Density Ratio Estimation with Stein Displacement Fields
Abstract
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.
Related
- Is Spectral Self-Supervised Learning a Minimax Optimal Method of Density Ratio Estimation?
- Learning under Target Shift: Optimal Density Ratio Estimation and Importance-Weighted Regression
- Local polynomial density ratio estimation
- Stein degree over fields of characteristic zero
- A unified approach for estimation of unnormalized models via Stein operators
BibTeX
@misc{liu2026density,
title = {{Density Ratio Estimation with Stein Displacement Fields}},
author = {Song Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12437},
eprint = {2610.12437},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12437},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Density Ratio Estimation with Stein Displacement Fields
AU - Song Liu
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.12437
UR - https://arxiv.org/abs/2610.12437
AB - 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.
ER -