Efficient Bootstrap Calibration of Generalized Posteriors with Importance Reweighting

The learning rate of a generalized posterior governs posterior uncertainty but typically has no canonical value. Generalized posterior calibration (GPC) selects this rate by matching bootstrap-estimated credible-interval coverage to its nominal level, often requiring repeated posterior simulation. We propose GPC via importance reweighting (GPC-IRW), which reuses draws from a reference posterior fitted to the original data across bootstrap resamples and candidate learning rates. For quantile-based intervals of a scalar functional, we express the coverage criterion in terms of bootstrap posterior cumulative distribution functions, avoiding computation of individual bootstrap credible intervals. Safeguards based on importance-weight effective sample size and proximity to coverage-decision boundaries invoke reweighting from cached bootstrap-specific draws or direct posterior simulation when necessary. Under the stated regularity conditions, with numerical error tending to zero, we show that median-centered calibration preserves the first-order learning-rate target and achieves asymptotically nominal coverage. The approach extends to independent clusters through cluster-level resampling. Simulations yield coverage and interval lengths comparable to those of existing GPC algorithms at substantially lower computational cost. In an exoplanet mass-radius analysis, GPC-IRW achieves 6.7- and 9.6-fold speedups over MCMC-based calibration in the single-planet and full-sample analyses, respectively, while illustrating the effects of within-host dependence.