Confidence sequences with informative, bounded-influence priors

TL;DR

This paper develops confidence sequences for Gaussian data that leverage informative priors to improve tightness while maintaining robustness against prior misspecification, using mixture martingales and Ville's inequality.

Contribution

It introduces a method combining mixture martingales with global informative priors to produce confidence sequences that are both sharper and robust.

Findings

Confidence sequences are tighter with well-specified priors.

Sequences remain valid under prior misspecification.

Classical priors are used to illustrate the approach.

Abstract

Confidence sequences are collections of confidence regions that simultaneously cover the true parameter for every sample size at a prescribed confidence level. Tightening these sequences is of practical interest and can be achieved by incorporating prior information through the method of mixture martingales. However, confidence sequences built from informative priors are vulnerable to misspecification and may become vacuous when the prior is poorly chosen. We study this trade-off for Gaussian observations with known variance. By combining the method of mixtures with a global informative prior whose tails are polynomial or exponential and the extended Ville's inequality, we construct confidence sequences that are sharper than their non-informative counterparts whenever the prior is well specified, yet remain bounded under arbitrary misspecification. The theory is illustrated with several classical priors.