Confidence Estimation via Sequential Likelihood Mixing

TL;DR

This paper introduces a universal framework for constructing confidence sets using sequential likelihood mixing, applicable to various models and data types, with guarantees and integration of approximate inference techniques.

Contribution

It unifies several recent approaches, extends to non-i.i.d. and misspecified models, and provides tighter confidence sequences with simplified proofs.

Findings

Tighter confidence sequences for linear regression.

Framework applies to non-i.i.d. data and misspecified models.

Integrates standard approximate inference methods.

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.