Generalized Posterior Calibration via Sequential Monte Carlo Sampler

For robustness toward model misspecification, the generalized posterior inference approach modifies the likelihood term by raising it to the power of a learning rate, thereby adjusting the spread of the posterior. This paper proposes a computationally efficient strategy for selecting an appropriate learning rate. The proposed approach builds upon the generalized posterior calibration (GPC) algorithm introduced by Syring and Martin (2019) [Biometrika, Volume 106, Issue 2, pp. 479-486], which is designed to select the learning rate to achieve the nominal frequentist coverage. This algorithm, which evaluates the coverage probability based on bootstrap samples, suffers from high computational costs due to the need for repeated posterior simulations for bootstrap samples. To address this limitation, the study proposes an algorithm that combines elements of the GPC algorithm with the sequential Monte Carlo (SMC) sampler. By leveraging the similarity between the learning rate in generalized posterior inference and the inverse temperature in SMC sampling, the proposed algorithm efficiently calibrates the posterior distribution with less computational cost. For demonstration, the proposed algorithm was applied to several statistical learning models.