🤖 AI Summary
Generalized posterior credible sets typically lack asymptotic frequentist coverage guarantees. This work proposes calibrating their coverage by estimating coverage probabilities via the bootstrap and selecting a scalar learning rate to achieve the desired nominal level. Leveraging Edgeworth expansions, we show that coverage error arises from two sources: corrections to the sampling distribution of the estimator and posterior-induced adjustments to the center, boundary, and shape of the credible set. We further establish that a scalar learning rate can yield globally valid calibration only when the posterior and sampling covariance matrices are proportional. Under fixed-dimensional asymptotics, regularity conditions, and local identifiability, we prove—using Edgeworth expansions, classical asymptotic theory, and stochastic approximation—that the solution to the bootstrap coverage equation is consistent, revealing that the procedure essentially implements a scale correction tailored to a specific confidence level.
📝 Abstract
Generalized posteriors replace the likelihood by an exponentiated empirical criterion, but their credible sets generally lack asymptotic justification for frequentist coverage. General posterior calibration selects a scalar learning rate by estimating coverage with the bootstrap. Using Edgeworth expansions under regular fixed-dimensional asymptotics, we derive higher-order coverage expansions and analyze the stochastic approximation step used in the implemented algorithm. For a fixed nominal level, the root of the bootstrap coverage equation is consistent under a uniform coverage approximation and local identification. The higher-order expansions separate two sources of coverage error: the sampling Edgeworth correction for the estimator and the posterior Edgeworth correction for credible set boundaries, centres, and shapes. A scalar learning rate can calibrate all nominal levels in the Gaussian limit only when the posterior covariance and the sampling covariance are proportional. Hence, bootstrap calibration is a level-specific scale correction, not a remedy for general shape misspecification.