A Theory of Bootstrap Coverage Calibration for Generalized Posterior Credible Sets

Masahiro Tanaka

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arXiv · 2606.25729

A Theory of Bootstrap Coverage Calibration for Generalized Posterior Credible Sets

Masahiro Tanaka

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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.

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Masahiro Tanaka. 2026-06-24. A Theory of Bootstrap Coverage Calibration for Generalized Posterior Credible Sets. https://arxiv.org/abs/2606.25729

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