Robust Bayesian Inference for Unnormalized Models with Mixed-Domain Data
Abstract
Many statistical models involve parameter-dependent normalizing constants that are computationally intractable, creating substantial obstacles to standard Bayesian inference. Although existing likelihood-based algorithms can often circumvent these constants, their uncertainty quantification may be poorly calibrated under model misspecification. To address these challenges, we propose SME-BETEL, a semiparametric Bayesian framework that combines score matching estimating equations with Bayesian exponentially tilted empirical likelihood. The resulting posterior avoids evaluation of normalizing constants and does not require learning-rate calibration. Building on this framework, we develop a new score matching criterion for mixed-domain data, extending SME-BETEL to models whose observations combine components from different sample spaces. This construction enables robust Bayesian inference for mixed-domain doubly-intractable models. We establish consistency and asymptotic normality of the score matching estimator, and prove a Bernstein-von Mises theorem for the SME-BETEL posterior. These results show that SME-BETEL credible sets are asymptotically calibrated to the sampling variability of the score matching estimator, yielding valid frequentist coverage under model misspecification. Simulation studies show that SME-BETEL remains competitive under correct specification and substantially improves uncertainty quantification under misspecification. An ozone-monitoring application demonstrates the practical utility of the mixed-domain construction for spatial preferential modeling.
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BibTeX
@misc{wang2026robust,
title = {{Robust Bayesian Inference for Unnormalized Models with Mixed-Domain Data}},
author = {Jiongran Wang and Debdeep Pati and Anirban Bhattacharya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01783},
eprint = {2609.01783},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01783},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Robust Bayesian Inference for Unnormalized Models with Mixed-Domain Data
AU - Jiongran Wang
AU - Debdeep Pati
AU - Anirban Bhattacharya
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.01783
UR - https://arxiv.org/abs/2609.01783
AB - Many statistical models involve parameter-dependent normalizing constants that are computationally intractable, creating substantial obstacles to standard Bayesian inference. Although existing likelihood-based algorithms can often circumvent these constants, their uncertainty quantification may be poorly calibrated under model misspecification. To address these challenges, we propose SME-BETEL, a semiparametric Bayesian framework that combines score matching estimating equations with Bayesian exponentially tilted empirical likelihood. The resulting posterior avoids evaluation of normalizing constants and does not require learning-rate calibration. Building on this framework, we develop a new score matching criterion for mixed-domain data, extending SME-BETEL to models whose observations combine components from different sample spaces. This construction enables robust Bayesian inference for mixed-domain doubly-intractable models. We establish consistency and asymptotic normality of the score matching estimator, and prove a Bernstein-von Mises theorem for the SME-BETEL posterior. These results show that SME-BETEL credible sets are asymptotically calibrated to the sampling variability of the score matching estimator, yielding valid frequentist coverage under model misspecification. Simulation studies show that SME-BETEL remains competitive under correct specification and substantially improves uncertainty quantification under misspecification. An ozone-monitoring application demonstrates the practical utility of the mixed-domain construction for spatial preferential modeling.
ER -