Confidence sequences with informative, bounded-influence priors
Stefano Cortinovis, Valentin Kilian, François Caron · Electronic Journal of Statistics · 2026
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 prior–data conflict 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 under a prior with polynomial or exponential tails with 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 prior–data conflict. Along the way, we provide general sufficient conditions under which extended Ville confidence sequences are convex sets, and show that the Bayesian posterior mean always lies inside the confidence regions obtained by the method of mixtures, thereby providing a natural accompanying point estimator. The theory is illustrated through simulations with several classical priors.