# Density Ratio Estimation with Stein Displacement Fields

arXiv: https://arxiv.org/abs/2610.12437 · HTML: https://mlsanity.com/paper/2610.12437/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 80 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 2 of 2)

- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)
- [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024)

## Abstract (arXiv)

Density ratios quantify distribution shift from a probability-mass point of view, whereas displacement fields describe, from a dynamical point of view, how one distribution is transported onto another. Although both offer complementary insights, they are usually estimated separately, and converting one into the other requires post-processing. In this paper, we estimate the density ratio between a target and a base distribution by parametrizing it through a displacement field acting on the base: the log-ratio is modeled as minus the Stein operator of the base applied to the field, up to a normalizing constant. This gives both statistical and dynamical descriptions of the distribution shift through a single convex optimization problem. Iterating this estimate-and-move step gives two inference algorithms: push-forward moves the model and corrects a pretrained sampler without retraining it, whereas pull-back moves the data closer to the base and fits a transformation model one layer at a time. Applications to distribution shift in simulation-based inference and to nonlinear independent component analysis illustrate the benefits and limitations of the approach.