🤖 AI Summary
Under model misspecification, conventional Bayesian inference yields posterior credible sets with inadequate uncertainty quantification due to the failure of the information identity. This work addresses the learning rate calibration problem in generalized Bayesian inference by proposing an optimization criterion based on the weighted Fisher divergence. The proposed approach derives a closed-form solution for the learning rate by minimizing the discrepancy between the asymptotic distribution of the generalized posterior and a normal distribution with sandwich covariance. This solution encompasses the Fisher information-matching learning rate as a special case and is shown to be no greater than this benchmark in important scenarios. Theoretical analysis, supported by numerical experiments and real-data applications, demonstrates the superior performance of the proposed method in posterior calibration and uncertainty quantification.
📝 Abstract
The general Bayesian approach provides a flexible modeling framework by introducing a loss-based likelihood. A general posterior has a learning rate, which controls the relative weight of the loss-based likelihood and the prior. The flexibility of general Bayesian inference comes with an important calibration problem, especially under model misspecification. In such cases, the conventional Bayesian information identity fails, and credible sets derived from an uncalibrated Gibbs posterior need not have the desired uncertainty interpretation. This paper aims to select the learning rate used to calibrate a general posterior. By introducing the weighted Fisher divergence between the asymptotic distribution of the general posterior and a normal distribution with sandwich-type variance, we provide a closed-form expression for the selected learning rate. The selected learning rate includes the Fisher information matching learning rate as a special case and is no larger than it in an important special case. Numerical examples and a real data analysis demonstrate the usefulness of the proposed method.