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

arXiv: https://arxiv.org/abs/2502.16888 · HTML: https://mlsanity.com/paper/2502.16888/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 75 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Builds on (most cited first, 19 of 19)

- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Bayesian Regression Tree Ensembles that Adapt to Smoothness and Sparsity](https://mlsanity.com/paper/1707.09461/) (2017)
- [Functional Regression](https://mlsanity.com/paper/1406.4068/) (2014)
- [bartMachine: Machine Learning with Bayesian Additive Regression Trees](https://mlsanity.com/paper/1312.2171/) (2013)
- [On Theory for BART](https://mlsanity.com/paper/1810.00787/) (2018)
- [Functional Data Analysis of Amplitude and Phase Variation](https://mlsanity.com/paper/1512.03216/) (2015)
- [Bayesian Additive Regression Trees with Model Trees](https://mlsanity.com/paper/2006.07493/) (2020)
- [Bayesian Monotone Regression using Gaussian Process Projection](https://mlsanity.com/paper/1306.4041/) (2013)
- [Variable Selection with ABC Bayesian Forests](https://mlsanity.com/paper/1806.02304/) (2018)
- [Random Tessellation Forests](https://mlsanity.com/paper/1906.05440/) (2019)
- [Adaptive Conditional Distribution Estimation with Bayesian Decision Tree Ensembles](https://mlsanity.com/paper/2005.02490/) (2020)
- [Bayesian Function-on-Scalars Regression for High Dimensional Data](https://mlsanity.com/paper/1808.06689/) (2018)
- [High-Dimensional Spatial Quantile Function-on-Scalar Regression](https://mlsanity.com/paper/2012.10579/) (2020)
- [Statistical learning methods for neuroimaging data analysis with applications](https://mlsanity.com/paper/2210.09217/) (2022)
- [Functional Autoregression for Sparsely Sampled Data](https://mlsanity.com/paper/1603.02982/) (2016)
- [Shape-constrained Estimation in Functional Regression with Bernstein Polynomials](https://mlsanity.com/paper/2209.04476/) (2022)
- [Monotone Cubic B-Splines with a Neural-Network Generator](https://mlsanity.com/paper/2307.01748/) (2023)
- [A Unified Analysis of Multi-task Functional Linear Regression Models with Manifold Constraint and Composite Quadratic Penalty](https://mlsanity.com/paper/2211.04874/) (2022)
- [Decomposition-Based Intrinsic Modeling of Shape-Constrained Functional Data](https://mlsanity.com/paper/2406.12817/) (2024)

## Abstract (arXiv)

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.