Confidence Sequence

Definition

A confidence sequence (CS) $(C_t)_{t\ge1}$ is a sequence of confidence intervals with time-uniform coverage: $P\big(\forall t\ge 1:\ \theta\in C_t\big)\ge 1-\alpha.$ It is constructed by inverting an e-process / test supermartingale. The nonparametric construction (Howard et al. 2021) is based on line-crossing and mixture supermartingales, and its width shrinks roughly like $\sqrt{\log\log t/t}$ (the LIL rate).

Intuitive Understanding

A CI whose error does not inflate even under continuous monitoring — valid simultaneously at every sample size. It solves the “peeking” problem of repeatedly inspecting a fixed-$n$ CI at every time point.

Related Concepts

• e-process — CS ↔ e-process duality
• Off-Policy Evaluation — anytime-valid OPE intervals

Key Papers

• Howard, Ramdas, McAuliffe & Sekhon, “Time-uniform, nonparametric, nonasymptotic confidence sequences”, Annals of Statistics 49(2):1055–1080, 2021
• Waudby-Smith, Wu, Ramdas, Karampatziakis & Mineiro, “Anytime-Valid Off-Policy Inference for Contextual Bandits”, ACM/IMS J. Data Science 1(3), 2024