Confidence Estimation via Sequential Likelihood Mixing
Johannes Kirschner, Andreas Krause, Michele Meziu, Mojmir Mutny
TL;DR
This work tackles non-i.i.d. uncertainty quantification by introducing a universal framework for anytime confidence sequences based on sequential likelihood mixing. It builds a bridge between Bayesian inference and frequentist coverage, showing how mixing over parameter priors or online predictions yields valid confidence sets even under misspecification. The framework subsumes standard Bayesian and variational approaches, enabling practical, tractable confidence sets through ELBO bounds, posterior sampling, or online regret analyses, and yields tighter sequences in settings like sequential linear regression and sparse recovery. The results have broad implications for robust, interactive learning systems, kernelized and nonparametric regression, and online decision-making, with extensions to tempered likelihoods and misspecified model classes.
Abstract
We present a universal framework for constructing confidence sets based on sequential likelihood mixing. Building upon classical results from sequential analysis, we provide a unifying perspective on several recent lines of work, and establish fundamental connections between sequential mixing, Bayesian inference and regret inequalities from online estimation. The framework applies to any realizable family of likelihood functions and allows for non-i.i.d. data and anytime validity. Moreover, the framework seamlessly integrates standard approximate inference techniques, such as variational inference and sampling-based methods, and extends to misspecified model classes, while preserving provable coverage guarantees. We illustrate the power of the framework by deriving tighter confidence sequences for classical settings, including sequential linear regression and sparse estimation, with simplified proofs.