FAB-PPI applies frequentist-assisted-by-Bayes confidence regions to the PPI rectifier, yielding shorter asymptotic confidence intervals when predictions are accurate and reverting to standard PPI under a horseshoe prior when they are not.
Therefore, (S41) and (S42) hold almost surely by sandwiching
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FAB-PPI: Frequentist, Assisted by Bayes, Prediction-Powered Inference
FAB-PPI applies frequentist-assisted-by-Bayes confidence regions to the PPI rectifier, yielding shorter asymptotic confidence intervals when predictions are accurate and reverting to standard PPI under a horseshoe prior when they are not.