Location--Scale Calibration for Generalized Posterior

TL;DR

This paper introduces a post-processing method for generalized Bayesian posteriors that ensures uncertainty quantification remains stable and invariant to the learning rate, improving reliability across various settings.

Contribution

It proposes a location-scale calibration technique that aligns generalized posterior draws with their asymptotic target, extending the open-faced sandwich adjustment and providing theoretical guarantees.

Findings

Calibrated posteriors maintain stable coverage and bias.

Calibration closely tracks frequentist benchmarks.

Method is robust to different learning rates.

Abstract

General Bayesian updating replaces the likelihood with a loss scaled by a learning rate, but posterior uncertainty can depend sharply on that scale. We propose a simple post-processing that aligns generalized posterior draws with their asymptotic target, yielding uncertainty quantification that is invariant to the learning rate. We prove total-variation convergence for generalized posteriors with an effective sample size, allowing sample-size-dependent priors, non-i.i.d. observations, and convex penalties under model misspecification. Within this framework, we justify and extend the open-faced sandwich adjustment (Shaby, 2014), provide general theoretical guarantees for its use within generalized Bayes, and extend it from covariance rescaling to a location--scale calibration whose draws converge in total variation to the target for any learning rate. In our empirical illustration, calibrated draws maintain stable coverage, interval width, and bias over orders of magnitude in the learning rate and closely track frequentist benchmarks, whereas uncalibrated posteriors vary markedly.