Generalized Posterior Calibration via Sequential Monte Carlo Sampler

TL;DR

This paper introduces an efficient sequential Monte Carlo-based method for calibrating generalized posterior distributions, improving computational speed while maintaining accurate coverage in statistical learning models.

Contribution

It develops a novel SMC sampler approach that reduces computational costs in generalized posterior calibration compared to existing bootstrap-based methods.

Findings

Significantly faster calibration than the original GPC algorithm.

Maintains nominal frequentist coverage in various models.

Efficiently leverages the similarity between learning rate and inverse temperature.

Abstract

As the amount and complexity of available data increases, the need for robust statistical learning becomes more pressing. To enhance resilience against model misspecification, the generalized posterior inference method adjusts the likelihood term by exponentiating it with a learning rate, thereby fine-tuning the dispersion of the posterior distribution. This study proposes a computationally efficient strategy for selecting an appropriate learning rate. The proposed approach builds upon the generalized posterior calibration (GPC) algorithm, which is designed to select a learning rate that ensures nominal frequentist coverage. This algorithm, which evaluates the coverage probability using bootstrap samples, has high computational costs because of the repeated posterior simulations needed 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 a reduced computational cost. For demonstration, the proposed algorithm was applied to several statistical learning models and shown to be significantly faster than the original GPC.