Estimating means of bounded random variables by betting

TL;DR

This paper introduces a novel betting-based method for deriving confidence intervals and sequences for estimating the mean of bounded variables, outperforming existing approaches and applicable to sampling with or without replacement.

Contribution

It presents a general framework using composite nonnegative martingales that improves concentration bounds and extends to sampling without replacement.

Findings

Outperforms Hoeffding and Bernstein-based bounds in empirical tests.

Provides adaptive bounds that account for unknown variance.

Establishes new state-of-the-art confidence sequences and intervals for bounded means.

Abstract

This paper derives confidence intervals (CI) and time-uniform confidence sequences (CS) for the classical problem of estimating an unknown mean from bounded observations. We present a general approach for deriving concentration bounds, that can be seen as a generalization and improvement of the celebrated Chernoff method. At its heart, it is based on a class of composite nonnegative martingales, with strong connections to testing by betting and the method of mixtures. We show how to extend these ideas to sampling without replacement, another heavily studied problem. In all cases, our bounds are adaptive to the unknown variance, and empirically vastly outperform existing approaches based on Hoeffding or empirical Bernstein inequalities and their recent supermartingale generalizations. In short, we establish a new state-of-the-art for four fundamental problems: CSs and CIs for bounded means, when sampling with and without replacement.