# A Unified Framework for Multi-distribution Density Ratio Estimation

arXiv: https://arxiv.org/abs/2112.03440 · HTML: https://mlsanity.com/paper/2112.03440/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 54 references that are in the map.
- First posted on arXiv 7 Dec 2021.

## Builds on (most cited first, 14 of 14)

- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Learning deep representations by mutual information estimation and maximization](https://mlsanity.com/paper/1808.06670/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Massively Multilingual Neural Machine Translation](https://mlsanity.com/paper/1903.00089/) (2019)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [Bias Correction of Learned Generative Models using Likelihood-Free Importance Weighting](https://mlsanity.com/paper/1906.09531/) (2019)
- [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)
- [Optimal Off-Policy Evaluation from Multiple Logging Policies](https://mlsanity.com/paper/2010.11002/) (2020)
- [Generative Adversarial Nets from a Density Ratio Estimation Perspective](https://mlsanity.com/paper/1610.02920/) (2016)
- [Trimmed Density Ratio Estimation](https://mlsanity.com/paper/1703.03216/) (2017)
- [Inverse Density as an Inverse Problem: The Fredholm Equation Approach](https://mlsanity.com/paper/1304.5575/) (2013)
- [A scaled Bregman theorem with applications](https://mlsanity.com/paper/1607.00360/) (2016)

## Built on by (most cited first, 5 of 5)

- [Offline Imitation Learning with Suboptimal Demonstrations via Relaxed Distribution Matching](https://mlsanity.com/paper/2303.02569/) (2023)
- [Contrastive Predictive Coding Done Right for Mutual Information Estimation](https://mlsanity.com/paper/2510.25983/) (2025)
- [Density-Ratio Losses for Post-Hoc Learning to Defer](https://mlsanity.com/paper/2605.19557/) (2026)
- [Some notes concerning a generalized KMM-type optimization method for density ratio estimation](https://mlsanity.com/paper/2309.07887/) (2023)
- [Mutual Information Multinomial Estimation](https://mlsanity.com/paper/2408.09377/) (2024)

## Abstract (arXiv)

Binary density ratio estimation (DRE), the problem of estimating the ratio p1/p2 given their empirical samples, provides the foundation for many state-of-the-art machine learning algorithms such as contrastive representation learning and covariate shift adaptation. In this work, we consider a generalized setting where given samples from multiple distributions p1, ldots, pk (for k > 2), we aim to efficiently estimate the density ratios between all pairs of distributions. Such a generalization leads to important new applications such as estimating statistical discrepancy among multiple random variables like multi-distribution f-divergence, and bias correction via multiple importance sampling. We then develop a general framework from the perspective of Bregman divergence minimization, where each strictly convex multivariate function induces a proper loss for multi-distribution DRE. Moreover, we rederive the theoretical connection between multi-distribution density ratio estimation and class probability estimation, justifying the use of any strictly proper scoring rule composite with a link function for multi-distribution DRE. We show that our framework leads to methods that strictly generalize their counterparts in binary DRE, as well as new methods that show comparable or superior performance on various downstream tasks.