Calibration-Free Robust Bayesian Inference for Discrete Unnormalized Models

2026-10-06

AI Summary

This study addresses the computational challenges of Bayesian inference for discrete unnormalized models and the poor calibration of uncertainty quantification under model misspecification by proposing the DFD-BETEL framework. The method constructs moment conditions via the first-order optimality conditions of the discrete Fisher divergence and integrates them with semiparametric Bayesian exponentially tilted empirical likelihood for inference, thereby eliminating the need to evaluate normalizing constants or tune learning rates. Theoretical analysis establishes consistency and a Bernstein–von Mises theorem. Numerical experiments and an application to automobile insurance data demonstrate that the proposed framework achieves asymptotically calibrated uncertainty quantification under model misspecification, substantially enhancing inferential reliability.

Abstract

Bayesian inference for discrete models with intractable normalizing constants is computationally challenging, particularly when the sample space is large or countably infinite. Although some likelihood-based Bayesian methods can circumvent the intractable normalizing constant, their uncertainty quantification may be poorly calibrated under model misspecification. Generalized Bayesian approaches can address this calibration issue, but typically require careful tuning of a learning rate. In this paper, we propose DFD-BETEL, a semiparametric Bayesian framework that uses the first-order optimality conditions of discrete Fisher divergence as moment conditions in Bayesian exponentially tilted empirical likelihood. The resulting posterior avoids evaluation of the normalizing constant and requires no learning-rate calibration. We establish consistency and asymptotic normality of the discrete Fisher divergence estimator and prove a Bernstein-von Mises theorem for the DFD-BETEL posterior, showing that its limiting covariance matches the sampling covariance of the estimator and yields asymptotically calibrated uncertainty quantification. Numerical studies show that DFD-BETEL remains competitive with likelihood-based Bayesian inference under correct specification and provides more reliable uncertainty quantification under model misspecification. An application to automobile-insurance claim data further illustrates the practical utility of DFD-BETEL.

Problem

• Bayesian inference
• discrete unnormalized models
• intractable normalizing constant
• model misspecification
• uncertainty quantification

Innovation

• Discrete Fisher Divergence
• Bayesian Exponentially Tilted Empirical Likelihood
• Calibration-Free Inference
• Unnormalized Models
• Bernstein-von Mises Theorem