Beyond Tweedie's Formula: Conditional Score Modeling for Empirical Bayes Inference
Abstract
We propose conditional f-modeling (Cf-modeling), a framework for empirical Bayes inference with covariates. A central identity shows that the conditional marginal score function determines not only the posterior mean through Tweedie's formula, but also the posterior moment-generating function, providing a basis for recovering posterior quantities without explicit prior modeling. Motivated by this observation, we treat the conditional marginal score as the primary object of inference and estimate it directly using an energy-based representation and Hyvärinen score matching, thereby avoiding potentially intractable covariate-dependent normalizing constants. The resulting framework flexibly accommodates covariate effects and heteroscedasticity and provides a practical approach to posterior moment estimation and uncertainty quantification. We demonstrate the effectiveness of the proposed method through simulations and an RNA-seq application.
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BibTeX
@misc{sugasawa2026beyond,
title = {{Beyond Tweedie's Formula: Conditional Score Modeling for Empirical Bayes Inference}},
author = {Shonosuke Sugasawa and Zhigen Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11136},
eprint = {2609.11136},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11136},
}
RIS
TY - UNPB
TI - Beyond Tweedie's Formula: Conditional Score Modeling for Empirical Bayes Inference
AU - Shonosuke Sugasawa
AU - Zhigen Zhao
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11136
UR - https://arxiv.org/abs/2609.11136
AB - We propose conditional f-modeling (Cf-modeling), a framework for empirical Bayes inference with covariates. A central identity shows that the conditional marginal score function determines not only the posterior mean through Tweedie's formula, but also the posterior moment-generating function, providing a basis for recovering posterior quantities without explicit prior modeling. Motivated by this observation, we treat the conditional marginal score as the primary object of inference and estimate it directly using an energy-based representation and Hyvärinen score matching, thereby avoiding potentially intractable covariate-dependent normalizing constants. The resulting framework flexibly accommodates covariate effects and heteroscedasticity and provides a practical approach to posterior moment estimation and uncertainty quantification. We demonstrate the effectiveness of the proposed method through simulations and an RNA-seq application.
ER -