A probabilistic score estimation using the Bismut--Elworthy--Li formula
Abstract
Score-learning methods generally lack pointwise supervision, which affects the training accuracy, particularly in regions where SDE trajectories are rare. To address this challenge, we introduce a novel method called \emph{BEL anchors}, which uses Monte Carlo score estimates at user-selected space-time query points. By combining a Feynman--Kac representation of the Fokker--Planck equation with the Bismut--Elworthy--Li formula, our estimator requires neither closed-form transition densities, spatial discretization, nor derivatives of the initial density. We derive its asymptotic variance and use the anchors as sparse supervised targets for existing score-learning objectives. Experiments show that BEL anchors improve implicit score matching in poorly sampled regions, prevent score-PINN collapse, and enable accurate sample regeneration in several numerical examples.
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BibTeX
@misc{sun2026probabilistic,
title = {{A probabilistic score estimation using the Bismut--Elworthy--Li formula}},
author = {Jiatong Sun and Long Nguyen and Ning Jiang and Weiqi Chu and Yao Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03915},
eprint = {2611.03915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03915},
}
RIS
TY - UNPB
TI - A probabilistic score estimation using the Bismut--Elworthy--Li formula
AU - Jiatong Sun
AU - Long Nguyen
AU - Ning Jiang
AU - Weiqi Chu
AU - Yao Li
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03915
UR - https://arxiv.org/abs/2611.03915
AB - Score-learning methods generally lack pointwise supervision, which affects the training accuracy, particularly in regions where SDE trajectories are rare. To address this challenge, we introduce a novel method called \emph{BEL anchors}, which uses Monte Carlo score estimates at user-selected space-time query points. By combining a Feynman--Kac representation of the Fokker--Planck equation with the Bismut--Elworthy--Li formula, our estimator requires neither closed-form transition densities, spatial discretization, nor derivatives of the initial density. We derive its asymptotic variance and use the anchors as sparse supervised targets for existing score-learning objectives. Experiments show that BEL anchors improve implicit score matching in poorly sampled regions, prevent score-PINN collapse, and enable accurate sample regeneration in several numerical examples.
ER -