints, resulting in smoother, more interpretable functions with improved predictive accuracy in monotone relationships.

Contribution

This paper introduces mBART, a novel constrained version of BART that allows for flexible monotonicity in predictors while maintaining nonparametric advantages.

Findings

mBART produces smoother, more interpretable estimates.

It improves out-of-sample predictive performance.

It reduces post-data uncertainty.

Abstract

For the discovery of regression relationships between Y and a large set of p potential predictors x 1 , . . . , x p , 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.