# Two-sample comparison through additive tree models for density ratios

arXiv: https://arxiv.org/abs/2508.03059 · HTML: https://mlsanity.com/paper/2508.03059/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 60 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Builds on (most cited first, 17 of 17)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [On Contrastive Learning for Likelihood-free Inference](https://mlsanity.com/paper/2002.03712/) (2020)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Likelihood-free inference by ratio estimation](https://mlsanity.com/paper/1611.10242/) (2016)
- [Density Ratio Estimation via Infinitesimal Classification](https://mlsanity.com/paper/2111.11010/) (2021)
- [Assigning a value to a power likelihood in a general Bayesian model](https://mlsanity.com/paper/1701.08515/) (2017)
- [Non-Negative Bregman Divergence Minimization for Deep Direct Density Ratio Estimation](https://mlsanity.com/paper/2006.06979/) (2020)
- [Featurized Density Ratio Estimation](https://mlsanity.com/paper/2107.02212/) (2021)
- [Calibrating general posterior credible regions](https://mlsanity.com/paper/1509.00922/) (2015)
- [Off-Policy Evaluation and Learning for External Validity under a Covariate Shift](https://mlsanity.com/paper/2002.11642/) (2020)
- [bartMachine: Machine Learning with Bayesian Additive Regression Trees](https://mlsanity.com/paper/1312.2171/) (2013)
- [Unsupervised tree boosting for learning probability distributions](https://mlsanity.com/paper/2101.11083/) (2021)
- [A generalized Bayes framework for probabilistic clustering](https://mlsanity.com/paper/2006.05451/) (2020)
- [Dirichlet-tree multinomial mixtures for clustering microbiome compositions](https://mlsanity.com/paper/2008.00400/) (2020)

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

- [Distributional Evaluation of Generative Models via Relative Density Ratio](https://mlsanity.com/paper/2510.25507/) (2025)

## Abstract (arXiv)

The ratio of two densities provides a direct characterization of their differences. We consider the two-sample comparison problem by estimating this ratio given i.i.d. observations from two distributions. To this end, we propose additive tree models for density ratio estimation along with efficient algorithms using a new loss function, the balancing loss. The loss allows tree-based models to be trained using several algorithms originally designed for supervised learning, such as forward-stagewise optimization and gradient boosting. Moreover, the balancing loss resembles an exponential family kernel, and it can serve as a pseudo-likelihood with conjugate priors. This property enables generalized Bayesian inference on the density ratio using backfitting samplers designed for Bayesian additive regression trees (BART). Our Bayesian strategy provides uncertainty quantification for the inferred density ratio, which is critical for applications involving high-dimensional and data-limited distributions with potentially substantial uncertainty. We further show connections of the balancing loss to the exponential loss in binary classification and to the variational form of f-divergence, particularly the squared Hellinger distance. Numerical experiments demonstrate that our method achieves both accuracy and computational efficiency, while uniquely providing uncertainty quantification. Finally, we demonstrate its application to assessing the quality of generative models for microbiome compositional data.