Posterior covariance information criterion for general loss functions

We propose a novel computationally low-cost method for estimating a general predictive measure of generalised Bayesian inference. The proposed method utilises posterior covariance and provides estimators of the Gibbs and the plugin generalisation errors. We present theoretical guarantees of the proposed method, clarifying the connection to the Bayesian sensitivity analysis and the infinitesimal jackknife approximation of Bayesian leave-one-out cross validation. We illustrate several applications of our methods, including applications to differential privacy-preserving learning, the Bayesian hierarchical modeling, the Bayesian regression in the presence of influential observations, and the bias reduction of the widely-applicable information criterion. The applicability in high dimensions is also discussed.