Functional BART with Shape Priors: A Bayesian Tree Approach to Constrained Functional Regression

Motivated by the remarkable success of Bayesian additive regression trees (BART) in regression modelling, we propose a novel nonparametric Bayesian method, termed Functional BART (FBART), tailored specifically for function-on-scalar regression. FBART leverages spline-based representations for functional responses coupled with a flexible tree-based partitioning structure, effectively capturing complex and heterogeneous relationships between response curves and scalar predictors. To facilitate efficient posterior inference, we develop a customized Bayesian backfitting algorithm. Additionally, we extend FBART by introducing shape constraints (e.g., monotonicity or convexity) on the response curves, enabling enhanced estimation and prediction when prior shape information is available. The use of shape priors ensures that posterior samples respect the specified functional constraints. Under mild regularity conditions, we establish posterior convergence rates for both FBART and its shape-constrained variant, demonstrating rate adaptivity to unknown smoothness. Extensive simulation studies and analyses of two real datasets illustrate the superior estimation accuracy and predictive performance of our proposed methods compared to existing state-of-the-art alternatives.