# Robust Bayesian Inference for Unnormalized Models with Mixed-Domain Data

arXiv: https://arxiv.org/abs/2609.01783 · HTML: https://mlsanity.com/paper/2609.01783/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 66 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [The Principles of Diffusion Models](https://mlsanity.com/paper/2510.21890/) (2025)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [On Graphical Models via Univariate Exponential Family Distributions](https://mlsanity.com/paper/1301.4183/) (2013)
- [Estimating Density Models with Truncation Boundaries using Score Matching](https://mlsanity.com/paper/1910.03834/) (2019)
- [On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods](https://mlsanity.com/paper/1306.4032/) (2013)
- [Robust and Scalable Bayesian Online Changepoint Detection](https://mlsanity.com/paper/2302.04759/) (2023)
- [General Bayesian Loss Function Selection and the use of Improper Models](https://mlsanity.com/paper/2106.01214/) (2021)
- [Generalised Bayesian Inference for Discrete Intractable Likelihood](https://mlsanity.com/paper/2206.08420/) (2022)
- [Bayesian model comparison with the Hyv"arinen score: computation and consistency](https://mlsanity.com/paper/1711.00136/) (2017)
- [Score matching for compositional distributions](https://mlsanity.com/paper/2012.12461/) (2020)
- [Bayesian Estimation and Comparison of Moment Condition Models](https://mlsanity.com/paper/1606.02931/) (2016)
- [A new scope of penalized empirical likelihood with high-dimensional estimating equations](https://mlsanity.com/paper/1704.00566/) (2017)
- [Regularized Exponentially Tilted Empirical Likelihood for Bayesian Inference](https://mlsanity.com/paper/2312.17015/) (2023)
- [Amortised and provably-robust simulation-based inference](https://mlsanity.com/paper/2602.11325/) (2026)
- [A Delayed Acceptance Auxiliary Variable MCMC for Spatial Models with Intractable Likelihood Function](https://mlsanity.com/paper/2504.17147/) (2025)
- [A correlated pseudo-marginal approach to doubly intractable problems](https://mlsanity.com/paper/2210.02734/) (2022)
- [Marked spatial point processes: current state and extensions to point processes on linear networks](https://mlsanity.com/paper/2309.01511/) (2023)
- [Contrastive Bayesian Inference for Unnormalized Models](https://mlsanity.com/paper/2603.09306/) (2026)
- [A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models](https://mlsanity.com/paper/2605.06829/) (2026)

## Abstract (arXiv)

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.