# Bayesian Additive Distribution Regression

arXiv: https://arxiv.org/abs/2603.06462 · HTML: https://mlsanity.com/paper/2603.06462/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 47 references that are in the map.
- First posted on arXiv 6 Mar 2026.

## Builds on (most cited first, 16 of 16)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Automated versus do-it-yourself methods for causal inference: Lessons learned from a data analysis competition](https://mlsanity.com/paper/1707.02641/) (2017)
- [Discussion on "Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects" by Hahn, Murray and Carvalho](https://mlsanity.com/paper/2108.02836/) (2021)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Bayesian Regression Tree Ensembles that Adapt to Smoothness and Sparsity](https://mlsanity.com/paper/1707.09461/) (2017)
- [bartMachine: Machine Learning with Bayesian Additive Regression Trees](https://mlsanity.com/paper/1312.2171/) (2013)
- [Distribution-Free Distribution Regression](https://mlsanity.com/paper/1302.0082/) (2013)
- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [The Mondrian Kernel](https://mlsanity.com/paper/1606.05241/) (2016)
- [Bayesian Approaches to Distribution Regression](https://mlsanity.com/paper/1705.04293/) (2017)
- [Model interpretation through lower-dimensional posterior summarization](https://mlsanity.com/paper/1905.07103/) (2019)
- [Gibbs posterior concentration rates under sub-exponential type losses](https://mlsanity.com/paper/2012.04505/) (2020)
- [On the Gaussian process limit of Bayesian Additive Regression Trees](https://mlsanity.com/paper/2410.20289/) (2024)
- [Improved learning theory for kernel distribution regression with two-stage sampling](https://mlsanity.com/paper/2308.14335/) (2023)
- [Bayesian Nonparametric Quasi Likelihood](https://mlsanity.com/paper/2405.20601/) (2024)

## Built on by (most cited first, 1 of 1)

- [Seeing the Forest for the Trees: The Gaussian Process Limit of BART](https://mlsanity.com/paper/2607.28844/) (2026)

## Abstract (arXiv)

Distribution regression, where the goal is to predict a scalar response from a distribution-valued predictor, arises naturally in settings where observations are grouped and outcomes depend on group-level characteristics rather than on individual measurements. We introduce DistBART, a Bayesian nonparametric approach to distribution regression that models the regression function as a linear functional with the Riesz representer assigned a Bayesian additive regression trees (BART) prior. We argue that shallow decision tree ensembles encode reasonable inductive biases for tabular data, making them appropriate in settings where the functional depends primarily on low-dimensional marginals of the distributions. We show this both empirically on synthetic and real data and theoretically through an adaptive posterior concentration result. We also establish connections to kernel methods, and use this connection to motivate variants of DistBART that can learn nonlinear functionals. To enable scalability to large datasets, we develop a random-feature approximation that samples trees from the BART prior and reduces inference to sparse Bayesian linear regression, achieving computational efficiency while retaining uncertainty quantification.