Learning rate selection via weighted Fisher divergence

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.